The geodesic flow on the unit cotangent bundle {SL(2,\mathbb{R})/SL(2,\mathbb{Z})} of the modular surface {\mathbb{H}^2/SL(2,\mathbb{Z})} is intimately related to the continued fraction algorithm (see e.g. this article of Series).

In this context, the entries {(a_n)_{n\in\mathbb{N}}} of the continued fraction expansion {\alpha=\frac{1}{a_1+\frac{1}{\ddots}}} of an irrational number are related to cusp excursions of typical geodesics in the modular surface (i.e., visits to regions {\{z\in\mathbb{H}: \textrm{Im}z>T\}/SL(2,\mathbb{Z})} for {T} large).

By exploiting this relationship, Vaibhav Gadre analysed cusp excursions on the modular surface to obtain a proof of the following theorem originally due to Diamond–Vaaler:

Theorem 1 For Lebesgue almost every {\alpha=\frac{1}{a_1+\frac{1}{\ddots}}\in [0,1]}, one has

\displaystyle \lim\limits_{n\rightarrow\infty}\frac{\sum\limits_{j=1}^n a_j - \max\limits_{1\leq i\leq n} a_i}{n\log n}= \frac{1}{\log 2}

Furthermore, as it is explained in Gadre’s paper here, his analysis of cusp excursions generalize to geodesic flows on complete non-compact finite-area hyperbolic surfaces and to Teichmüller geodesic flows on moduli spaces of flat surfaces.

As the reader can infer from Section 2 of Gadre’s paper, an important ingredient in his investigation of cusp excursions is the exponential mixing property for the corresponding geodesic flows.

Partly motivated by this fact, Vaibhav Gadre and I asked ourselves if the exponential mixing result for “Weil–Petersson like” geodesic flows on surfaces obtained by Burns–Masur–M.–Wilkinson could be explored to control cusp excursions of typical geodesics.

In this post, we record lower and upper bounds obtained together with V. Gadre on the depth of cusp excursions of typical “Weil–Petersson like” geodesics.

Remark 1 As usual, all errors/mistakes are my sole responsibility.

Remark 2 Our exposition follows closely Section 2 of Gadre’s paper.

1. Ergodic averages of exponentially mixing flows

Let {(g_t)_{t\in\mathbb{R}}} be a flow on {X} preserving a probability measure {\mu}.

Suppose that {g_t} has exponential decay of correlations, i.e., there are constants {C>0} and {\delta>0} such that

\displaystyle |\int_X u_1 \cdot u_2\circ g_t \, d\mu - \int_X u_1 \, d\mu \int_X u_2 \, d\mu|\leq C e^{-\delta t} \|u_1\|_{B} \|u_2\|_{B} \ \ \ \ \ (1)

for all {t\geq 0} and all “smooth” real-valued observables {u_1, u_2\in B} in a Banach space {B\subset L^1(\mu)} containing all constant functions (e.g., {B} is a Hölder or Sobolev space).

Lemma 2 Any observable {u\in B} with {\int_X u \, d\mu = 0} satisfies

\displaystyle \int_X\left(\int_0^T u(g_t x) \, dt \right)^2\,d\mu(x)\leq \frac{2C}{\delta} T \|u\|_{B}^2

Proof: We write

\displaystyle \begin{array}{rcl} \int_X\left(\int_0^T u(g_t x) \, dt \right)^2\,d\mu(x) &=& \int_X\int_0^T\int_0^T u(g_t x) u(g_s x) \, dt \, ds \, d\mu(x) \\ &=& \int_0^T\int_0^T \left(\int_X u(g_t x) u(g_s x) \, d\mu(x)\right) \, dt \, ds \end{array}

By {g_t}-invariance of {\mu}, we get

\displaystyle \int_X\left(\int_0^T u(g_t x) \, dt \right)^2\,d\mu(x) = \int_0^T\int_0^T \left(\int_X u(g_{|t-s|} x) u(x) \, d\mu(x)\right) \, dt \, ds

The exponential decay of correlations (1) implies that

\displaystyle \int_X\left(\int_0^T u(g_t x) \, dt \right)^2\,d\mu(x) \leq C\|u\|_{B}^2 \int_0^T\int_0^T e^{-\delta |t-s|} \, dt \, ds\leq \frac{2C}{\delta}T \|u\|_{B}^2

This proves the lemma. \Box

2. Effective ergodic theorem for fast mixing flows

Suppose that {(g_t)_{t\in\mathbb{R}}} is an exponentially mixing flow on {(X,\mu)} (i.e., {g_t} satisfies (1)).

Fix {1/2<\alpha<1} and denote {T_k=T_k(\alpha)=k^{2\alpha/(2\alpha-1)}}.

Theorem 3 Given {m>1}, a function {n:\mathbb{R}\rightarrow\mathbb{N}} such that {n(T)=n(T_k)} for each {T_k\leq T < T_{k+1}}, and a sequence {\{f_j\}_{j\in\mathbb{N}}\subset B} of non-negative functions, we have for {\mu}-almost every {x\in X} that

\displaystyle \frac{1}{m} T \|f_{n(T)}\|_{L^1} - 2 T^{\alpha}\|f_{n(T)}\|_{B} \leq \int_0^T f_{n(T)}(g_t x) dt \leq mT\|f_{n(T)}\|_{L^1} + 2 T^{\alpha}\|f_{n(T)}\|_B

for all {T} sufficiently large (depending on {x}).

Proof: Given {f\in B}, let {F=f-\int_X f \, d\mu\in B}. Since {\|F\|_{B}\leq 2\|f\|_{B}}, we get from Lemma 2 that

\displaystyle \int_X\left(\int_0^T F(g_t x) \, dt \right)^2\,d\mu(x)\leq \frac{2C}{\delta} T \|F\|_{B}^2\leq \frac{8C}{\delta} T \|f\|_{B}^2


\displaystyle \mu\left(\left\{x\in X: \left(\int_0^T F(g_t x) \, dt \right)^2 \geq R\right\}\right)\leq \frac{8C}{\delta} \frac{T}{R} \|f\|_{B}^2

By setting {R=T^{2\alpha}\|f\|_{B}^2}, we obtain

\displaystyle \mu\left(\left\{x\in X: \left(\int_0^T F(g_t x) \, dt \right)^2 \geq T^{2\alpha}\|f\|_{B}^2\right\}\right)\leq \frac{8C}{\delta} T^{1-2\alpha} \ \ \ \ \ (2)

Consider the sequence {\{f_j\}_{j\in\mathbb{N}}\subset B} and let {F_j:= f_j-\int_X f_j\,d\mu}. From the estimate (2) with {T=T_k} and {F=F_{n(T_k)}}, and {T=T_{k+1}} and {F=F_{n(T_k)}}, we get

\displaystyle \mu\left(\left\{x\in X: \left(\int_0^{T_k} F_{n(T_k)}(g_t x) \, dt \right)^2 \geq T_k^{2\alpha}\|f_{n(T_k)}\|_{B}^2\right\}\right)\leq \frac{8C}{\delta} T_k^{1-2\alpha} = \frac{8C}{\delta} \frac{1}{k^{2\alpha}}


\displaystyle \mu\left(\left\{x\in X: \left(\int_0^{T_{k+1}} F_{n(T_k)}(g_t x) \, dt \right)^2 \geq T_{k+1}^{2\alpha}\|f_{n(T_k)}\|_{B}^2\right\}\right)\leq \frac{8C}{\delta} \frac{1}{(k+1)^{2\alpha}}

By Borel–Cantelli lemma, the summability of the series {\sum\limits_{i=1}^{\infty}\frac{1}{i^{2\alpha}}<\infty} for {\alpha>1/2} and the previous inequalities imply that for {\mu}-almost every {x\in X}

\displaystyle \left(\int_0^{T_k} F_{n(T_k)}(g_t x) \, dt \right)^2 \leq T_k^{2\alpha}\|f_{n(T_k)}\|_{B}^2


\displaystyle \left(\int_0^{T_{k+1}} F_{n(T_k)}(g_t x) \, dt \right)^2 \leq T_{k+1}^{2\alpha}\|f_{n(T_k)}\|_{B}^2

for all {k} sufficiently large (depending on {x}).

On the other hand, the non-negativity of the functions {f_j} says that

\displaystyle \int_0^{T_k} f_{n(T_k)}(g_t x) \, dt \leq \int_0^{T} f_{n(T_k)}(g_t x) \, dt \leq \int_0^{T_{k+1}} f_{n(T_k)}(g_t x) \, dt

for all {T_k\leq T < T_{k+1}}. Hence,

\displaystyle \begin{array}{rcl} \int_0^{T_k} F_{n(T_k)}(g_t x)\, dt &=& \int_0^{T_k} f_{n(T_k)}(g_t x)\, dt - T_k\int_X f_{n(T_k)} \, d\mu \\ &\leq& \int_0^{T} f_{n(T_k)}(g_t x)\, dt - T_k\int_X f_{n(T_k)} \, d\mu \\ &\leq& \int_0^{T_{k+1}} f_{n(T_k)}(g_t x)\, dt - T_k\int_X f_{n(T_k)} \, d\mu \\ &=& \int_0^{T_{k+1}} f_{n(T_k)}(g_t x)\, dt - T_{k+1}\int_X f_{n(T_k)} \, d\mu + (T_{k+1}-T_k)\int_X f_{n(T_k)} \, d\mu \\ &=& \int_0^{T_{k+1}} F_{n(T_k)}(g_t x)\, dt + (T_{k+1}-T_k)\int_X f_{n(T_k)} \, d\mu \end{array}

for all {T_k\leq T < T_{k+1}}.

It follows from this discussion that for {\mu}-almost every {x\in X} and all {k} sufficiently large (depending on {x})

\displaystyle T_k \|f_{n(T_k)}\|_{L^1} - T_k^{\alpha}\|f_{n(T_k)}\|_B \leq \int_0^T f_{n(T_k)}(g_t x) \, dt \leq T_{k+1} \|f_{n(T_k)}\|_{L^1} + T_{k+1}^{\alpha}\|f_{n(T_k)}\|_B

whenever {T_k\leq T < T_{k+1}}. Because {\frac{T_{k+1}}{T_k} = \left(\frac{k+1}{k}\right)^{2\alpha/(2\alpha-1)}\rightarrow 1} as {k\rightarrow\infty} and {n(T)=n(T_k)} for {T_k\leq T<T_{k+1}}, given {m>1}, the previous estimate says that for {\mu}-almost every {x\in X}

\displaystyle \frac{1}{m}T \|f_{n(T)}\|_{L^1} - 2T^{\alpha}\|f_{n(T)}\|_B \leq \int_0^T f_{n(T)}(g_t x) \, dt \leq mT \|f_{n(T)}\|_{L^1} + 2T^{\alpha}\|f_{n(T)}\|_B

for all {T} sufficiently large (depending on {x} and {m>1}). This proves the theorem. \Box

3. Bounds for certain cusp excursions

Let {S} be a compact surface with finitely many punctures equipped with a negatively curved Riemannian metric which is “asymptotically modelled” by surfaces of revolutions of profiles {y=x^r}, {r>2}, near the punctures: see this paper here for details.

In this setting, it was shown by Burns, Masur, M. and Wilkinson that the geodesic flow {g_t} on {X=T^1S} is exponentially mixing with respect to the Liouville (volume) measure {\mu}, i.e., for each {0< \theta\leq 1}, the estimate (1) holds for the space {B=C^{\theta}} of {\theta}-Hölder functions.

In this section, we want to explore the exponential mixing result of Burns–Masur–M.–Wilkinson to study the cusp excursions of {g_t}.

For the sake of simplicity of exposition, we are going to assume that there is only one cusp where the metric is isometric to the surface of revolution of the profile {y=x^r} for {r>2} and {0<x\leq 2}.

Remark 3 The general case can be deduced from the arguments below after replacing the geometric facts about the surfaces of revolution of {y=x^r} (e.g., Clairaut’s relations, etc.) by their analogs in Burns–Masur–M.–Wilkinson article (e.g., quasi-Clairaut’s relation in Proposition 3.2, etc.).

Given a vector {v\in T^1 S} with base point near the cusp, let {\phi(v)} be the angle between {v} and the direction pointing straight into the cusp of the surface of revolution of {y=x^r}. Denote by {C} the collar in {S} around the cusp consisting of points whose {x}-coordinate satisfies {1/2\leq x\leq 3/2}.

3.1. Good initial positions for deep excursions

Given a parameter {R>0}, let {X_R:=\{v\in T^1C: |\phi(v)|\leq 1/R\}}. The next proposition says that any vector in {X_R} generates a geodesic making a {\frac{3}{2R^{1/r}}}deep excursion into the cusp in bounded time.

Proposition 4 If {v\in X_R}, then the base point of {g_t(v)} has {x}-coordinate {\leq \frac{3}{2R^{1/r}}} for a certain time {0\leq t\leq a} where {a=a(r)} depends only on {r}.

Proof: By Clairaut’s relation, the {x}-coordinate along {g_t(v)} satisfies

\displaystyle x(g_t(v))^r\sin\phi(g_t(v)) = x(v)^r \sin\phi(v)

for all {t} (during the cusp excursion).

Thus, the value of the {x}-coordinate along {g_t(v)} is minimized when {\phi(g_{t_0}(v))=\pi/2}: at this instant {x(g_{t_0}(v))=x(v)(\sin\phi(v))^{1/r}}. Since {v\in X_R} implies that {x(v)\leq 3/2} and {|\phi(v)|\leq 1/R}, the proof of the proposition will be complete once we can bound {t_0} by a constant {a=a(r)}. As it turns out, this fact is not hard to establish from classical facts about geodesics on surfaces of revolution: see, for example, Equation (6) in Pollicott–Weiss paper. \Box

3.2. Smooth approximations of characteristic functions

Take {b} a smooth non-negative bump function equal to {1} on {3/4\leq x\leq 4/3} and supported on {1/2\leq x\leq 3/2} such that {\|b\|_{C^1}\leq 10}. Similarly, take {q_R} a smooth non-negative bump function equal to {1} on {|\phi|\leq 1/2R} and supported on {|\phi|\leq 1/R} such that {\|\phi\|_{C^1}\leq 3R}.

The non-negative function {f_R(v):=b(x(v))\cdot q_R(\phi(v))} is a smooth approximation of the characteristic function of {X_R}:

  • {f_R} is supported on {X_R};
  • there exists a constant {d=d(r)\geq 1} depending only on {r>2} such that
    • {\frac{1}{d}\leq R\int_S f_R \, d\mu \leq d} and
    • {\|f_R\|_{C^{\theta}}\leq d R^{\theta}}.

3.3. Deep cusp excursions of typical geodesics

At this point, we are ready to use the effective ergodic theorem to show that typical geodesics perform deep cusp excursions:

Theorem 5 For {\mu}-almost every {v\in T^1 S} and for all {T} sufficiently large (depending on {v} and {r>2}), the base point of {g_t(v)} has {x}-coordinate {\leq T^{-\frac{1}{2r}+}} for a certain time {0\leq t\leq T}. (Here, {-\frac{1}{2r}+} denotes any quantity slightly larger than {-\frac{1}{2r}}.)

Proof: Fix {\frac{1}{2}<\alpha<1}, {m=2}, {\theta>0}. Let {\xi>0} be a parameter to be chosen later and consider the function {n:\mathbb{R}\rightarrow\mathbb{N}}, {n(T)=T_k^{\xi}} for {T_k\leq T < T_{k+1}} (where {T_j:=j^{2\alpha/(2\alpha-1)}}).

The effective ergodic theorem (cf. Theorem 3) applied to the functions {f_R} introduced in the previous subsection says that, for {\mu}-almost every {v\in X} and all {T} sufficiently large (depending on {v} and {r>2}),

\displaystyle \int_0^T f_{n(T)}(g_t v) \, dt\geq \frac{1}{2}T\|f_{n(T)}\|_{L^1} - 2 T^{\alpha}\|f_{n(T)}\|_{C^{\theta}}

On the other hand, by construction, {\|f_{n(T)}\|_{L^1}\geq\frac{1}{d\, T_k^{\xi}}} and {\|f_{n(T)}\|_{C^{\theta}}\leq d \,T_k^{\theta\xi}} for a certain constant {d=d(r)>1} and for all {T_k\leq T < T_{k+1}}.

It follows that, for {\mu}-almost every {v\in X} and all {T} sufficiently large,

\displaystyle \int_0^T f_{n(T)}(g_t v) \, dt\geq \frac{1}{2d}T^{1-\xi} - 2 d T^{\alpha+\theta\xi}

If {1-\xi>\alpha+\theta\xi}, i.e., {\frac{1-\alpha}{1+\theta}>\xi}, the right-hand side of this inequality is strictly positive for all {T} sufficiently large. Since the function {f_{n(T)}} is supported on {X_{T_k^{\xi}}}, we deduce that if {\frac{1-\alpha}{1+\theta}>\xi} then, for {\mu}-almost every {v\in X} and all {T} sufficiently large, {g_{t_0}(v)\in X_{T_k^{\xi}}} (where {T_k\leq T<T_{k+1}}) for some {0\leq t_0\leq T}.

By plugging this information into Proposition 4, we conclude that, if

\displaystyle \frac{1-\alpha}{1+\theta}>\xi

then, for {\mu}-almost every {v\in X} and all {T} sufficiently large, the {x}-coordinate of {g_t(v)} is {\leq \frac{3}{2T_k^{\xi/r}}\leq 2/T^{\xi/r}} for some time {0\leq t_1\leq T+a} (where {a=a(r)} is a constant).

This proves the desired theorem: indeed, we can take the parameter {\xi} arbitrarily close to {1/2} in the previous paragraph because {\frac{1-\alpha}{1+\theta}\rightarrow 1/2} as {\alpha\rightarrow 1/2} and {\theta\rightarrow 0}. \Box

3.4. Very deep cusp excursions are atypical

Closing this post, let us now show that an elementary argument à la Borel–Cantelli implies that a typical geodesic doesn’t perform very deep cusp excursions:

Theorem 6 For {\mu}-almost every {v\in T^1 S} and for all {T} sufficiently large (depending on {v} and {r>2}), the base point of {g_t(v)} has {x}-coordinate {>T^{-\frac{1}{r}-}} for all times {0\leq t\leq T}. (Here, {-\frac{1}{r}-} denotes any quantity slightly smaller than {-\frac{1}{r}}.)

Proof: Let {\xi>0} and {\beta>0} be parameters to be chosen later, and denote {T_k=k^{\beta}}.

By elementary geometrical considerations about surfaces of revolution (similar to the proof of Proposition 4), we see that if the base point of {w\in T^1S} has {x}-coordinate {x=T_k^{-\xi}}, then the base point of {g_s(v)} has {x}-coordinate in {[(1/2)T_k^{-\xi}, 2T_k^{-\xi}]} for all {|s|\sim T_k^{-\xi}}.

Therefore, if we divide {[0,T_k]} into {\sim T_k^{1+\xi}} intervals {I_j^{(k)}=[a_j^{(k)}, b_j^{(k)}]} of sizes {\sim T_k^{-\xi}}, then

\displaystyle \{v: \exists t\in I_j^{(k)} \textrm{ with } x(g_t(v)) = T_k^{-\xi} \} \subset \{v: x(g_{a_j^{(k)}}(v))\in [\frac{1}{2}T_k^{-\xi}, 2T_k^{-\xi}]\}

Since the Liouville measure {\mu} is {g_t}-invariant and the surface of revolution of the profile {y=x^r} has the property that the volume of the region {\{w\in T^1S: x(w)\in [\frac{1}{2R}, \frac{2}{R}]\}} is {O(R^{r+1})}, we deduce that

\displaystyle \mu(\{v\in T^1S: \exists t\in I_j^{(k)} \textrm{ with } x(g_t(v)) = T_k^{-\xi} \})=O(1/T_k^{\xi(r+1)})

for all {j}. Because we need {\sim T_k^{1+\xi}} indices {j} to cover the time interval {[0,T_k]}, we obtain that

\displaystyle \mu(\{v\in T^1S: \exists t\in [0,T_k] \textrm{ with } x(g_t(v)) = T_k^{-\xi} \})=O(1/T_k^{\xi r-1}) \ \ \ \ \ (3)

We want to study the set {A_k=\{v\in T^1S: \exists t\in [0,T_k] \textrm{ with } x(g_t(v)) \leq T_k^{-\xi}\}}. We divide {A_k} into {B_k:=A_k\cap\{v\in T^1S: x(v)\leq 2T_k^{-\xi}\}} and {C_k:=A_k\setminus B_k}. Because {\mu(B_k) \leq \mu(\{v\in T^1S: x(v)\leq 2T_k^{-\xi}\}) = O(1/T_k^{\xi(r+1)})}, we just need to compute {\mu(C_k)}. For this sake, we observe that

\displaystyle C_k\subset \{v: \exists t\in [0,T_k] \textrm{ with } x(g_t(v)) = T_k^{-\xi}\}

and, a fortiori, {\mu(C_k)=O(1/T_k^{\xi r-1})} thanks to (3). In particular,

\displaystyle \mu(\{v\in T^1S: \exists t\in [0,T_k] \textrm{ with } x(g_t(v)) \leq T_k^{-\xi}\})=\mu(A_k) = O(1/T_k^{\xi r-1})

Note that the series {\sum\limits_{k=1}^{\infty}1/T_k^{\xi r - 1} = \sum\limits_{k=1}^{\infty}1/k^{\beta(\xi r - 1)}} is summable when {\beta(\xi r-1)>1}, i.e., {\xi>\frac{1}{r}(1+\frac{1}{\beta})} In this context, Borel–Cantelli lemma implies that, for {\mu}-almost every {v\in T^1S}, the {x}-coordinate {g_t(v)} is {>T_k^{-\xi}} for all {t\in [0, T_k]} and all {T_k=k^{\beta}} sufficiently large (depending on {v}). Since {\frac{T_{k+1}}{T_k}\rightarrow 1} as {k\rightarrow\infty}, we conclude that if

\displaystyle \xi>\frac{1}{r}(1+\frac{1}{\beta})

then for {\mu}-almost every {v\in T^1S}, the {x}-coordinate {g_t(v)} is {>T^{-\xi}} for all {t\in [0, T]} and all {T} sufficiently large (depending on {v}).

This ends the proof of the theorem: in fact, by letting {\beta\rightarrow\infty}, we can take {\xi>1/r} arbitrarily close to {1/r} in the previous paragraph. \Box

Remark 4 By Theorems 5 and 6, a typical geodesic {\{g_t(v)\}_{t\in\mathbb{R}}} enters the region {\{w: x(w)\leq T^{-1/2r+}\}} while avoiding the region {\{w: x(w)\leq T^{-1/r-}\}} during the time interval {[0,T]} (for all {T} sufficiently large).Of course, the presence of a gap between {T^{-1/2r+}} and {T^{-1/r-}} motivates the following question: is there an optimal exponent {\frac{1}{2r}\leq \xi\leq \frac{1}{r}} such that a typical geodesic {\{g_t(v)\}_{t\in\mathbb{R}}} enters {\{w: x(w)\leq T^{-\xi+}\}} while avoiding {\{w: x(w)\leq T^{-\xi-}\}} during the time interval {[0,T]} (for all {T} sufficiently large)?

Remark 5 Contrary to the cases discussed in Gadre’s paper, we can’t show that typically there is only one “maximal” cusp excursion in our current setting (of geodesic flows on negatively curved surfaces with cusps modelled by surfaces of revolution of profiles {y=x^r}). In fact, the exponential decay of correlations estimate proved by Burns–Masur–M.–Wilkinson is not strong enough to provide good quasi-independence estimates for consecutive cusp excursions (with the same quality of Lemma 2.11 in Gadre’s paper).

Rodolfo Gutiérrez-Romo has just uploaded to the arXiv his preprint Zariski density of Rauzy–Veech groups: proof of the Zorich conjecture.

This article is part of the PhD thesis project of Rodolfo (under the supervision of Anton Zorich and myself), which started last September 2016. (In fact, one of my motivations to obtain a “Habilitation à Diriger des Recherches” degree last June 2, 2017 was precisely to be able to formally co-supervise Rodolfo’s PhD thesis project.)

In this (short) blog post, we discuss some aspects of Rodolfo’s solution to Zorich conjecture (and we refer to the preprint for the details).

1. Statement of Zorich conjecture

The study of Lyapunov exponents of the Kontsevich-Zorich cocycle (and, more generally, variations of Hodge structures) found many applications since the pioneer works of Zorich and Forni in the late nineties:

  • Zorich and Forni described the deviations of ergodic averages of typical interval exchange maps and translation flows in terms of Lyapunov exponents;
  • Avila and Forni used in 2007 the positivity of second Lyapunov of the Kontsevich-Zorich cocycle with respect to Masur-Veech measures (among many other ingredients) to show that typical interval exchange transformations and translation flows are weak mixing;
  • Delecroix, Hubert and Lelièvre confirmed in 2014 a conjecture of Hardy and Weber on the abnormal rate of diffusion of typical trajectories on {\mathbb{Z}^2}-periodic Ehrenfest wind-tree models of Lorenz gases;
  • Kappes and Möller completed in 2016 the classification of commensurability classes of non-arithmetic lattices of {PU(1,n)}, {n\geq 2}, constructed by Deligne and Mostow in the eighties thanks to new invariants coming from Lyapunov exponents;
  • etc.

The success of Zorich in describing such deviations of ergodic averages together with many numerical experiments led Kontsevich and him to conjecture that the Lyapunov exponents of the Kontsevich-Zorich (KZ) cocycle with respect to Masur-Veech measures are simple (i.e., their multiplicities are {1}).

Moreover, Zorich had in mind a specific way to establish the Kontsevich-Zorich conjecture: first, he conjectured that the so-called Rauzy-Veech groups (associated to a certain combinatorial description of the matrices of the KZ cocycle appearing along typical trajectories for the Masur-Veech measures) are Zariski-dense in the symplectic groups {Sp(2g,\mathbb{R})}, {g\geq 1}; then, he noticed that the works of Guivarc’h-Raugi and Goldsheid-Margulis on the simplicity of Lyapunov exponents for random products of matrices forming a Zariski-dense subgroup could be useful to deduce the “Kontsevich-Zorich simplicity conjecture” from his “Zariski density conjecture”.

After an important partial result of Forni in 2002, Avila and Viana famously established the Kontsevich-Zorich conjecture in 2007. Nevertheless, the arguments of Avila and Viana were slightly different from the scheme outline by Zorich: indeed, as they pointed out in Remark 6.12 of their paper, Avila and Viana avoided discussing the Zariski closure of Rauzy-Veech groups by showing that Rauzy-Veech groups are pinching and twisting, and that these two properties suffice to get the simplicity of the Lyapunov spectrum (i.e., Kontsevich-Zorich conjecture).

Remark 1 It is worth to notice that Zariski density implies pinching and twisting, but the converse is not true in general.

In summary, the solution of the Kontsevich-Zorich conjecture by Avila and Viana via the pinching and twisting properties for Rauzy-Veech groups left open Zorich’s conjecture on the Zariski density of Rauzy-Veech groups.

Remark 2 Besides giving stronger information about Rauzy-Veech groups (and, in particular, a new proof of Avila and Viana theorem), Zorich’s conjecture has other applications: for example, Magee recently showed that the validity of Zorich’s conjecture implies that the spectral gap / rate of mixing of the geodesic flow on congruence covers of connected components of the strata of moduli spaces of unit area translation surfaces is uniform.

2. Hyperelliptic Rauzy-Veech groups

As we already discussed in this blog, Avila, Yoccoz and myself were able to prove Zorich’s conjecture in the particular case of hyperelliptic Rauzy-Veech groups by showing the stronger statement that such groups contain an explicit finite-index subgroup of {Sp(2g,\mathbb{Z})}: roughly speaking, the Rauzy-Veech group is the subgroup of {Sp(2g,\mathbb{Z})} consisting of matrices whose reduction modulo two permute the basis vectors {e_1,\dots, e_{2g}} and {\sum\limits_{k=1}^{2g} e_k}.

As it turns out, the hyperelliptic Rauzy-Veech groups are associated to one of the three connected components of the so-called minimal strata {\mathcal{H}(2g-2)} (consisting of translation surfaces of genus {g} with a unique conical singularity of total angle {2\pi(2g-1)}): in fact, it was proved by Kontsevich and Zorich in 2003 that the minimal strata (of genus {g\geq 4}) have three connected components called hyperelliptic {\mathcal{H}(2g-2)^{hyp}}, even spin {\mathcal{H}(2g-2)^{even}} and odd spin {\mathcal{H}(2g-2)^{odd}}.

3. Rauzy-Veech groups of minimal strata

As a warm-up problem, we asked Rodolfo to perform numerical experiments with the Rauzy-Veech groups of the odd connected component {\mathcal{H}(4)^{odd}} of the minimal stratum in genus 3 and the even and odd connected components of the minimal stratum in genus 4. In particular, we told him to “compute” the indices of the reductions modulo two of such a Rauzy-Veech group in {Sp(6,\mathbb{Z}/2\mathbb{Z})} and {Sp(8,\mathbb{Z}/2\mathbb{Z})}.

After playing a bit with the matrices in his computer, Rodolfo announced (among many other things) that the index in {Sp(6,\mathbb{Z}/2\mathbb{Z})} of the Rauzy-Veech group of {\mathcal{H}(4)^{odd}} was 28.

This number ringed a bell because (as it is briefly explained here for instance) {Sp(2g,\mathbb{F}_2)} contains two orthogonal subgroups {O^{even}}, resp. {O^{odd}}, of indices {2^{g-1}(2^g+1)}, resp. {2^{g-1}(2^g-1)}, consisting of matrices stabilizing a quadratic form with even, resp. odd Arf invariant of representing the reduction modulo two of the symplectic form. In particular, the fact that the number {28 = 2^{3-1}(2^3-1)} matches the index of {O^{odd}} suggest the conjecture that Rauzy-Veech groups of {\mathcal{H}(2g-2)^{odd}}, resp. {\mathcal{H}(2g-2)^{even}}, is the pre-image of {O^{odd}}, resp. {O^{even}} in {Sp(2g,\mathbb{Z})} under the reduction modulo two map {Sp(2g,\mathbb{Z})\rightarrow Sp(2g,\mathbb{Z}/2\mathbb{Z})}.

Once we convinced ourselves about the plausibility of this conjecture, Rodolfo started working on the geometry of the corresponding Rauzy diagrams (graphs underlying the structure of the Rauzy-Veech groups) in order to figure out a systematic way of producing many particular matrices generating the desired candidate groups above.

As it turns out, Rodolfo did this in two steps (which occupy most [15 pages] of his preprint):

  • first, he exploits the fact that the level two congruence subgroup {\Gamma_2(2g)} of {Sp(2g,\mathbb{Z})} (i.e., the kernel of the natural map {Sp(2g,\mathbb{Z})\rightarrow Sp(2g,\mathbb{Z}/2\mathbb{Z})}) is generated by the squares of certain symplectic transvections to show that the Rauzy-Veech groups of the odd and even components of {\mathcal{H}(2g-2)} contain {\Gamma_2(2g)}; for this sake, he exhibits a rich set of loops in Rauzy diagrams inducing appropriate Dehn twists (giving “most” of the desired symplectic transvections).
  • secondly, he proves that the reduction modulo two of the Rauzy-Veech groups of the odd, resp. even components of {\mathcal{H}(2g-2)} coincides with {O^{odd}}, resp. {O^{even}}, by using the fact that {O^{odd}} and {O^{even}} are generated by orthogonal transvections.

In summary, Rodolfo showed that the Rauzy-Veech groups of the odd and even components of {\mathcal{H}(2g-2)} are explicit subgroups of {Sp(2g,\mathbb{Z})} of indices {2^{g-1}(2^g-1)} and {2^{g-1}(2^g+1)}.

By putting this result together with the result by Avila, Yoccoz and myself for hyperelliptic Rauzy-Veech groups, we conclude that the Rauzy-Veech group of any connected component of a minimal stratum {\mathcal{H}(2g-2)} is a finite-index subgroup of {Sp(2g,\mathbb{Z})} and, a fortiori, a Zariski-dense subgroup of {Sp(2g,\mathbb{R})}.

4. Rauzy-Veech groups of general strata

Philosophically speaking, a general translation surface {X} of genus {g} “differs” from a translation surface {Y} in the minimal stratum {\mathcal{H}(2g-2)} because of the relative homology produced by the presence of many conical singularities. In particular, if we “merge” conical singularities of {X}, we should find a translation surface {Y\in\mathcal{H}(2g-2)}.

From the geometrical point of view, this philosophy was made rigorous by Kontsevich and Zorich in 2003: indeed, they formalized the notions of “merging” and “breaking” zeroes in order to “reduce” the classification of connected components of general strata to the case of connected components of minimal strata!

From the combinatorial point of view, Avila and Viana obtained the following combinatorial analog of Kontsevich-Zorich geometrical statement: we can merge conical singularities until we end up with a component of a minimal stratum in such a way that a “copy” of the Rauzy-Veech group of a component of a minimal stratum shows up inside any Rauzy-Veech group.

Hence, this result of Avila-Viana (or rather its variant stated as Lemma 6.3 in Rodolfo’s preprint) allows Rodolfo to conclude his proof of (a statement slightly stronger than) Zorich’s conjecture: the Rauzy-Veech group of any connected component of any stratum of the moduli space of unit area translation surfaces of genus {g} is a finite-index subgroup of {Sp(2g,\mathbb{Z})} simply because the same is true for the connected components of the minimal stratum {\mathcal{H}(2g-2)}.

Posted by: matheuscmss | April 5, 2017

New numbers in M-L

In 1973, Freiman found an interesting number {\alpha_{\infty}} in the complement {M\setminus L} of the Lagrange spectrum {L} in the Markov spectrum {M}.

By carefully analyzing Freiman’s argument, Cusick and Flahive constructed in 1989 a sequence {\alpha_n} converging to {\alpha_{\infty}} as {n\rightarrow\infty} such that {\alpha_n\in M\setminus L} for all {n\geq 4}, and, as it turns out, {\alpha_4} was the largest known element of {M\setminus L}.

In our recent preprint, Gugu and I described the structure of the complement {M\setminus L} of the Lagrange spectrum {L} in the Markov spectrum {M} near {\alpha_{\infty}}, and this led us to wonder if our description could be used to find new numbers in {M\setminus L} which are larger than {\alpha_4}.

As it turns out, Gugu and I succeeded in finding such numbers and we are currently working on the combinatorial arguments needed to extract the largest number given by our methods. (Of course, we plan to include a section on this matter in a forthcoming revised version of our preprint.)

In order to give a flavour on our construction of new numbers in {M\setminus L}, we will prove in this post that a certain number

\displaystyle x=3.2930444288\dots > \alpha_4 = 3.2930442719\dots

belongs to {M\setminus L}.

1. Preliminaries


\displaystyle [a_0; a_1, a_2,\dots] = a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{\ddots}}}

be the usual continued fraction expansion.

We abbreviate periodic continued fractions by putting a bar over the period: for instance, {[2; \overline{1,1,2,2,2,1,2}] = [2; 1,1,2,2,2,1,2,1,1,2,2,2,1,2,\dots]}. Moreover, we use subscripts to indicate the multiplicity of a digit in a sequence: for example, {[2; 1,1,2,2,2,1,2, \dots] = [2; 1_2,2_3,1,2,\dots]}.

Given a bi-infinite sequence {A=(a_n)_{n\in\mathbb{Z}}\in(\mathbb{N}^*)^{\mathbb{Z}}} and {i\in\mathbb{Z}}, let

\displaystyle \lambda_i(A) := [a_i; a_{i+1}, a_{i+2}, \dots] + [0; a_{i-1}, a_{i-2}, \dots]

In this context, recall that the classical Lagrange and Markov spectra {L} and {M} are the sets

\displaystyle L=\{\ell(A)<\infty: A\in(\mathbb{N}^*)^{\mathbb{Z}}\} \quad \textrm{and} \quad M=\{m(A)<\infty: A\in(\mathbb{N}^*)^{\mathbb{Z}} \}


\displaystyle \ell(A)=\limsup\limits_{i\rightarrow\infty}\lambda_i(A) \quad \textrm{and} \quad m(A) = \sup\limits_{i\in\mathbb{Z}} \lambda_i(A)

As we already mentioned, Freiman proved that

\displaystyle \alpha_{\infty} = [2;\overline{1_2, 2_3, 1, 2}] + [0; 1, 2_3, 1_2, 2, 1, \overline{2}] = 3.2930442654\dots\in M\setminus L,

and Cusick and Flahive extended Freiman’s argument to show that the sequence

\displaystyle \alpha_n := [2;\overline{1_2, 2_3, 1, 2}] + [0; 1, 2_3, 1_2, 2, 1, 2_n, \overline{1, 2, 1_2, 2_3}]

accumulating on {\alpha_{\infty}} has the property that {\alpha_n\in M\setminus L} for all {n\geq 4}. In particular,

\displaystyle \alpha_4 = 3.2930442719\dots

was the largest known number in {M\setminus L}.

2. A new number in {M\setminus L}

In what follows, we will show that

Theorem 1

\displaystyle x:=[2;\overline{1_2, 2_3, 1, 2}]+[0;1, 2_3, 1_2, 2, 1, 2_2,1,\overline{1_2,2}] = 3.2930444288\dots \in M\setminus L

Remark 1 {x} is a “good” variant of {\alpha_2 = 3.2930444886\dots} in the sense that it falls in a certain interval {I} which can be proved to avoid the Lagrange spectrum: see Proposition 5 below.

Remark 2 Note that {\frac{x-\alpha_{\infty}}{\alpha_4-\alpha_{\infty}}=24.8321\dots}, i.e., if we center our discussion at {\alpha_{\infty}}, then {x} is almost 25 times bigger than {\alpha_4}.

Similarly to the arguments of Freiman and Cusick-Flahive, the proof of Theorem 1 starts by locating an appropriate interval {I} centered at {\alpha_{\infty}} such that {I} does not intersect the Lagrange spectrum.

In this direction, one needs the following three lemmas:

Lemma 2 If {B\in\{1,2\}^{\mathbb{Z}}} contains any of the subsequences

  • (a) {1^*}
  • (b) {22^*}
  • (c) {1_2 2^* 1_2}
  • (d) {2_2 1 2^* 1_2 2 1}
  • (e) {1 2_2 1 2^* 1_2 2}
  • (f) {2_4 1 2^* 1_2 2_3}

then {\lambda_j(B)<\alpha_{\infty} - 10^{-5}} where {j} indicates the position in asterisk.

Proof: See Lemma 2 in Chapter 3 of Cusick-Flahive’s book. \Box

Lemma 3 If {B\in\{1,2\}^{\mathbb{Z}}} contains any of the subsequences:

  • (i) {212^*12}
  • (ii) {212^*1_3}
  • (iii) {1212^*1_2}
  • (iv) {2_3 1 2^* 1_2 2_2 1}
  • (v) {2 1 2_3 1 2^* 1_2 2_3}
  • (vi) {1_2 2_3 1 2^* 1_2 2_4}
  • (vii) {1_2 2_3 1 2^* 1_2 2_3 1_2}
  • (viii) {1_3 2_3 1 2^* 1_2 2_3 1 2}
  • (ix) {2 1_2 2_3 1 2^* 1_2 2_3 1 2_2}
  • (x) {2_2 1_2 2_3 1 2^* 1_2 2_3 1 2 1}
  • (xi) {1_2 2 1_2 2_3 1 2^* 1_2 2_3 1 2 1_2 2}

then {\lambda_j(B)>\alpha_{\infty}+10^{-6}} where {j} indicates the position in asterisk.

Proof: See Lemma 1 in Chapter 3 of Cusick-Flahive’s book and also Lemma 3.2 of our preprint with Gugu. \Box

Lemma 4 If {B\in\{1,2\}^{\mathbb{Z}}} contains the subsequence:

  • (xii) {2_2 1 2 1_2 2_3 1 2^* 1_2 2_3 1 2 1_2 2 1}

then {\lambda_j(B)>\alpha_{\infty}+1.968\times 10^{-7}} where {j} indicates the position in asterisk.

Proof: In this situation,

\displaystyle \begin{array}{rcl} \lambda_j(B) &=& [2;1_2,2_3,1,2,1_2,2,1,\dots] + [0;1,2_3,1_2,2,1,2_2,\dots] \\ &\geq& [2;1_2,2_3,1,2,1_2,2,1,\overline{1,2}] + [0;1,2_3,1_2,2,1,2_2,\overline{2,1}] \\ &=& 3.2930444624\dots \end{array}

thanks to the standard fact that if

\displaystyle \alpha=[a_0; a_1,\dots, a_n, a_{n+1},\dots]


\displaystyle \beta=[a_0; a_1,\dots, a_n, b_{n+1},\dots]

with {a_{n+1}\neq b_{n+1}}, then {\alpha>\beta} if and only if {(-1)^{n+1}(a_{n+1}-b_{n+1})>0}. \Box

As it is explained in Chapter 3 of Cusick-Flahive’s book and also in the proof of Proposition 3.7 of our preprint with Gugu, Lemmas 2, 3 and 4 allow to show that:

Proposition 5 The interval

\displaystyle I=[\alpha_{\infty}-10^{-8}, \alpha_{\infty}+ 1.968\times 10^{-7})

does not intersect the Lagrange spectrum {L}.

Of course, this proposition gives a natural strategy to exhibit new numbers in {M\setminus L}: it suffices to build elements of {M\cap I} as close as possible to the right endpoint of {I}.

Remark 3 As the reader can guess from the statement of the previous proposition, the right endpoint of {I} is intimately related to Lemma 4. In other terms, the natural limit of this method for producing the largest known numbers in {M\setminus L} is given by how far we can push to the right the boundary of {I}.Here, Gugu and I are currently trying to optimize the choice of {I} by exploiting the simple observation that the proof of Lemma 4 is certainly not sharp in our situation: indeed, we used the sequence

\displaystyle B':=\overline{1,2},2_2,1,2,1_2,2_3,1,2;1_2,2_3,1,2,1_2,2,1,\overline{1,2}

to bound {\lambda_j(B)}, but we could do better by noticing that this sequence provides a pessimistic bound because {\overline{1,2}} contains a copy of the word {21212} (and, thus, {m(B')>\alpha_{\infty}+10^{-6}}, i.e., {B} can’t coincide with {B'} on a large chunk when {m(B)<\alpha_{\infty}+10^{-6}}).

Anyhow, Proposition 5 ensures that

\displaystyle x:=[2;\overline{1_2, 2_3, 1, 2}]+[0;1, 2_3, 1_2, 2, 1, 2_2,1,\overline{1_2,2}] = 3.2930444288\dots

does not belong to the Lagrange spectrum {L}.

At this point, it remains only to check that {x} belongs to the Markov spectrum.

For this sake, let us verify that

\displaystyle x= [2;\overline{1_2, 2_3, 1, 2}]+[0;1, 2_3, 1_2, 2, 1, 2_2,1,\overline{1_2,2}]=:\lambda_0(A) =m(A)

By items (a), (b), (c) and (e) of Lemma 2,

\displaystyle \lambda_i(A)<\alpha_{\infty}-10^{-5}

except possibly for {i=7k} with {k\geq 1}. Since

\displaystyle \begin{array}{rcl} \lambda_{7k}(A) &=& [2; \overline{1_2, 2_3, 1, 2}] + [0;\underbrace{1, 2_3, 1_2, 2, \dots, 1, 2_3, 1_2, 2}_{k \textrm{ times }}, 1, 2_3, 1_2, 2, 1, 2_2,\dots] \\ &<& [2; \overline{1_2, 2_3, 1, 2}] + [0; 1, 2_3, 1_2, 2, 1, 2_3, 1_2, 2, 1,\overline{2,1}] \\ &<& [2; \overline{1_2, 2_3, 1, 2}] + [0; 1, 2_3, 1_2, 2, 1, 2_3,\overline{2,1}] \\ &<& [2; \overline{1_2, 2_3, 1, 2}] + [0; 1, 2_3, 1_2, 2, 1, 2_2,1,\overline{1_2,2}] = \lambda_0(A), \end{array}

we have

\displaystyle \begin{array}{rcl} \lambda_0(A)-\lambda_{7k}(A)&>&[0; 1, 2_3, 1_2, 2, 1, 2_3,\overline{2,1}] - [0; 1, 2_3, 1_2, 2, 1, 2_3, 1_2, 2, 1,\overline{2,1}] \\ &>& 2.817153\dots\times 10^{-8} \end{array}

for all {k\geq 1}. This proves that {x=\lambda_0(A)=m(A)} belongs to the Markov spectrum.

Posted by: matheuscmss | March 14, 2017

HD(M-L) > 0.353

My friend Gugu and I have just uploaded to arXiv our paper {0.353 < HD(M\setminus L)}.

In this article, we study the complement {M\setminus L} of the Lagrange spectrum {L} in the Markov spectrum {M} near a non-isolated point {\alpha_{\infty}} found by Freiman, and, as a by-product, we prove that its Hausdorff dimension {HD(M\setminus L)} is

\displaystyle HD(M\setminus L) > 0.353

Remark 1 Currently, this paper deals exclusively with lower bounds on {HD(M\setminus L)}. In its next version, Gugu and I will include upper bounds on {HD(M\setminus L)}.

In what follows, we present a streamlined version of our proof of {HD(M\setminus L) > 0.353} based on the construction of an explicit Cantor set {K\subset M\setminus L} with {HD(K)>0.353}.

Remark 2 W e refer to our paper for more refined informations about the structure of {M\setminus L} near {\alpha_{\infty}}.

1. Perron’s characterization of the classical spectra

Given a bi-infinite sequence {A=(a_n)_{n\in\mathbb{Z}}\in(\mathbb{N}^*)^{\mathbb{Z}}} and {i\in\mathbb{Z}}, let

\displaystyle \lambda_i(A) := [a_i; a_{i+1}, a_{i+2}, \dots] + [0; a_{i-1}, a_{i-2}, \dots]


\displaystyle [a_0; a_1, a_2,\dots] = a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{\ddots}}}

is the usual continued fraction expansion, and

\displaystyle [a_0; a_1,\dots, a_n] := a_0+\frac{1}{a_1+\frac{1}{\ddots+\frac{1}{a_n}}} := [a_0; a_1,\dots, a_n,\infty,\dots]

is the {n}th convergent.

In 1921, Perron showed that the classical Lagrange and Markov spectra {L} and {M} are the sets

\displaystyle L=\{\ell(A)<\infty: A\in(\mathbb{N}^*)^{\mathbb{Z}}\} \quad \textrm{and} \quad M=\{m(A)<\infty: A\in(\mathbb{N}^*)^{\mathbb{Z}} \}


\displaystyle \ell(A)=\limsup\limits_{i\rightarrow\infty}\lambda_i(A) \quad \textrm{and} \quad m(A) = \sup\limits_{i\in\mathbb{Z}} \lambda_i(A)

2. Freiman’s number {\alpha_{\infty}}

In 1973, Freiman showed that

\displaystyle \alpha_{\infty}:=\lambda_0(A_{\infty}):=[2; \overline{1_2, 2_3, 1, 2}] + [0; 1, 2_3, 1_2, 2, 1, \overline{2}]\in M \setminus L

In a similar vein, Theorem 4 in Chapter 3 of Cusick-Flahive book asserts that

\displaystyle \alpha_n:=\lambda_0(A_n):= [2; \overline{1_2, 2_3, 1, 2}] + [0; 1, 2_3, 1_2, 2, 1, 2_n, \overline{1, 2, 1_2, 2_3}]\in M\setminus L

for all {n\geq 4}. In particular, {\alpha_{\infty}} is not isolated in {M\setminus L}.

Remark 3 As it turns out, {\alpha_4} is the largest known number in {M\setminus L}: see page 35 of Cusick-Flahive book.

In what follows, we shall revisit Freiman’s arguments as described in Chapter 3 of Cusick-Flahive book in order to prove the following result:

Theorem 1 Consider the alphabet {B=\{\beta_1, \beta_2\}} consisting of the words {\beta_1 = 1\in\mathbb{N}^*} and {\beta_2 = 2_2 = (2,2)\in(\mathbb{N}^*)^2}. Then,

\displaystyle K:=\{[2; \overline{1_2, 2_3, 1, 2}] + [0; 1, 2_3, 1_2, 2, 1, 2_4,\gamma_1,\gamma_2,\dots]: \gamma_i\in B \,\,\,\,\forall\,i\geq 1\}\subset M\setminus L

3. A standard comparison tool

In the sequel, we use the following standard comparison tool for continued fractions is the following lemma (cf. Lemmas 1 and 2 in Chapter 1 of Cusick-Flahive book):

Lemma 2 Let {\alpha=[a_0; a_1,\dots, a_n, a_{n+1},\dots]} and {\beta=[a_0; a_1,\dots, a_n, b_{n+1},\dots]} with {a_{n+1}\neq b_{n+1}}. Then:

  • {\alpha>\beta} if and only if {(-1)^{n+1}(a_{n+1}-b_{n+1})>0};
  • {|\alpha-\beta|<1/2^{n-1}}.

Remark 4 For later use, note that Lemma 2 implies that if {a_0\in\mathbb{Z}} and {a_i\in\mathbb{N}^*} for all {i\geq 1}, then {[a_0; a_1,\dots, a_n,\dots]<[a_0; a_1,\dots, a_n,\infty, ...]:=[a_0; a_1,\dots, a_n]} when {n\geq 1} is odd, and {[a_0; a_1,\dots, a_n,\dots]>[a_0; a_1,\dots, a_n]} when {n\geq 0} is even.

4. Proof of Theorem 1

Similarly to the discussions in Cusick-Flahive book, we shall use the next lemma (extracted from Lemma 2 in Chapter 3 of this book):

Lemma 3 If {B\in\{1,2\}^{\mathbb{Z}}} contains any of the subsequences

  • (a) {1^*}
  • (b) {22^*}
  • (f) {2_4 1 2^* 1_2 2_3}

then {\lambda_j(B)<\alpha_{\infty} - 10^{-5}} where {j} indicates the position in asterisk.

Proof: If (a) occurs, then {\lambda_j(B) = 1+[0;\dots]+[0;\dots]<3<\alpha_{\infty}-10^{-1}}.

If (b) occurs, then Remark 4 implies that

\displaystyle \lambda_j(B) = [2;2,\dots]+[0;\dots]<[2; 1, 2, 1] + [0; 2, 2, 1] = \frac{89}{28} < \alpha_{\infty}-10^{-1}

If (f) occurs, then Remark 4 implies that

\displaystyle \begin{array}{rcl} \lambda_j(B) &=& [2; 1_2, 2_3, \dots]+[0; 1, 2_4, \dots] \\ &<& [2; 1_2, 2_4, 1] + [0; 1, 2_5, 1] = \frac{45641}{13860} < \alpha_{\infty} - 10^{-5} \end{array}


We shall also need the following fact:

Lemma 4 If {A\in(\mathbb{N}^*)^{\mathbb{Z}}} is a bi-infinite sequence such that

\displaystyle \alpha_{\infty}-10^{-8} < m(A) < \alpha_{\infty} + 10^{-8}

then {m(A)\in M\setminus L}.

Proof: See the proof of Theorem 4 in Chapter 3 of Cusick-Flahive book (especially the last paragraph at page 40). \Box

These lemmas allow us to conclude the proof of Theorem 1 along the following lines.

Proposition 5 Given a bi-infinite sequence

\displaystyle B=\dots,\gamma_2^T,\gamma_1^T, 2_4, 1, 2, 1_2, 2_3, 1, 2; \overline{1_2, 2_3, 1, 2}

where {\gamma_i\in \{1, 2_2\}} for all {i\geq 1} and {;} serves to indicate the zeroth position, then

\displaystyle m(B) = [2; \overline{1_2, 2_3, 1, 2}] + [0; 1, 2_3, 1_2, 2, 1, 2_4,\gamma_1,\gamma_2,\dots]\in [\alpha_{\infty}-10^{-8}, \alpha_{\infty}+10^{-8}]

Proof: On one hand, Remark 4 implies that

\displaystyle \lambda_0(B)\leq [2; \overline{1_2, 2_3, 1, 2}] + [0; 1, 2_3, 1_2, 2, 1, 2_4, 1, 2, 1] < \alpha_{\infty}+10^{-8}


\displaystyle \lambda_0(B)\geq [2; \overline{1_2, 2_3, 1, 2}] + [0; 1, 2_3, 1_2, 2, 1, 2_4, 2, 1] > \alpha_{\infty} - 10^{-8},

and items (a), (b) and (f) of Lemma 3 imply that

\displaystyle \lambda_i(B)<\alpha_{\infty}-10^{-5}

for all positions {i} except possibly for {i=7k} with {k\geq 1}.

On the other hand,

\displaystyle \begin{array}{rcl} \lambda_{7k}(B) &=& [2; \overline{1_2, 2_3, 1, 2}] + [0;\underbrace{1, 2_3, 1_2, 2, \dots, 1, 2_3, 1_2, 2}_{k \textrm{ times }}, 1, 2_3, 1_2, 2, 1, 2_4,\dots] \\ &<& [2; \overline{1_2, 2_3, 1, 2}] + [0; 1, 2_3, 1_2, 2, 1, 2_3, 1_2, 2, 1] \\ &<& [2; \overline{1_2, 2_3, 1, 2}] + [0; 1, 2_3, 1_2, 2, 1, 2_4] \\ &<& [2; \overline{1_2, 2_3, 1, 2}] + [0; 1, 2_3, 1_2, 2, 1, 2_4,\dots] = \lambda_0(B), \end{array}

so that {\lambda_0(B)-\lambda_{7k}(B)>[0; 1, 2_3, 1_2, 2, 1, 2_4] - [0; 1, 2_3, 1_2, 2, 1, 2_3, 1_2, 2, 1]> 10^{-9}} for all {k\geq 1}. This proves the proposition. \Box

At this point, the proof of Theorem 1 is complete: in fact, Proposition 5 and Lemma 4 together imply that

\displaystyle K:=\{[2; \overline{1_2, 2_3, 1, 2}] + [0; 1, 2_3, 1_2, 2, 1, 2_4,\gamma_1,\gamma_2,\dots]: \gamma_i\in B \,\,\,\,\forall\,i\geq 1\}

is contained in {M\setminus L}.

5. Lower bounds on {HD(M\setminus L)}

The Gauss map {G:(0,1)\rightarrow[0,1]}, {G(x):=\{1/x\}} (where {\{y\}} is the fractional part of {y}) acts on continued fractions as a shift operator:

\displaystyle G([0;a_1, a_2, a_3, \dots]) = [0; a_2, a_3, \dots]

Therefore, we can use the iterates of the Gauss map {G} to build a bi-Lipschitz map between the Cantor set {K} introduced above and the dynamical Cantor set

\displaystyle K(\{1, 2_2\}):=\{[0;\gamma_1,\gamma_2,\dots]: \gamma_i\in B \,\,\,\,\forall\,i\geq 1\}

Since the Hausdorff dimension is preserved by bi-Lipschitz maps, an immediate corollary of Theorem 1 is:

Corollary 6 One has {HD(M\setminus L)\geq HD(K) = HD(K(\{1, 2_2\}))}.

On the other hand, the Hausdorff dimension {HD(K(\{1, 2_2\}))} was estimated in Subsection 2.2 of this previous post here. In particular, it was shown that:

Proposition 7 One has {HD(K(\{1, 2_2\})) > 0.353}.

By putting Corollary 6 and Proposition 7, we conclude the desired estimate

\displaystyle HD(M\setminus L)>0.353

in the title of this post.

Posted by: matheuscmss | February 4, 2017

Soft bounds on the Hausdorff dimension of dynamical Cantor sets

Many problems in several areas of Mathematics (including Dynamical Systems and Number Theory) can “reduced” to the analysis of dynamical Cantor sets: for instance, the theorems of Newhouse, Palis and Takens on homoclinic bifurcations of surfaces diffeomorphisms, and the theorems of Hall, Freiman and Moreira on the structure of the classical Lagrange and Markov spectra rely on the study of dynamical Cantor sets of the real line.

The wide range of applicability of dynamical Cantor sets is partly explained by the fact that several natural examples of Cantor sets are defined in terms of Dynamical Systems: for example, Cantor’s ternary set {C} is

\displaystyle C=\bigcap\limits_{n\in\mathbb{N}} T^{-n}([0,1])

where {T:[0,1/3]\cup [2/3,1]\rightarrow [0,1]} is {T(x)=3x (\textrm{ mod } 1)}.

In some applications of dynamical Cantor sets, it is important to dispose of estimates on their Hausdorff dimensions: for instance, the celebrated work of Bourgain and Kontorovich on Zaremba’s conjecture needs particular types of dynamical Cantor sets with Hausdorff dimension close to one.

For this reason, a considerable literature on this topic was developed. Among the diverse settings covered by many authors, one finds the articles of Bumby, Hensley, …, Jenkinson-Pollicott, Falk-Nussbaum, where the so-called thermodynamical methods are exploited to produce approximations for the Hausdorff dimension of Cantor sets defined in terms of continued fraction expansions (i.e., Cantor sets of number-theoretical nature).

In general, the thermodynamical methods quoted above provide a sequence {s_n} of fast-converging approximations for the Hausdorff dimension {HD(K)} of dynamical Cantor sets: for instance, the algorithm described by Jenkinson-Pollicott here gives a sequence {s_n} converging to {HD(E_2)} at super-exponential speed, i.e., {|s_n-HD(E_2)|\leq C \theta^{n^2}} for some constants {C>0} and {0<\theta<1}, where {E_2} is the Cantor set of real numbers whose continued fraction expansions contain only {1} and {2}.

In particular, the thermodynamical methods give good heuristics for the first several digits of the Hausdorff dimension of dynamical Cantor sets (e.g., if we list {s_n} for {1\leq n\leq 10} and the first three digits of {s_n} coincide for all {5\leq n\leq 10}, then it is likely that one has found the first three digits of {HD(K)}).

The heuristic bounds provided by the thermodynamical methods can be turned into rigorous estimates: indeed, one of the goals of the recent work of Jenkinson-Pollicott consists into rigorously computing the first 100 digits of {HD(E_2)}.

However, the conversion of heuristic bounds into rigorous estimates is not always easy, and, for this reason, sometimes a slowly converging method producing two sequences {\alpha_n} and {\beta_n} of rigorous bounds (i.e., {\alpha_n<HD(K)<\beta_n} for all {n\in\mathbb{N}}) might be interesting for practical purposes.

In this post, we explain a method described in pages 68 to 70 of Palis-Takens book giving explicit sequences {\alpha_n\leq HD(K)\leq \beta_n} converging slowly (e.g., {\beta_n-\alpha_n\leq C/n} for some constant {C>0} and all {n\in\mathbb{N}}) towards {HD(K)}, and, for the sake of comparison, we apply it to exhibit crude bounds on the Hausdorff dimensions of some Cantor sets defined in terms of continued fraction expansions.

Read More…

I have just upload to the arXiv the memoir of my `Habilitation à Diriger des Recherches’ (HDR) dossier.

This memoir is a preliminary step towards obtaining a HDR diploma (a general requirement in the French academic system to supervise PhD students, etc.) and it summarizes some of my researches after my PhD thesis (or, more specifically, my researches on the dynamics of the Teichmüller flow).

Despite the title and abstract in French, the main part of this memoir is in English.

The first chapter of the memoir recalls many basic facts on the SL(2,\mathbb{R}) action on the moduli spaces of translation surfaces. In particular, this chapter is a general introduction to all subsequent chapters of the memoir. Here, the exposition is inspired from Zorich’s survey, Yoccoz’s survey and our survey with Forni.

After reading the first chapter, the reader is free to decide the order in which the remainder of the memoir will be read: indeed, the subsequent chapters are independent from each other.

The second chapter of the memoir is dedicated to the main result in my paper with Avila and Yoccoz on the Eskin-Kontsevich-Zorich regularity conjecture. Of course, the content of this chapter borrows a lot from my four blog posts on this subject.

The third chapter of the memoir discusses my paper with Schmithüsen on complementary series (and small spectral gap) for explicit families of arithmetic Teichmüller curves (i.e., SL(2,\mathbb{R})-orbits of square-tiled surfaces).

The fourth chapter of the memoir is dedicated to my paper with Wright on the applications of the notion of Hodge-Teichmüller planes to the question of classification of algebraically primitive Teichmüller curves. Evidently, some portions of this chapter are inspired by my blog posts on this topic.

The fifth chapter of the memoir is consacrated to Lyapunov exponents of the Kontsevich-Zorich cocycle over arithmetic Teichmüller curves. Indeed, after explaining the results in my paper with Eskin about the applications of Furstenberg boundaries to the simplicity of Lyapunov exponents, we spend a large portion of the chapter discussing the Galois-theoretical criterion for simplicity of Lyapunov exponents developed in my paper with Möller and Yoccoz. Finally, we conclude this chapter with an application of these results (obtained together with Delecroix in this paper here) to a counter-example to a conjecture of Forni.

The last chapter of the memoir is based on my paper with Filip and Forni on the construction of examples of `exotic’ Kontsevich-Zorich monodromy groups and it is essentially a slightly modified version of this blog post here.

Closing this short post, let me notice that the current version of this memoir still has no acknowledgements (except for a dedicatory to Jean-Christophe Yoccoz’s memory) because I plan to add them only after I get the referee reports. Logically, once I get the feedback from the referees, I’m surely going to include acknowledgements to my friends/coauthors who made this memoir possible! 😀

Posted by: matheuscmss | November 28, 2016

“Mesures stationnaires absolument continues”

About 3+1/2 weeks ago, Jean-François Quint gave a very nice talk (with same title as this post) during Paris 6 and 7 “Journées de dynamique” about his joint work with Yves Benoist on the regularity properties of stationary measures.

In what follows, I’m reproducing my notes for Jean-François Quint’s lecture. (As usual, all errors/mistakes in the sequel are my responsibility.)

1. Introduction

1.1. Limit sets of semigroups of matrices

Let {\Gamma\subset GL_d(\mathbb{R})} be a semigroup of invertible {d\times d} real matrices.

Recall that:

  • {\Gamma} is irreducible if there are no non-trivial {\Gamma}-invariant subspaces, i.e., {V\subset\mathbb{R}^d} and {\Gamma(V)=V} imply {V=\{0\}} or {\mathbb{R}^d};
  • {\Gamma} is proximal if it contains a proximal element {g\in\Gamma}, i.e., {g} has an unique eigenvalue with maximal modulus which has multiplicity one in the characteristic polynomial of {g}; equivalently, {\mathbb{R}^d = \mathbb{R} x_g^+ \oplus V_g^{<}}, {g(x_g^+)=\lambda x_g^+}, {g(V_g^{<})=V_g^{<}} and {g|_{V_g^{<}}} has spectral radius {<|\lambda|} or, in other terms, the action of {g} on the projective space {\mathbb{P}^{d-1}} has an attracting fixed point.

Proposition 1 Let {\Gamma\subset GL_d(\mathbb{R})} be a irreducible and proximal semigroup. Then, the action of {\Gamma} on {\mathbb{P}^{d-1}} admits a smallest non-empty invariant closed subset {\Lambda_{\Gamma}} called the limit set of {\Gamma}.

Proof: Let {\Lambda_{\Gamma}:=\overline{\{\mathbb{R}x_g^+: g\in\Gamma \textrm{ proximal}\}}}. It is clear that {\Lambda_{\Gamma}} is non-empty, closed and invariant. Moreover, {\Lambda_{\Gamma}} is the smallest subset with these properties thanks to the following argument. Let {g\in\Gamma} be a proximal element. If {x\notin\mathbb{P}(V_g^{<})}, then {g^n(x)} converges to {\mathbb{R}x_g^+} as {n\rightarrow\infty}. If {x\in\mathbb{P}(V_g^{<})}, we use the irreducibility of {\Gamma} to find an element {\gamma\in\Gamma} such that {\gamma(x)\notin\mathbb{P}(V_g^{<})} and, a fortiori, {g^n(\gamma(x))} converges to {\mathbb{R}x_g^+} as {n\rightarrow\infty}. \Box

1.2. Stationary measures

Suppose that {\mu} is a probability measure on a semigroup {G} acting on a space {X}. We say that a probability measure {\nu} on {X} is {\mu}stationary if it is {G}-invariant on average, i.e.,

\displaystyle \mu\ast\nu:=\int_G g_{\ast}(\nu) d\mu(g)

is equal to {\nu}.

In the case of irreducible and proximal semigroups of matrices, the following theorem of Furstenberg and Kesten ensures the existence and uniqueness of stationary measures for the corresponding projective actions:

Theorem 2 (Furstenberg-Kesten) Let {\mu} be a Borel probability measure on {GL_d(\mathbb{R})} and denote by {\Gamma_{\mu}} the subsemigroup generated by the elements in the support {\textrm{supp}(\mu)} of {\mu}. Suppose that {\Gamma_{\mu}} is irreducible and proximal. Then, {\mu} has an unique {\mu}-stationary measure on {\mathbb{P}^{d-1}} and {\nu(\Lambda_{\Gamma_{\mu}})=1}.

In what follows, we shall also assume that {\Gamma_{\mu}} is strongly irreducible, i.e., {\nu(\mathbb{P}V)=0} for all non-trivial proper subspaces {V\subset \mathbb{R}^d}, and we will be interested in the nature of {\nu} in Furstenberg-Kesten theorem.

It is possible to show that if {\mu} is absolutely continuous with respect to the Lebesgue (Haar) measure (on {GL_d(\mathbb{R})}), then {\nu} is absolutely continuous with respect to the Lebesgue measure (on {\mathbb{P}^{d-1}}).

For this reason, we shall focus in the sequel on the following question:

Can {\nu} be absolutely continuous when {\mu} is finitely supported?

It was shown by Kaimanovich and Le Prince that the answer to this question is not always positive:

Theorem 3 (Kaimanovich-Le Prince) There exists {S\subset SL_2(\mathbb{R})} finite (actually, {\# S=2}) such that {S} spans a Zariski dense subsemigroup of {SL_2(\mathbb{R})}, but {S} is the support of a probability measure {\mu} such that the associated stationary measure {\nu} on {\mathbb{P}^1} is singular with respect to the Lebesgue measure.

On the other hand, Bárány-Pollicott-Simon and Bourgain showed that the answer to this question is sometimes positive:

Theorem 4 (Bárány-Pollicott-Simon, Bourgain) There exists {S\subset SL_2(\mathbb{R})} finite supporting a probability measure {\mu} such that the corresponding stationary measure {\nu} is absolutely continuous with respect to Lebesgue.

Remark 1 As it was pointed out by Quint, the examples produced by Bourgain are explicit, but it would be desirable to get simpler explicit examples of sets {S\subset SL_2(\mathbb{R})} satisfying the previous theorem. In this direction, he asked the following question. Denote by {R_{\theta} = \left(\begin{array}{cc} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array}\right)}, {0<\theta<\pi/2} and {g_t = \left(\begin{array}{cc} e^t & 0 \\ 0 & e^{-t} \end{array}\right)}, {t\in\mathbb{R}}, and consider the probability measures

\displaystyle \mu_{t,\theta}=\frac{1}{2}\left(\delta_{g_t} + \delta_{R_{\theta} g_t R_{\theta}^{-1}}\right)

Is it true that, for each fixed {\theta}, if {t} is small enough (and typical?), then the stationary measure {\nu_{t,\theta}} associated to {\mu_{t,\theta}} is absolutely continuous with respect to the Lebesgue measure? (Note that if {t} is very large, then we are in the regime described by Kaimanovich-Le Prince theorem 3.)

1.3. Statement of the main result

In a recent paper, Benoist and Quint extended Theorem 4 to higher dimensions:

Theorem 5 (Benoist-Quint) For any {d\geq 3}, there exists {S\subset GL_d(\mathbb{R})} finite and a probability measure {\mu} with {\textrm{supp}(\mu)=S} and {\Gamma_{\mu}=\Gamma_S} proximal and strongly irreducible such that the corresponding stationary measure {\nu} on {\mathbb{P}^{d-1}} is absolutely continuous with respect to Lebesgue.

The remainder of this post is dedicated to the proof of this result.

Read More…

Gabriela Weitze-Schmithüsen is currently visiting me in Paris and I took the opportunity to revisit some of my favorite questions about square-tiled surfaces / origamis.

Last week, we spent a couple of days revisiting the content of my blog post on Sarnak’s question about thin KZ monodromies and we realized that the origami {\mathcal{O}_1} of genus 3 discussed in this post turns out to exhibit arithmetic KZ monodromy! (In particular, this answers my Mathoverflow question here.)

In this very short post, we show the arithmeticity of the KZ monodromy of {\mathcal{O}_1}.

1. Description of the KZ monodromy of {\mathcal{O}_1}

The KZ monodromy {\Gamma_{\mathcal{O}_1}} of {\mathcal{O}_1} is the subgroup of {\mathrm{Sp}(4,\mathbb{Z})} generated by the matrices

\displaystyle A=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -1 & 1 \\ 0 & 0 & -1 & 0 \end{array}\right) \quad \textrm{and} \quad B=\left(\begin{array}{cccc} -1 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & -1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)

of order three: see Remark 9 in this post here.

For the sake of exposition, we are going to permute the second and fourth vectors of the canonical basis using the permutation matrix

\displaystyle P=\left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right),

so that the KZ monodromy {\Gamma_{\mathcal{O}_1}} is the subgroup {P\cdot\langle A, B\rangle\cdot P}.

Remark 1 This change of basis is purely cosmetical: it makes that the symplectic form preserved by these matrices in {P\cdot\langle A, B\rangle\cdot P} is

\displaystyle \Upsilon=\left(\begin{array}{cccc} 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right)

Denote by {U(\mathbb{Z})} the subgroup of unipotent upper triangular matrices in {\mathrm{Sp}(4,\mathbb{Z})}.

2. Arithmeticity of the KZ monodromy of {\mathcal{O}_1}

A result of Tits says that a Zariski-dense subgroup {\Gamma\subset \mathrm{Sp}(4,\mathbb{Z})} such that {\Gamma\cap U(\mathbb{Z})} has finite-index in {U(\mathbb{Z})} must be arithmetic (i.e., {\Gamma} has finite-index in {\mathrm{Sp}(4,\mathbb{Z})}).

Since we already know that {\Gamma_{\mathcal{O}_1}} is Zariski-dense (cf. Proposition 3 in this post here), it suffices to check that {\Gamma_{\mathcal{O}_1}\cap U(\mathbb{Z})} has finite-index in {U(\mathbb{Z})}.

For this sake, we follow the strategy in Section 2 of this paper of Singh and Venkataramana, namely, we study matrices in {\Gamma_{\mathcal{O}_1}} fixing the first basis vector {e_1} and, a fortiori, stabilizing the flag {\mathbb{Q} e_1\subset e_1^{\perp}:=\{v\in\mathbb{Q}^4:\Upsilon(v,e_1)=0\}\subset \mathbb{Q}^4}.

After asking Sage to compute a few elements of {\Gamma_{\mathcal{O}_1}} (conjugates under {P} of words on {A}, {B}, {A^2} and {B^2} of size {\leq 10}) fixing the basis vector {e_1}, we found the following interesting matrices:

\displaystyle x:=P\cdot (A^2B)^2 (AB^2)^2 \cdot P = \left(\begin{array}{cccc} 1 & 0 & 3 & -3 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right),

\displaystyle y:=P\cdot A B A^2 B A (AB^2)^2\cdot P = \left(\begin{array}{cccc} 1 & 3 & 0 & 3 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 1 \end{array}\right),


\displaystyle z:=P\cdot A^2 B A^2 (B^2 A)^2 B\cdot P = \left(\begin{array}{cccc} 1 & 0 & 3 & 0 \\ 0 & 1 & -3 & 3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right).

In order to check that {\Gamma_{\mathcal{O}_1}\cap U(\mathbb{Z})} has finite index in {U(\mathbb{Z})}, we observe that

\displaystyle \alpha = [y,x] = yxy^{-1}x^{-1} = \left(\begin{array}{cccc} 1 & 0 & 0 & 18 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right),

\displaystyle \beta = x^6[y,x] = \left(\begin{array}{cccc} 1 & 0 & 18 & 0 \\ 0 & 1 & 0 & 18 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)

\displaystyle \gamma = y^6[y,x]^{-1} = \left(\begin{array}{cccc} 1 & 18 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -18 \\ 0 & 0 & 0 & 1 \end{array}\right)

\displaystyle \delta = z^6 \beta^{-1} = z^6 (x^6 [y,x])^{-1} = \left(\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & -18 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)

are elements in {\Gamma_{\mathcal{O}_1}\cap U(\mathbb{Z})} generating the positive root groups of {\textrm{Sp}(4,\mathbb{R})}. In particular, {\Gamma_{\mathcal{O}_1}\cap U(\mathbb{Z})} has finite-index in {U(\mathbb{Z})}, so that the argument is complete.

Remark 2 It is worth noticing that all non-arithmetic Veech surfaces in genus two provide examples of thin KZ monodromy, but this is not the case for origamis (arithmetic Veech surfaces) of genus 2 in the stratum {\mathcal{H}(2)} with tiled by {\leq 6} squares (as well as for the origami {\mathcal{O}_1} of genus three mentioned above). In particular, this indicates that Sarnak’s question about existence and/or abundance of thin KZ monodromies among origamis might have an interesting answer…

Finite groups generated by two elements are a rich source of examples of origamis (square-tiled surfaces). Indeed, given a finite group {G} generated by {h} and {v}, we take a collection of unit squares {Sq(g)\subset\mathbb{R}^2} indexed by the elements {g\in G} and we glue by translations the rightmost vertical, resp. topmost horizontal, side of {Sq(g)} with the leftmost vertical, resp. bottommost horizontal, side of {Sq(gh)}, resp. {Sq(gv)}, to obtain an origami {M(G,h,v)} naturally associated to the data of {(G, h, v)}.

Such origamis were baptized regular origamis by David Zmiaikou in his PhD thesis and the study of the so-called Kontsevich-Zorich over regular (and quasi-regular) origamis was pursued in our joint article with Jean-Christophe Yoccoz.

In this short post, I would like discuss some informal questions posed by Jean-Christophe right after the completion of our joint paper with David Zmiaikou (in order to keep a trace of them in case someone [e.g., myself] wants to think about this matter in the future).

1. Regular origamis and commutators

The commutator {c=[h,v]=hvh^{-1}v^{-1}} determines the nature of the conical singularities of the origami {M(G,h,v)}: in fact, {M(G, h, v)} has exactly {\#G/\textrm{order}(c)} such singularities and the total angle around each of them singularities is {2\pi \cdot \textrm{order}(c)}.

Also, Yoccoz, Zmiaikou and I showed that the “potential blocks” of the Kontsevich-Zorich cocycle over the {SL(2,\mathbb{R})}-orbit of {M(G,h,v)} are completely determined by the commutator {c=[h,v]} (cf. Theorem 1.1 of our paper).

Given this scenario, Jean-Christophe asked whether the Lyapunov exponents of the Kontsevich-Zorich cocycle of {M(G,h,v)} were also completely determined from the knowledge of {c=[h,v]}.

2. Lyapunov exponents and commutators

By the time we finished our joint paper, David Zmiaikou pointed out to me that the commutator is not a complete invariant for many algebraic question about {(G,h,v)}: for example, Daniel Stork proved (among other things) that the pairs of permutations {\sigma:=(136), \tau:=(12345)} and {\sigma'=(13)(26), \tau'=(12645)} have the same commutator {c=(13642)=[\sigma,\tau]=[\sigma',\tau']} but they generate distinct T-systems of the alternate group {A_6} (cf. Section 2.6 of David Zmiaikou’s PhD thesis for more comments).

This remark led me to play with the origamis {\mathcal{O}:=M(A_6,\sigma,\tau)} and {\mathcal{O}':=M(A_6,\sigma',\tau')} in an attempt to answer Jean-Christophe’s question.

First, note that both of them have {\#A_6/\textrm{order}(c) = 72} conical singularities and the total angle around each of them is {2\pi\cdot \textrm{order}(c)=10\pi}. In particular, both {\mathcal{O}} and {\mathcal{O}'} have genus {145=(\frac{72\times 4}{2}+1)}.

On the other hand, a short computation with the aid of Sage (or a long calculation by hand) reveals that the {SL(2,\mathbb{Z})}-orbit of {\mathcal{O}} has cardinality {15} and the {SL(2,\mathbb{Z})}-orbit of {\mathcal{O}'} has cardinality {10} (this is closely related to the comments in Section 5 of Stork’s paper).

Remark 1 Recall that it is easy to algorithmically compute {SL(2,\mathbb{Z})}-orbits of origamis described by two permutations {h} and {v} of a finite collection of squares because {SL(2,\mathbb{Z})} is generated by {A=\left(\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}\right)} and {B=\left(\begin{array}{cc} 1 & 0 \\ 1 & 1 \end{array}\right)}, and these matrices act on pairs of permutations by {A(h,v)=(h,vh^{-1})} and {B(h,v)=(hv^{-1},v)} (and the permutations {(h,v)} and {(shs^{-1},svs^{-1})} generate the same origami).

Moreover, this calculation also reveals that

  • the {SL(2,\mathbb{R})}-orbit of {\mathcal{O}} decomposes into four {A}-orbits:
    • two {A}-orbits have size {3} and all origamis in these orbits decompose into {120} horizontal cylinders of width {3} and height {1};
    • one {A}-orbit has size {4} and all origamis in this orbit decompose into {90} horizontal cylinders of width {4} and height {1};
    • one {A}-orbit has size {5} and all origamis in this orbit decompose into {72} horizontal cylinders of width {5} and height {1}.
  • the {SL(2,\mathbb{R})}-orbit of {\mathcal{O}'} decomposes into three {A}-orbits:
    • one {A}-orbit contains a single origami decomposing into {180} horizontal cylinders of width {2} and height {1};
    • one {A}-orbit has size {4} and all origamis in this orbit decompose into {90} horizontal cylinders of width {4} and height {1};
    • one {A}-orbit has size {5} and all origamis in this orbit decompose into {72} horizontal cylinders of width {5} and height {1}.

This information can be plugged into the Eskin-Kontsevich-Zorich formula to determine the sums {L(\mathcal{O})} and {L(\mathcal{O}')} of the non-negative Lyapunov exponents of the Kontsevich-Zorich cocycle over the {SL(2,\mathbb{R})}-orbits of {\mathcal{O}} and {\mathcal{O}'}.

Indeed, if {M} is an origami with {\kappa} conical singularities whose total angles around them are {2\pi (k_i+1)}, {i=1,\dots,\kappa}, then Alex Eskin, Maxim Kontsevich and Anton Zorich showed that the sum {L(M)} of all non-negative Lyapunov exponents of the Kontsevich-Zorich cocycle over {SL(2,\mathbb{R})\cdot M} is

\displaystyle L(M) = \frac{1}{12}\sum\limits_{i=1}^{\kappa}\frac{k_i(k_i+2)}{k_i+1} + \frac{1}{\# SL(2,\mathbb{Z})\cdot M} \sum\limits_{M_i\in SL(2,\mathbb{Z}) M}\sum\limits_{M_i=\cup \textrm{cyl}_{ij}} \frac{h_{ij}}{w_{ij}}

where {M_i=\cup\textrm{cyl}_{ij}} is the decomposition of {M_i} into horizontal cylinders and {h_{ij}}, resp. {w_{ij}} is the height, resp. width, of the horizontal cylinder {\textrm{cyl}_{ij}}.

In our setting, this formula gives

\displaystyle L(\mathcal{O}) = \frac{1}{12}\frac{72\cdot 4\cdot 6}{5} + \frac{1}{15}\left(2\cdot 3\cdot 120\cdot\frac{1}{3} + 1\cdot 4\cdot 90\cdot \frac{1}{4} + 1\cdot 5\cdot 72\cdot \frac{1}{5}\right)


\displaystyle L(\mathcal{O}') = \frac{1}{12}\frac{72\cdot 4\cdot 6}{5} + \frac{1}{10}\left(1\cdot 1\cdot 180\cdot\frac{1}{2} + 1\cdot 4\cdot 90\cdot \frac{1}{4} + 1\cdot 5\cdot 72\cdot \frac{1}{5}\right)

that is,

\displaystyle L(\mathcal{O}) = \frac{278}{5} \neq 54 = L(\mathcal{O}')

In particular, this allowed me to answer Jean-Christophe’s informal question: the knowledge of the commutator {c=[\sigma,\tau]=[\sigma',\tau']} is not sufficient to determine the Lyapunov exponents.

3. Lyapunov exponents and T-systems?

Logically, this discussion led Jean-Christophe to ask a new question: how does the Lyapunov exponents of {M(G,h,v)} relate to algebraic invariants of {(G,h,v)}? For example, is the `Lyapunov exponent invariant’ equivalent to `T-systems invariant’ (or is the `Lyapunov exponent invariant’ a completely new invariant? [and, if so, can it be used to derive new interesting algebraic consequences for {(G,h,v)}?])

Remark 2 André Kappes and Martin Möller used Lyapunov exponents in another (but not completely unrelated) setting as `new invariants’ to solve an algebraic problem (namely, classify commensurability classes of non-arithmetic lattices constructed by Pierre Deligne and George Mostow).

(Jean-Christophe and) I plan(ned) to come back to this question at some point in the future, but for now I will close this post here since I have nothing to say about this matter.

Posted by: matheuscmss | October 6, 2016

Jota-Cê …

Last September 3, Jean-Christophe Yoccoz passed away after a long battle against illness. He was an incredible man whose huge influence on this blog and, more generally, on my mathematical life, is certainly evident to some readers.

Nevertheless, as my close friends know, Jean-Christophe’s influence on me goes way beyond Mathematics, and, after spending the last month trying to cope with this terrible loss, I believe that today I’m now ready from the emotional point of view to talk a little bit about the extraordinary role model he provided to me.

Since there is no doubt that Jean-Christophe’s mathematical work will be always a recurring theme in this blog, for now I prefer to postpone all purely mathematical discussions/hommages. Instead, this post contains some `souvenirs’ of my 9 years of friendship with Jean-Christophe as a way to pay a modest tribute for all fantastic and unforgettable moments (both inside and outside Mathematics) shared with him.

1. Prologue

I saw Jean-Christophe for the first time during the International Conference on Dynamical Systems held at IMPA in July 2000. Indeed, Jean-Christophe entered the auditorium Ricardo Mañé accompanied by his son (coincidently born in 1984 like me) and wife to deliver a talk about his joint work (of 217 pages) with Jacob Palis (published in 2009).

I remember that I got lost after the first 5 minutes of Jean-Christophe’s lecture, mainly because at that time my mathematical background (of a first year PhD student) was not appropriate.

Remark 1 Retrospectively, I find it funny that, even though I could not understand this talk at the time, I would join Jean-Christophe and Jacob in 2009 in the investigation of some open questions left open by their paper.Indeed, I started working with them on this subject by accident: I was present at Collège de France when the offprints of this paper of Jean-Christophe and Jacob arrived by regular mail, and we started right away to “celebrate” it by drinking a coffee while informally discussing the problems left open in this work; at some point, our informal conversation became serious because we noticed an argument allowing to improve the information on the stable sets of the so-called non-uniformly hyperbolic horseshoes.

After this initial success, Jean-Christophe laughed out while saying (as a joke) that we should ask the journal to stop printing the article so that we could add to it an appendix with our brand new result, and, in a more serious tone, he proposed to use this “low-hanging fruit” as the beginning of a separate article (still in prepartion …) on this topic.

2. My first conversation with Jean-Christophe

After finishing my PhD in 2004, I started looking for post-doctoral positions. In 2006, Jean-Christophe was visiting IMPA (and also attending ICMP 2006), and my friend Gugu (an IMO gold medalist who finished his PhD at the age of 20 years-old) strongly encouraged me to ask Jean-Christophe whether he could eventually supervise my post-doctoral `sejour’ in France.

Despite Gugu’s advice, I was hesitating to approach Jean-Christophe. In fact, even though a transition from Brazil to France would be relatively smooth under Jean-Christophe’s supervision (partly because his wife was Brazilian and, thus, he spoke Portuguese fluently), I was not sure that I could handle the pressure of becoming the third Brazilian post-doctoral student of a brilliant French mathematician whose expectations about Brazilian post-doctoral students were very high: indeed, his previous experience with Brazilians was to supervise the post-doctoral `sejours’ of Gugu himself and Artur Avila.

After many conversations with Gugu and my wife, I finally got sufficiently motivated to knock the door of Jean-Christophe’s temporary office at IMPA, introduce myself and ask him to be my post-doctoral supersivor: after all, at worst I would simply get a `no’ as an answer.

So, I knocked the door of his office and he called me in (by saying `Oui, entrez’). I told him that I was looking for a post-doctoral advisor, I asked if he could advise me, and, when I was about to give him more details about the kind of Mathematics I have done so far, he gave me a big smile and he said: “There is no need for formal presentations: Gugu talked to me about you and it would be an honor to me to be your post-doctoral advisor.”

Such a kind and humble response was certainly not what I was expecting. In particular, I was still under the `shock’ when I told him that I was very happy to hear his answer. Then, I quickly left the office, and my wife (who was waiting for me) saw my puzzled face and asked me: “So, what he said? Do you think the conversation went well?”. Here, I simply replied: “I’m not sure, but I think the conversation went well: if I heard it correctly, he told that it would be an honor to him to be my advisor.”

Of course, these two traits (kindness and humbleness) of Jean-Christophe’s personality are well-known among those who met him: in particular, Gugu was not surprised by his answer to me (and this is probably why Gugu insisted that I should talk to Jean-Christophe in the first place!).

3. My post-doctoral `sejour’ in France

Several bureaucratic details made that I started my post-doctoral `sejour’ in France in September 17, 2007, i.e., almost one year after my first conversation with Jean-Christophe.

Since the first day of my post-doctoral `sejour’, Jean-Christophe was always very kind to me. For example, he offered me an office next to his own office, so that we could drink coffee, have lunch together and talk (in Portuguese) about many topics (including Mathematics) regularly.

Also, he would never hesitate to stop his research activities to help me with daily problems (e.g., openning a bank account to get my first salary): indeed, his constant support made that my first two years in France were quite smooth and, of course, this gave me the time needed to learn French.

Remark 2 Here, he gave the following precious advice: I would learn French more easily by seeing the news on TV because it is easier to absorb the information when just one person is talking calmly at a reasonably constant pace (indeed, this is what he did to learn Portuguese). Moreover, he put me in close contact with the secretaries, so that I would be forced to practice my French (because it was not possible to shift to Portuguese with them).

On the other hand, despite all his attention towards me, he left me completely free during the first year of my post-doctoral `sejour’. In fact, as he told me later, he thought that the post-doctoral `sejour’ was an important moment to develop our own ideas about Mathematics and, hence, it was not a good idea to impose to me any specific problem / research topic. In particular, he would limit himself to ask what I was doing recently and periodically invite me to give a talk in the Eliasson-Yoccoz seminar (so that the community would know what I was working on).

Consequently, we would start working in our first joint paper only in 2008…

4. Our first collaboration

Upon my return to Paris from a conference in Trieste (in August 2008), I talked to Jean-Christophe about the recently discovered example by Giovanni Forni and myself — later baptized Ornithorynque by Vincent Delecroix and Barak Weiss — of a translation surface with peculiar properties.

He got interested by the subject, and I started to explain to him the main features of the example: “the symmetries of this example are very particular because the Hodge structure on its cohomology …” After letting me end the description of the example, Jean-Christophe replied that, even though my explanation was mathematically correct, he was not happy because it does not allow to `put your hands in the example’. In fact, his phrase (who made a profound impact on me) was: “I don’t like to work on extremely abstract theories with highly sophisticated arguments. I prefer to understand things from a concrete point of view, by working with many concrete examples before reaching the final result. In particular, my `tactics’ is to cover the ground slowly via basic examples before exploring general theories.”

So, he thought that it could be a good idea to work together on a paper giving a explanation for the example in such a way that it could reveal more examples (ideally infinite families) with similar properties.

Frankly speaking, I also found that the original (Hodge-theoretical) arguments obtained with Giovanni had the drawback that they did not allow one to `touch’ the example, and, for this reason, I accepted Jean-Christophe’s offer to investigate more closely the Ornithorynque.

The first meetings related to our work in the Ornithorynque were quite curious: while he would never refuse to meet anytime in the morning (9h, 8h, 7h30, …), we usually would stop our discussions by 3:30 PM or 4 PM because, as he liked to say, by this time “he had runned out of energy”.

At some point, I asked him what time he used to arrive at his office (so that I could try to maximize the span of our conversations). He simply smiled and said: “Normally I wake up around 4:30 or 5 AM, I take the first train and I arrive here around 6 AM.” (Of course, this explains why he could meet me in the morning at any time.) In fact, thanks to his metabolism, Jean-Christophe just needed to sleep about 6 hours per night.

After struggling a bit with my own metabolism, I managed to adapt myself to Jean-Christophe’s rythm and this lead us to a small `competition’ to know how would arrive first at the office. Normally, I would arrive by 6:15 AM at Collège de France and, evidently, Jean-Christophe would be waiting for me with a bottle of fresh coffee (that he had prepared a few minutes ago at the kitchen of Collège), so that we would drink coffee together and talk about the latest news (on a variety of subjects: politics, chess, soccer, etc.) before starting our mathematical conversations. (In general, we would talk from 7 AM to 11:30 AM, take a break to have lunch together, and then come back to work until 3:30 PM or 4 PM.) However, it happened a couple of times that I managed somehow to arrive first in Collège: in every such occasion, Jean-Christophe would spend a couple of minutes explaining why he was `late’. Indeed, this was the natural attitude to him because Jean-Christophe was someone who liked to do his best in everything regardless it was a `small thing’ (e.g., arriving before me at Collège) or a `big thing’ (e.g., proving theorems).

Our work on the `Ornithorynque project’ was going well: the concrete approach of Jean-Christophe (computing tons of particular cases [including multiplying many 4×4 matrices by hand] at an extremely fast pace [which always forced me to be extremely concentrated to be able to follow him up…] for several hours in a row, always keeping an eye for `symmetries’ to reduce the sizes of calculations, etc.) introduced me to a whole new way of doing Mathematics. In particular, there is no doubt that my own vision of Mathematics completely changed by seeing so closely how the mind of a brilliant mathematician works. Moreover, despite the enormous differences in our mathematical skills, Jean-Christophe was always open to hear my ideas and suggestions (and I will always be grateful to him for such an humble attitude towards me).

The rapid and steady progress in `dissecting the Ornithorynque’ made that I did not want to completely stop working on it during the summer vacations in August 2009. So, I had no doubts that I should accept Jean-Christophe’s invitation to spend 10 days in his vacations house in Loctudy (in French Brittany). Since the vacations are sacred in France (it is usually very hard to contact French friends in August …), Jean-Christophe said that the amount of mathematical work in these 10 days would be decided by `chance’, or, more accurately, by the weather. More precisely, we did the following agreement: we would start our day around 5h30 and work until 8h30; if the weather was nice (i.e., not raining too much) outside, we would stop talking about Mathematics and we would go for a walking in the forest or a boat trip in nearby islands, etc.; otherwise, we stay at home working on the Ornithorynque.

In principle, this agreement meant that we would work about {50\%} of the time because the weather in French Brittany is very unstable: one might see several `seasons’ within a given day …. However, as Jean-Christophe pointed out later, I was extremely lucky that it rained only in three days, so that we had plenty of leisure: we played pétanque, visited Île-de-Sein, etc.

The photos below illustrate such moments (during my second visit [in 2011] to Loctudy): the first picture displays Jean-Christophe plotting a strategy to improve the performance of our participation in a pétanque tournement in Loctudy (but my bad skills in this game made that we were kicked out of the tournement after three matches) and the next two photos show him in Île-de-Sein.

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Remark 3 In Loctudy’s pétanque club, almost everyone knew that the well-known joke that `you can give mathematicians a hard time by asking them to split a bill in a restaurant’ did not apply to Jean-Christophe. In fact, the `standard procedure’ in the end of the tournement was to go to the bar, ask for a drink and wait for Jean-Christophe to compute (within 2 or 3 minutes) the rankings for the 42 or so participants, sizes of prizes, etc. Also, I overheard a conversation between a newcomer asking an organizer of the tournement: the newcomer was puzzled why everybody went take a drink instead to doing laborious calculations for setting rankings of participants, and the organizer smiled and said: “There is no need to worry. That guy over there will take care of all calculations in 2 minutes, and he knows what he is doing: indeed, he is a sort of `Nobel prize’ in Mathematics…”

5. Some anecdotes of Jean-Christophe

Closing this post, I would like to share some anecdotes about Jean-Christophe: in fact, instead of describing more details of our subsequent collaborations, I believe that the reader might get a better idea of Jean-Christophe’s personality via a few anecdotes about other aspects of his life.

Jean-Christophe participated twice in Samba school parades in Rio de Janeiro, and he always laughed at the fact he had all kinds of experiences in these parades: in his first participation, his Samba school won the parade, but in his second participation, his Samba school got the last place…

Jean-Christophe liked to laugh at the fact that if Internet existed before, then the failed attempts of many countries to name the bird `Turkey’ after its country of origin could have been avoided: indeed, the bird is called `Turkey‘ in English, ‘Dinde‘ (a variant of `D’Inde’, i.e., `from India’) in French, `Peru‘ in Portuguese, but the bird is originally Mexican.

Jean-Christophe was a strong chess player: his ELO rating was 2200+ at some point and we followed together many chess tournements via Chessbomb Arena (with particular attention to TCEC, a tournement for chess engines) during some breaks in our mathematical conversations, and he has many interesting anecdotes involving chess.

For example, he wrote here that his PhD advisor (Michel Herman) was so afraid that Jean-Christophe was spending too much time with Chess instead of Mathematics that he called Jean-Christophe’s parents for a conversation about his worries.

Also, once I invited Jean-Christophe, Gugu and Artur Avila for a barbecue at the vacations house of my wife’s family in Vargem Grande, Rio de Janeiro. At some point, we decided to play chess and, as usual, Gugu wanted to play against Jean-Christophe. For some reason, we thought that it could be a good idea to ask Jean-Christophe to play blindfold (with Artur and I moving the pieces in his place), while Gugu would play normally. After 25 moves or so, Gugu announces his move. Jean-Christophe takes a deep breath, spend a couple of minutes thinking, and then asks: `Gugu, are you sure about your move?’. Gugu says `yes’ and Jean-Christophe announces a checkmate in a couple of moves. Gugu is puzzled by Jean-Christophe’s claim, so he looks at the board and he comes up with the genius phrase in response to his blindfold friend: `Is it already checkmate? I could not see it!’ Of course, Jean-Christophe was right and, after laughing for a couple of minutes at Gugu’s phrase, we passed to the next match (Jean-Christophe versus me…).

Jean-Christophe liked sports in general: during his youth, he played rugby and he sailed together with his friends from École Normale Supérieure, and he followed on TV and newspapers many tournements (soccer, rugby, golf, snooker, …). In particular, we saw together several finals of the UEFA Champions League while drinking his favorite beer (namely, Guinness) and he followed on the hospital’s TV the Rio 2016 Olympic Games (which was especially interesting to him because he knew `Rio’s Zona Sul’ [where rowing and cycling took place]).

Finally, Jean-Christophe was an avid reader and an erudite person: he read (in French, English and Portuguese) many books and novels per month, he saw many movies, and he liked to go to museums. For example:

  • he was happy when he learned that I read The Murders in the Rue Morgue (partly because Edgar Allan Poe was one of his favorite writers),
  • he borrowed me Crimes Cèlebres by Alexandre Dumas and Ne le dis à personne (French version of a thriller novel) by Harlan Corben, so that I could improve my French,
  • besides the classical in Brazilian literature (by Machado de Assis, José de Alencar, etc.), Jean-Christophe liked `Agosto’ by Rubem Fonseca. In fact, Jean-Christophe told me that he read the original book in Portuguese because he was not sure that the French translation was very good: indeed, the title of `Agosto’ — literally `August’ in English — was translated as `Un été brésilien‘ — `a Brazilian summer’ — reveals that the editors probably did a job of questionable quality since it is winter (and not summer) in Brazil in August …
  • he was a big fan of Alfred Hitchock (among many other film directors): in particular, I will always remember with affection a `Hitchcock session’ in Jean-Christophe’s vacations house in Loctudy.
  • after trying to read (unsuccessfully) `À la recherche du temps perdu‘ during his youth, Jean-Christophe spent the last months of his life (doing Mathematics and) re-reading all seven volumes of Proust’s novel …

6. Epilogue

Since a picture is worth more than a thousand words, I’m sharing below a couple of photos of Jean-Christophe (holding my daughter, next to my wife [during vacations in French Brittany], working with me at Mittag-Leffler institute on our joint paper with Martin Moeller, and together with Gugu, Artur, Fernando Codá Marques, Susan Schommer and myself in Artur’s appartment, respectively).

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