Posted by: matheuscmss | August 9, 2017

Cusp excursions of typical Weil-Petersson like geodesics on surfaces

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).

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