Posted by: matheuscmss | October 4, 2017

Jean-Christophe Yoccoz mathematical archives

Almost one year ago, Yoccoz family gave me the immense honour of taking care of Jean-Christophe’s mathematical archives.

My general plan is to follow the same steps by Jean-Christophe when he became the responsible for Michel Herman archives, namely, I will make available at this webpage here all unpublished texts after selecting and revising them together with Jean-Christophe’s friends.

So far, the webpage dedicated to Jean-Christophe’s archives contains only an original text (circa 1986), a latex version of this text (typed by Alain Albouy, Alain Chenchiner, and myself), and some lecture notes taken by Alain Chenchiner of a talk by Jean-Christophe on the central configurations for the planar four-body problem.

Nevertheless, I hope that this webpage will be regularly updated in the forthcoming years: indeed, Jean-Christophe’s archives takes all cabinets and some corners of an entire office, and, thus, there is more than enough material to keep his friends occupied for some time. 😀

Closing this extremely short post, let me take the opportunity to announce also that Gazette des Mathématiciens (published by the French Mathematical Society) plans to publish in April 2018 a special volume (edited by P. Berger, S. Crovisier, P. Le Calvez and myself) dedicated to several aspects of Jean-Christophe’s mathematical life.

 

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Posted by: matheuscmss | August 22, 2017

HD(M\L) < 0.986927

My friend Gugu and I have just uploaded to the arXiv our paper {HD(M\setminus L) < 0.986927}. This article continues our investigations of the Hausdorff dimension {HD(M\setminus L)} of the complement of the Lagrange spectrum {L} in the Markov spectrum {M}. More precisely, we showed in a previous paper (see also this blog post here) that {HD(M\setminus L) > 0} and we prove now that {HD(M\setminus L)<1}.

The key dynamical idea to give upper bounds on {HD(M\setminus L)} is to show that any sufficiently large element {m\in M\setminus L} is realized by a sequence {\underline{\theta}\in(\mathbb{N}^*)^{\mathbb{Z}}} whose past or future dynamics lies in the gaps of an appropriate horseshoe.

Qualitately speaking, this idea is explained by the following lemma.

Lemma 1 Fix {\Lambda}horseshoe of a surface diffeomorphism {\varphi} and {f} a height function. For simplicity, let us denote the orbits of {\varphi} by {x_n:=\varphi^n(x)}. Denote by

\displaystyle M=\{\sup\limits_{n\in\mathbb{Z}}f(x_n): x_0\in\Lambda\} \quad \textrm{ and } \quad L=\{\limsup\limits_{n\rightarrow\infty}f(x_n): x_0\in\Lambda\}

the corresponding Markov and Lagrange spectra.Let {\widetilde{\Lambda}} a subhorseshoe of {\Lambda} and set

\displaystyle m(\widetilde{\Lambda}) = \max\limits_{y\in\widetilde{\Lambda}} m(y) \quad (= \max\limits_{z\in\widetilde{\Lambda}} f(z) )

where {m(a):=\sup\limits_{n\in\mathbb{Z}}f(a_n)} is the Markov value of {a}. Consider {m\in M\setminus L} such that {m>m(\widetilde{\Lambda})}, and denote by {x_0\in\Lambda} a point with

\displaystyle m=\sup\limits_{n\in\mathbb{Z}} f(x_n) = f(x_0)

Then, either {\alpha(x_0)\cap \overline{\Lambda}=\emptyset} or {\omega(x_0)\cap \overline{\Lambda}=\emptyset} (where {\alpha(x)} and {\omega(x)} denote the {\alpha} and {\omega} limit sets of the orbit of {x}).

Proof: By contradiction, suppose that {z\in\alpha(x)\cap\widetilde{\Lambda}} and {w\in\omega(x)\cap\widetilde{\Lambda}}.

Since {m\in M\setminus L} and {m>m(\widetilde{\Lambda})}, we can select {\varepsilon>0} and {N\in\mathbb{N}} such that {f(x_n)<m-\varepsilon} for all {|n|\geq N}, and {m(\widetilde{\Lambda})<m-\varepsilon}. Also, the definitions allow us to take {m_k\rightarrow-\infty} and {n_k\rightarrow\infty} such that {x_{m_k}\rightarrow z} and {x_{n_k}\rightarrow w}.

Fix {y\in\widetilde{\Lambda}} with dense orbit and consider pieces {y_{r_k}\dots y_{s_k}} of the orbit of {y} with {y_{r_k}\rightarrow w} and {y_{s_k}\rightarrow z}.

Consider the pseudo-orbits {x_0\dots x_{n_k} y_{r_k}\dots y_{s_k}x_{m_k}\dots x_0}. By the shadowing lemma, we obtain a sequence of periodic orbits accumulating {x_0} whose Markov values converge to {m}. In particular, {m\in L}, a contradiction. \Box

In simple terms, this lemma says that an element {m\in M\setminus L} with {m>m(\widetilde{\Lambda})} is associated to an orbit {(x_n)_{n\in\mathbb{Z}}} whose past dynamics (described by {\alpha(x_0)}) or future dynamics (described by {\omega(x_0)}) avoids {\widetilde{\Lambda}}. Thus, there exists {k\in\mathbb{N}} such that either the piece {(x_n)_{n\leq -k}} of past orbit or the piece {(x_n)_{n\geq k}} of future orbit avoids a neighborhood of {\widetilde{\Lambda}} in {\Lambda} (i.e., one of these pieces of orbit lives in the gaps of {\Lambda\setminus\widetilde{\Lambda}}).

Remark 1 As it turns out, this qualitative lemma is not sufficient for our purposes and, for this reason, Gugu and I end up using a quantitative version of this lemma (called Lemma 3.1) in our paper.

Once we got some constraints on the dynamics of orbits generating elements of {M\setminus L}, our strategy to estimate {HD(M\setminus L)} consists in careful choices of {\widetilde{\Lambda}} and {\Lambda}.

For the sake of exposition, let us explain how our strategy yields some bounds for {HD((M\setminus L)\cap [3.06, \sqrt{12}])}.

Perron proved that any {m\in M\cap(-\infty, \sqrt{12}]} has the form {m=m(\underline{\theta})} where {\underline{\theta}\in\{1,2\}^{\mathbb{Z}}=\Lambda}.

Consider the subhorseshoe {\widetilde{\Lambda} = \{11,22\}^{\mathbb{Z}}} (of sequences formed by concatenations of two consecutive 1’s and two consecutive 2’s).

By applying (a quantitative version of) Lemma 1 (cf. Remark 1), one concludes that if {m=m(\underline{\theta})\in (M\setminus L)\cap [3.06, \sqrt{12}]}, then the past or future dynamics of {\underline{\theta}} lives in the gaps of {\widetilde{\Lambda}}.

This means that, up to replacing {\underline{\theta}=(\theta_n)_{n\in\mathbb{Z}}} by {(\theta_{-n})_{n\in\mathbb{Z}}}, for all {n\in\mathbb{N}} sufficiently large:

  • either there is an unique extension of {\dots\theta_0\dots\theta_n} giving a sequence whose Markov value in {(M\setminus L)\cap [3.06,\sqrt{12}]};
  • or there are two continuations {\dots\theta_0\dots\theta_n1\alpha_{n+2}} and {\dots\theta_0\dots\theta_n2\beta_{n+2}} of {\dots\theta_0\dots\theta_n} so that the interval {[[0;2\beta_{n+1}], [0;1\alpha_{n+1}]]} is disjoint from the Cantor set

    \displaystyle K(\{11,22\}):=\{[0;\gamma]:\gamma\in\{11, 22\}^{\mathbb{N}}\}

    associated to {\widetilde{\Lambda} = \{11,22\}^{\mathbb{Z}}}.

(Here, {[a_0;a_1,\dots] = a_0+\frac{1}{a_1+\frac{1}{\ddots}}} denotes continued fraction expansions.)

Note that this dichotomy imposes severe restrictions on the future {(\theta_n)_{n\geq 0}} of {\underline{\theta}} because there are not many ways to build sequences associated to {(M\setminus L)\cap[3.06,\sqrt{12}]}. More precisely, we claim that at each sufficiently large step {n},

  • either we get a forced continuation {\dots\theta_0\dots\theta_n\rightarrow \dots\theta_0\dots\theta_n\theta_{n+1}};
  • or our possible continuations are {\dots\theta_0\dots\theta_n112\alpha_{n+4}} and {\dots\theta_0\dots\theta_n221\beta_{n+4}}.

Indeed, suppose that we have two possible continuations {\dots\theta_0\dots\theta_n1\alpha_{n+2}} and {\dots\theta_0\dots\theta_n2\beta_{n+2}}. If {I(a_1,\dots,a_n)} denotes the interval of numbers in {[0,1]} whose continued fraction expansion starts by {[0;a_1,\dots, a_n,\dots]}, then the intervals {I(a_1a_2)}, {(a_1,a_2)\in\{1,2\}^2} appear in the following order on the real line:

\displaystyle I(21), I(22), I(11), I(12)

Thus:

  • the continuation {\dots\theta_0\dots\theta_n12\alpha_{n+3}} is not possible (otherwise {[[0;2\beta_{n+2}]], [0;12\alpha_{n+3}]]} would contain {I(11)} and, a fortiori, intersect {K(\{11,22\})};
  • the continuation {\dots\theta_0\dots\theta_n21\beta_{n+3}} is not possible (otherwise {[[0;21\beta_{n+3}]], [0;1\alpha_{n+2}]]} would contain {I(22)} and, a fortiori, intersect {K(\{11,22\})}

so that our continuations are {\dots\theta_0\dots\theta_n11\alpha_{n+3}} and {\dots\theta_0\dots\theta_n22\beta_{n+3}}. Now, we observe that the intervals {I(11a_3)} and {I(22a_3)}, {a_3\in\{1,2\}}, appear in the following order on the real line:

\displaystyle I(222), I(221), I(112), I(111)

Hence:

  • the continuation {\dots\theta_0\dots\theta_n111\alpha_{n+4}} is not possible (otherwise {[[0;22\beta_{n+3}]], [0;111\alpha_{n+4}]]} would contain {I(112)} and, a fortiori, intersect {K(\{11,22\})};
  • the continuation {\dots\theta_0\dots\theta_n222\beta_{n+3}} is not possible (otherwise {[[0;222\beta_{n+4}]], [0;11\alpha_{n+3}]]} would contain {I(221)} and, a fortiori, intersect {K(\{11,22\})}

so that our claim is proved.

This claim allows us to bound the Hausdorff dimension of

\displaystyle K:=\{[\theta_0;\theta_1,\dots]: 3.06<m(\underline{\theta})<\sqrt{12} \textrm{ as above}\}

In fact, the claim says that we refine the natural cover of {K} by the intervals {I(\theta_1,\dots, \theta_n)} by replacing it by a “forced” {I(\theta_1,\dots, \theta_n,\theta_{n+1})} or by the couple of intervals

\displaystyle I(\theta_1,\dots, \theta_n, 1, 1, 2) \quad \textrm{ and } \quad I(\theta_1,\dots, \theta_n, 2, 2, 1)

Therefore, it follows from the definition of Hausdorff dimension that

\displaystyle HD(K)\leq s_0

for any parameter {0\leq s_0\leq 1} such that

\displaystyle |I(\theta_1,\dots,\theta_n,1,1,2)|^{s_0} + |I(\theta_1,\dots,\theta_n,2,2,1)|^{s_0} \leq |I(\theta_1,\dots,\theta_n)|^{s_0} \ \ \ \ \ (1)

Because we can assume that {m=m(\underline{\theta}) = [\theta_0;\theta_1,\dots]+[0;\theta_{-1},\dots]} with {[\theta_0;\theta_1,\dots]\in K} and {[0;\theta_{-1},\dots]\in C(2) := \{[0;\gamma]: \gamma\in\{1,2\}^{\mathbb{N}}\}}, our discussion so far can be summarized by following proposition:

Proposition 2 {(M\setminus L)\cap[3.06,\sqrt{12}]} is contained in the arithmetic sum

\displaystyle C(2)+K

where {HD(K)\leq s_0} for any parameter {0\leq s_0\leq 1} satisfying (1).

Since the arithmetic sum {C(2)+K} is the projection {\pi(C(2)\times K)}, {\pi(x,y)=x+y}, of the product set {C(2)\times K}, this proposition implies the following result:

Corollary 3 {HD((M\setminus L)\cap[3.06,\sqrt{12}])\leq HD(C(2))+s_0} where {0\leq s_0\leq 1} satisfies (1).

The Hausdorff dimension of {C(2)} was computed with high accuracy by Hensley among other authors: one has {HD(C(2))<0.531291}. In particular,

\displaystyle HD((M\setminus L)\cap[3.06,\sqrt{12}])<0.531291+s_0

where {s_0} verifies (1).

Closing this post, let us show that (1) holds for {s_0=0.174813} and, consequently,

\displaystyle HD((M\setminus L)\cap[3.06,\sqrt{12}])<0.706104

For this sake, recall that

\displaystyle |I(b_1,\dots,b_n)|=\frac{1}{q_n(q_n+q_{n-1})},

where {q_j} is the denominator of {[0;b_1,\dots,b_j]}.

Hence, if we set

\displaystyle g(s) := \frac{|I(a_1,\dots,a_n,1,1,2)|^s+|I(a_1,\dots,a_n,2,2,1)|^s}{|I(a_1,\dots,a_n)|^s}

then the recurrence formula {q_{j+2}=a_{j+2}q_{j+1}+q_j} implies that

\displaystyle g(s) = \left(\frac{r+1}{(3r+5)(4r+7)}\right)^s + \left(\frac{r+1}{(3r+7)(5r+12)}\right)^s

where {r=q_{n-1}/q_n\in (0,1)}.

Because {\frac{r+1}{(3r+5)(4r+7)}\leq \frac{1}{35}} and {\frac{r+1}{(3r+7)(5r+12)}<\frac{1}{81.98}} for all {0\leq r\leq 1}, we have

\displaystyle g(s)<\left(\frac{1}{35}\right)^s + \left(\frac{1}{81.98}\right)^s

This completes the argument because {\left(\frac{1}{35}\right)^{0.174813} + \left(\frac{1}{81.98}\right)^{0.174813} < 1}.

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

Therefore,

\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}}

and

\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

and

\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

Let

\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}} \}

where

\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]

and

\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]

Here,

\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}} \}

where

\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}

\Box

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}

and

\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),

and

\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…

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