The geodesic flow on the unit cotangent bundle of the modular surface is intimately related to the continued fraction algorithm (see e.g. this article of Series).

In this context, the entries of the continued fraction expansion of an irrational number are related to *cusp excursions* of typical geodesics in the modular surface (i.e., visits to regions for 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 , one has*

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 be a flow on preserving a probability measure .

Suppose that has *exponential decay of correlations*, i.e., there are constants and such that

for all and all “smooth” real-valued observables in a Banach space containing all constant functions (e.g., is a Hölder or Sobolev space).

**Lemma 2** *Any observable with satisfies*

*Proof:* We write

By -invariance of , we get

The exponential decay of correlations (1) implies that

This proves the lemma.

**2. Effective ergodic theorem for fast mixing flows**

Suppose that is an exponentially mixing flow on (i.e., satisfies (1)).

Fix and denote .

**Theorem 3** *Given , a function such that for each , and a sequence of non-negative functions, we have for -almost every that*

*for all sufficiently large (depending on ).*

*Proof:* Given , let . Since , we get from Lemma 2 that

Therefore,

By setting , we obtain

Consider the sequence and let . From the estimate (2) with and , and and , we get

and

By Borel–Cantelli lemma, the summability of the series for and the previous inequalities imply that for -almost every

and

for all sufficiently large (depending on ).

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

for all . Hence,

for all .

It follows from this discussion that for -almost every and all sufficiently large (depending on )

whenever . Because as and for , given , the previous estimate says that for -almost every

for all sufficiently large (depending on and ). This proves the theorem.

**3. Bounds for certain cusp excursions**

Let 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 , , 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 on is exponentially mixing with respect to the Liouville (volume) measure , i.e., for each , the estimate (1) holds for the space of -Hölder functions.

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

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 for and .

**Remark 3** *The general case can be deduced from the arguments below after replacing the geometric facts about the surfaces of revolution of (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 with base point near the cusp, let be the angle between and the direction pointing straight into the cusp of the surface of revolution of . Denote by the collar in around the cusp consisting of points whose -coordinate satisfies .

**3.1. Good initial positions for deep excursions**

Given a parameter , let . The next proposition says that any vector in generates a geodesic making a –*deep* excursion into the cusp in *bounded* time.

**Proposition 4** *If , then the base point of has -coordinate for a certain time where depends only on .*

*Proof:* By Clairaut’s relation, the -coordinate along satisfies

for all (during the cusp excursion).

Thus, the value of the -coordinate along is minimized when : at this instant . Since implies that and , the proof of the proposition will be complete once we can bound by a constant . 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.

**3.2. Smooth approximations of characteristic functions**

Take a smooth non-negative bump function equal to on and supported on such that . Similarly, take a smooth non-negative bump function equal to on and supported on such that .

The non-negative function is a smooth approximation of the characteristic function of :

- is supported on ;
- there exists a constant depending only on such that
- and
- .

**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 -almost every and for all sufficiently large (depending on and ), the base point of has -coordinate for a certain time . (Here, denotes any quantity slightly larger than .)*

*Proof:* Fix , , . Let be a parameter to be chosen later and consider the function , for (where ).

The effective ergodic theorem (cf. Theorem 3) applied to the functions introduced in the previous subsection says that, for -almost every and all sufficiently large (depending on and ),

On the other hand, by construction, and for a certain constant and for all .

It follows that, for -almost every and all sufficiently large,

If , i.e., , the right-hand side of this inequality is strictly positive for all sufficiently large. Since the function is supported on , we deduce that if then, for -almost every and all sufficiently large, (where ) for some .

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

then, for -almost every and all sufficiently large, the -coordinate of is for some time (where is a constant).

This proves the desired theorem: indeed, we can take the parameter arbitrarily close to in the previous paragraph because as and .

**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 -almost every and for all sufficiently large (depending on and ), the base point of has -coordinate for all times . (Here, denotes any quantity slightly smaller than .)*

*Proof:* Let and be parameters to be chosen later, and denote .

By elementary geometrical considerations about surfaces of revolution (similar to the proof of Proposition 4), we see that if the base point of has -coordinate , then the base point of has -coordinate in for all .

Therefore, if we divide into intervals of sizes , then

Since the Liouville measure is -invariant and the surface of revolution of the profile has the property that the volume of the region is , we deduce that

for all . Because we need indices to cover the time interval , we obtain that

We want to study the set . We divide into and . Because , we just need to compute . For this sake, we observe that

and, *a fortiori*, thanks to (3). In particular,

Note that the series is summable when , i.e., In this context, Borel–Cantelli lemma implies that, for -almost every , the -coordinate is for all and all sufficiently large (depending on ). Since as , we conclude that if

then for -almost every , the -coordinate is for all and all sufficiently large (depending on ).

This ends the proof of the theorem: in fact, by letting , we can take arbitrarily close to in the previous paragraph.

**Remark 4** *By Theorems 5 and 6, a typical geodesic enters the region while avoiding the region during the time interval (for all sufficiently large).**Of course, the presence of a gap between and motivates the following question: is there an ***optimal** exponent such that a typical geodesic enters while avoiding during the time interval (for all 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 ). 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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