For the last installment of this series, our goal is to discuss the rates of mixing of the Weil-Petersson (WP) geodesic flow on the unit tangent bundle of the moduli space of Riemann surfaces of genus with punctures for .

However, before entering into the mathematical discussion strictly speaking, let me take the opportunity to dedicate this blog post to the memory of two Russian mathematicians who passed away earlier this month: Dmitri Anosov and Nikolai Chernov. Among their several well-known contributions in Dynamical Systems, we can quote:

- Anosov’s proof of the ergodicity of -volume preserving of a large class of hyperbolic systems (nowadays called Anosov diffeomorphisms);
- Chernov’s proof of subexponential mixing for a large class of Anosov flows;
- …

Of course, the list of contributions of Anosov and Chernov to Dynamical Systems is vast: each of them wrote more than 90 research articles and books about the features of systems with some hyperbolicity (such as geodesic flows on negatively curved manifolds and chaotic billiards) among other topics.

In particular, it is out of the scope of this post to provide detailed descriptions of the works of these two very influential dynamicists.

On the other hand, as a form of “small compensation”, let me say that the second section of this post (about rates of the WP flow on the modular surface) briefly discusses some of the ideas advanced by these two mathematicians.

Concerning the rates of mixing of the WP flow, let us recall that, by Burns-Masur-Wilkinson theorem (cf. Theorem 1 in the first post of this series), the WP flow on is *mixing* with respect to the Liouville measure whenever .

By definition of the mixing property, this means that the correlation function converges to as for any given -integrable observables and . (See, e.g., the section “ formulation” in this Wikipedia article about the mixing property.)

Given this scenario, it is natural to ask how *fast* the correlation function converges to zero. In general, the correlation function can decay to (as a function of ) in a very slow way depending on the choice of the observables (see, e.g., this blog post of Climenhaga for some concrete examples). Nevertheless, it is often the case (for mixing flows with some hyperbolicity) that the correlation function decays to with a *definite* (e.g., polynomial, exponential, etc.) speed when restricting the observables to appropriate spaces of “reasonably smooth” functions.

In other words, given a mixing flow (with some hyperbolicity), it is usually possible to choose appropriate functional (e.g., Hölder, , Sobolev, etc.) spaces and such that

- for some constants , and for all (
*polynomial decay*), - or for some constants , and for all (
*exponential decay*).

Evidently, the “precise” rate of mixing of the flow (i.e., the sharp values of the constants , and/or above) depend on the choice of the functional spaces and (as they might change if we replace observables by observables say). On the other hand, the *qualitative* speed of decay of , that is, the fact that decays polynomially or exponentially as whenever and are “reasonably smooth”, remains *unchanged* if we select and from a well-behaved scale of functional (like spaces, , or spaces, ). In particular, this partly explains why in the Dynamical Systems literature one simply says that a given mixing flow has “polynomial decay” or “exponential decay”: usually we are interested in the qualitative behavior of the correlation function for reasonably smooth observables, but the particular choice of functional spaces and is normally treated as a “technical detail”.

After this brief description of the notion of rate of mixing (speed of decay of correlation functions), we are ready to state the main result of this post.

Theorem 1 (Burns-Masur-M.-Wilkinson)The rate of mixing of the WP flow on is:

- at most polynomial when ;
- rapid (faster than any polynomial) when .

Remark 1This result was announced as Theorem 2 in the first post of this series and also in this preprint here. Since then, Burns, Masur, Wilkinson and myself found some evidence indicating that the Weil-Petersson geodesic flow on is actually exponentially mixing when . The details will hopefully appear in the forthcoming paper (currently still in preparation).

Remark 2An open problem left by Theorem 1 is to determine the rate of mixing of the WP flow on for . Indeed, while this theorem provides a polynomial upper bound for the rate of mixing in this setting, it does not rule out the possibility that the actual rate of mixing of the WP flow is sub-polynomial (even for reasonably smooth observables). Heuristically speaking, we believe that the sectional curvatures of the WP metric control the time spend by WP geodesics near the boundary of . In particular, it seems that the problem of determining the rate of mixing of the WP flow (when ) is somewhat related to the issue of finding suitable (polynomial?) bounds for how close to zero the sectional curvatures of the WP metric can be (in terms of the distance to the boundary of ). Unfortunately, the best available bounds for the sectional curvatures of the WP metric (due to Wolpert) do not rule out the possibility that some of these quantities get extremely close to zero (see Remark 4 of this post here).

The difference in the rates of mixing of the WP flow on when or in Theorem 1 reflects the following simple (yet important) feature of the WP metric near the boundary of the Deligne-Mumford compactification of .

In the case , e.g., , the moduli space equipped with the WP metric looks like the surface of revolution of the profile near the cusp at infinity (see Remark 6 of this post here). In particular, even though a -neighborhood of the cusp is “polynomially large” (with area ), the Gaussian curvature approaches only near the cusp and, as it turns out, this strong negative curvature near the cusp makes that all geodesic not pointing directly towards the cusp actually come back to the compact part in bounded (say ) time. In other words, the excursions of infinite WP geodesics on near the cusp are so quick that the WP flow on is “close” to a classical Anosov geodesic flow on negatively curved compact surface. In particular, it is not entirely surprising that the WP flow on is rapid.

On the other hand, in the case , the WP metric on has *some* sectional curvatures close to *zero* near the boundary of the Deligne-Mumford compactification of (see Theorem 3 and Remark 5 of this post here). By exploiting this feature of the WP metric on for (that has no counterpart for or ), we will build a *non-neglegible* set of WP geodesics spending a *long* time near the boundary of before eventually getting into the compact part. In this way, we will deduce that the WP flow on takes a fair (polynomial) amount of time to mix certain parts of the boundary of with fixed compact subsets of .

In the remainder of this post, we will give some details of the proof of Theorem 1. In the next section, we give a fairly complete proof (assuming the results in this previous post, of course) of the polynomial upper bound on the rate of mixing of the WP flow on when . After that, in the final section, we provide a *sketch* of the proof of the rapid mixing property of the WP flow on . In fact, we decided (for pedagogical reasons) to explain some key points of the rapid mixing property *only* in the *toy model* case of a negatively curved surface with one cusp corresponding *exactly* to a surface of revolution of a profile , . In this way, since the WP metric near the cusp of can be thought as a “perturbation” of the surface of revolution of (thanks to Wolpert’s asymptotic formulas), the reader hopefully will get a flavor of the main ideas behind the proof of rapid mixing of the WP flow on without getting into the (somewhat boring) technical details needed to check that the arguments used in the toy model case are “sufficiently robust” so that they can be “carried over” to the “perturbative setting” of the WP flow on .

**1. Rates of mixing of the WP flow on . I **

In this section, our notations are the same as in this previous post here.

Given , let us consider the portion of consisting of such that a non-separating (homotopically non-trivial, non-peripheral) simple closed curve has hyperbolic length . The following picture illustrates this portion of as a -neighborhood of the stratum of the boundary of the Deligne-Mumford compactification where gets pinched (i.e., becomes zero).

Note that the stratum is non-trivial (that is, not reduced to a single point) when . Indeed, by pinching as above and by disconnecting the resulting node, we obtain Riemann surfaces of genus with punctures whose moduli space is isomorphic to . It follows that is a complex orbifold of dimension , and, a fortiori, is not trivial. Evidently, this argument breaks down when : for example, by pinching a curve as above in a once-punctured torus and by removing the resulting node, we obtain thrice punctured spheres (whose moduli space is trivial). In particular, our Figure 1 concerns *exclusively* the case .

We want to locate certain regions near taking a long time to mix with the compact part of . For this sake, we will exploit the geometry of the WP metric near — e.g., the fact provided by Wolpert’s formulas (cf. Theorem 3 in this post) that some sectional curvatures of the WP metric approach zero — to build nice sets of unit vectors traveling in an “almost parallel” way to for a significant amount of time.

More precisely, we consider the vectors and (where is the complex structure). By definition, they span a complex line . Intuitively, the complex line points in the normal direction to a “copy” of inside a level set of the function as indicated in the following picture:

Using the complex line , we formalize the notion of “almost parallel” vector to . Indeed, given , let us denote by the quantity (where is the WP metric). By definition, measures the size of the projection of the unit vector in the complex line . In particular, we can think of as “almost parallel” to whenever the quantity is very close to zero.

In this setting, we will show that unit vectors almost parallel to whose footprints are close to always generate geodesics staying near for a long time. More concretely, given , let us define the set

where and is the footprint of the unit vector . Equivalently, is the disjoint union of the pieces of spheres attached to points with . The following figure summarizes the geometry of :

We would like to prove that a geodesic originating at any stays in a -neighborhood of for an interval of time of size of order , so that the WP geodesic flow does *not* mix with any fixed ball in the compact part of of Riemann surfaces with systole :

In this direction, we will need the following lemma from the third post of this series (cf. Lemma 13 in this post here).

From this lemma, it is not hard to estimate the amount of time spent by a geodesic near for an arbitrary :

Lemma 3There exists a constant (depending only on and ) such that

for all and .

*Proof:* By definition, implies that . Thus, it makes sense to consider the maximal interval of time such that for all .

By Lemma 2, we have that , i.e., for some constant depending only on and . In particular, for all . From this estimate, we deduce that

for all . Since the fact that implies that , the previous inequality tell us that

for all .

Next, we observe that, by definition, . Hence,

By putting together the previous two inequalities and the fact that (as ), we conclude that

Since was chosen so that is the maximal interval with for all , we have that . Therefore, the previous estimate can be rewritten as

Because , it follows from this inequality that where .

In other words, we showed that , and, *a fortiori*, for all . This completes the proof of the lemma.

Once we have Lemma 3 in our toolbox, it is not hard to infer some upper bounds on the rate of mixing of the WP flow on when .

Proposition 4Suppose that the WP flow on has a rate of mixing of the form

for some constants , , for all , and for all choices of -observables and .Then, , i.e., the rate of mixing of the WP flow is at most polynomial.

*Proof:* Let us fix once and for all an open ball (with respect to the WP metric) contained in the compact part of : this means that there exists such that the systoles of all Riemann surfaces in are .

Take a function supported on the set of unit vectors with footprints on with values such that and : such a function can be easily constructed by smoothing the characteristic function of with the aid of bump functions. Next, for each , take a function supported on the set with values such that and : such a function can also be constructed by smoothing the characteristic function of after taking into account the description of the WP metric near given by Theorems 2 and 3 in this post here and the definition of (in terms of the conditions and ). Furthermore, this description of the WP metric near combined with the asymptotic expansion where and is a twist parameter (see the proof of Lemma 4 of this post here) says that : indeed, the condition on footprints of unit tangent vectors in provides a set of volume (cf. the proof of Lemma 4 of the aforementioned post for details) and the condition on unit tangent vectors in with a fixed footprint provides a set of volume comparable to the Euclidean area of the Euclidean ball (cf. Theorem 2 in this post here), so that

In summary, for each , we have a function supported on with , and for some constant depending only on and .

Our plan is to use the observables and to give some upper bounds on the mixing rate of the WP flow . For this sake, suppose that there are constants and such that

for all and .

By Lemma 3, there exists a constant such that whenever . Indeed, since is a symmetric set (i.e., if and only if ), it follows from Lemma 3 that all Riemann surfaces in the footprints of have a systole . Because we took in such a way that all Riemann surfaces in have systole , we obtain , that is, , as it was claimed.

Now, let us observe that the function is supported on because is supported on and is supported on . By putting together this fact and the claim in the previous paragraph (that for ), we deduce that whenever . Thus,

By plugging this identity into the polynomial decay of correlations estimate , we get

whenever and .

We affirm that the previous estimate implies that . In fact, recall that our choices were made so that where is a fixed ball, , for some constant and . Hence, by combining these facts and the previous mixing rate estimate, we get that

that is, , for some constant and for all sufficiently small (so that and ). It follows that , as we claimed. This completes the proof of the proposition.

Remark 3In the statement of the previous proposition, the choice of -norms to measure the rate of mixing of the WP flow is not very important. Indeed, an inspection of the construction of the functions in the argument above reveals that for any , . In particular, the proof of the previous proposition is sufficiently robust to show also that a rate of mixing of the form

for some constants , , for all , and for all choices of -observables and holds only if .In other words, even if we replace -norms by (stronger, smoother) -norms in our measurements of rates of mixing of the WP flow (on for ), our discussions so far will always give polynomial upper bounds for the decay of correlations.

At this point, our discussion of the proof of the first item of Theorem 1 is complete (thanks to Proposition 4 and Remark 3). So, we will now move on to the next section we give some of the key ideas in the proof of the second item of Theorem 1.

**2. Rates of mixing of the WP flow on . II **

Let us consider the WP flow on when , that is, when or .

Actually, we will restrict our attention to the case because the remaining case is very similar to .

Indeed, the moduli space of four-times punctured spheres is a *finite* cover of the moduli space : this can be seen by sending each four-punctured sphere to the elliptic curve , so that becomes naturally isomorphic to where is a congruence subgroup of of level with index . Since all arguments towards rapid mixing of geodesic flows in this section still work after taking finite covers, it suffices to prove the second item of Theorem 1 to the WP flow on .

The rate of mixing of a geodesic flow on the unit tangent bundle of a negatively curved *compact* surface is known to be *fast*: indeed, Chernov used his technique of “Markov approximations” to show *stretched exponential* decay of correlations, and Dolgopyat added a new crucial ingredient (“Dolgopyat’s estimate”) to Chernov’s work to prove *exponential* decay of correlations.

Evidently, these works of Chernov and Dolgopyat can not be applied to the Wp flow on because of the non-compactness of due to the presence of a (single) cusp (at infinity). Nevertheless, this suggests that we should be able to determine the rate of mixing of the WP flow on provided we have enough control of the geometry of the WP metric near the cusp.

Fortunately, as we mentioned in Example 5 of this post here, Wolpert showed that the WP metric on has an *asymptotic* expansion at a point . Thus, the WP metric on neighborhoods (with ) of the cusp at infinity of becomes closer (as ) to the metric of surface of revolution of the profile on neighborhoods of the cusp at (as ).

Partly motivated by the scenario of the previous paragraph, from now on we will *pretend* that the WP metric on looks *exactly* like the metric at all points for some . In other words, instead of studying the WP flow on , we will focus on the rates of mixing of the following *toy model*: the geodesic flow on a negatively curved surface with a single cusp possessing a neighborhood where the metric is isometric to the surface of revolution of a profile for a fixed real number .

Remark 4The surface of revolution modeling the WP metric on is obtained by rotating the profile . In other words, we see that the study of rates of mixing of the surface of revolution approximating the WP metric on is a “borderline case” in our subsequent discussion.

Here, our main motivations to replace the WP flow on by the toy model described above are:

- all important ideas for the study of rates of mixing of are also present in the case of the toy model, and
- even though the WP metric on is a perturbation of a surface of revolution, the verification of the fact that the arguments used to estimate the decay of correlations of the geodesic flow on the toy model surfaces are robust enough so that they can be carried over the WP metric situation is somewhat boring: basically, besides performing a slight modification of the proofs to include the borderline case , one has to introduce “error terms” in the whole discussion below and, after that, one has to check that these errors terms do not change the qualitative nature of all estimates.

In summary, the remainder of this section will contain a proof of the following “toy model version” of the second item of Theorem 1.

Theorem 5Then, the geodesic flow (associated to ) on is rapid (faster than polynomial) mixing in the sense that, for all , one can choose an adequate Banach space of “reasonably smooth” observables and a constant so thatLet be a compact surface and fix . Suppose that is equipped with a negatively curved Riemannian metric such that the restriction of to a neighborhood of is isometric to a surface of revolution of a profile (for some choices of and ).

for all .

Remark 5The arguments below show that the statement above also holds when is equipped with a negatively curved metric that is isometric to a surface of revolution , , near for each .

Remark 6The Riemannian metric is incomplete because the surface of revolution of is incomplete when (as the reader can check via a simple calculation).

Recall that, in the setting of Theorem 5, we want to understand the dynamics of the excursions of the geodesic flow near the cusp (in order to get rapid mixing). For this sake, we describe these excursions by rewriting the geodesic flow (near ) as a *suspension flow*.

** 2.1. Excursions near the cusp and suspension flows **

Consider a small neighborhood in of where the metric is isometric to the surface of revolution of the profile , i.e.,

Next, take a small parameter and consider the parallel . We parametrize unit tangent vectors to the surface of revolution with footprints in as follows.

Given , we denote by the unique unit tangent vector pointing towards to the cusp at . Equivalently, is the unit vector tangent to the meridian at time , or, alternatively, where is the distance function from the cusp to a point . Also, we let be the unit vector obtained by rotating by in the counterclockwise sense (i.e., by applying the natural almost complex structure ).

In this setting, an unit vector pointing towards the cusp is completely determined by a real number such that and , i.e.,

The *qualitative* behavior of the excursion of a geodesic starting at can be easily determined in terms of the parameter thanks to the classical results in Differential Geometry about surfaces of revolutions. Indeed, it is well-known (see, e.g., Do Carmo’s book) that such a geodesic satisfies

and

for a certain constant , and, furthermore, these relations imply the famous Clairaut’s relation:

where is the parameter attached to (i.e., ). In particular, except for the geodesic going directly to the cusp (i.e., the geodesic starting at associated to ), all geodesics (starting at with ) behave qualitatively in a simple way. In the first part of its excursion towards the cusp, the angle increases (resp. decreases) from to (resp. from to ) while the value of diminishes in order to keep up with Clairaut’s relation. Then, the geodesic reaches its closest position to the cusp at time : here, (i.e., is tangent to the parallel containing ) and, hence,

Finally, in the second part , does the “opposite” from the first part: the angle goes from to and increases from back to . The following picture summarizes the discussion of this paragraph:

Remark 7Note that the time taken by the geodesic to go from the parallel to and then from back to isindependentof the basepoint . Indeed, this is a direct consequence of the rotational symmetry of our surface. Alternatively, this can be easily seen from the formula

deduced by integration of the ODE satisfied by . Observe that this formula also shows that is uniformly bounded, i.e., for all . Geometrically, this means that all geodesics starting at must return to in bounded time unless they go directly into the cusp.

This description of the excursions of geodesics near the cusp permits to build a *suspension-flow* model of the geodesic flow near . Indeed, let us consider the *cross-section* . As we saw above, an element of the surface is parametrized by two angular coordinates and : the value of determines a point and the value of determines an unit tangent vector making angle with . The subset of consisting of those elements with angular coordinate corresponds to the unit vectors with footprint in pointing towards the cusp at . The equation determines a circle inside corresponding to geodesics going straight into the cusp, and, furthermore, we have a natural “first-return map” defined by where is the geodesic starting at at time .

In this setting, the orbits , are modeled by the “suspension flow” if , over the *base map* with *roof* function , .

Remark 8Technically speaking, one needs to “complete” the definition of and by including the dynamics of the geodesic flow on the compact part of in order to properly write the geodesic flow on as a suspension flow. Nevertheless, since the major technical difficulty in the proof of Theorem 5 comes from the presence of the cusp, we will ignore the excursions of geodesics in the compact part and we will pretend that the (partially defined) flow is a “genuine” suspension flow model.

** 2.2. Rapid mixing of contact suspension flows **

One of the advantages about thinking of the geodesic flow on as a suspension flow comes from the fact that several authors have previously studied the interplay between the rates of mixing of this class of flows and the features of and : see, e.g., these papers of Avila-Gouëzel-Yoccoz and Melbourne for some results in this direction (and also for a precise definition of suspension flows).

For our current purposes, it is worth to recall that Bálint and Melbourne (cf. Theorem 2.1 [and Remarks 2.3 and 2.5] of this paper here) proved the rapid mixing property for *contact* suspension flows whose base map is modeled by a *Young tower* with *exponential tails* and whose roof function is bounded and *uniformly piecewise Hölder continuous* on each subset of the basis of the Young tower. In particular, the proof of Theorem 5 is complete once we prove that the base map is modeled by Young towers and the roof function is bounded and uniformly piecewise Hölder continuous on each element of the basis of the Young tower (whatever this means).

As it turns out, the theory of Young towers (introduced by Young in these papers here and here) is a *double-edged sword*: while it provides an adequate setup for the study of statistical properties of systems with some hyperbolicity *once* the so-called *Young towers* were built, it has the *drawback* that the construction of Young towers (satisfying all five natural but technical axioms in Young’s definition) is usually a delicate issue: indeed, one has to find a countable Markov partition of a positive measure subset (working as the basis of the Young tower) so that the return maps associated to this Markov partition verify several hyperbolicity and distortion controls, and it is not always clear where one could possibly find such a Markov partition for a given dynamical system.

Fortunately, Chernov and Zhang gave a list of *sufficient* geometric properties for a *two-dimensional* map like to be modeled by Young towers with exponential tails: in fact, Theorem 10 in Chernov-Zhang paper is a sort of “black-box” producing Young towers with exponential tails whenever seven *geometrical* conditions are fulfilled. For the sake of exposition, we will not attempt to check all seven conditions for : instead, we will focus on two main conditions called *distortion bounds* and *one-step growth condition*.

Before we discuss the distortion bounds and the one-step growth condition, we need to recall the concept of *homogeneity strips* (originally introduced by Bunimovich-Chernov-Sinai). In our setting, we take and (to be chosen later) and we make a partition of a neighborhood of the *singular set* (of geodesics going straight into the cusp) into countably many strips:

for all , . (Actually, has two connected components, but we will slightly abuse of notation by denoting these connected components by .)

Intuitively, the partition into polynomial scales in the parameter is useful in our context because the relevant quantities (such as Gaussian curvature, first and second derivatives, etc.) for the study of the geodesic flow of the surface of revolution blows up with a polynomial speed as the excursions of geodesics get closer the cusp (that is, as ). Thus, the important quantities for the analysis of the geodesic flow near the cusp become “almost constant” when restricted to one of the homogeneity strips .

Also, another advantage of the homogeneity strips is the fact that they give a rough control of the elements of the countable Markov partition at the basis of the Young tower produced by Chernov-Zhang: indeed, the arguments of Chernov-Zhang show that each element of the basis of their Young tower is completely contained in a homogeneity strip. In particular, the verification of the uniform piecewise Hölder continuity of the roof function follows once we prove that the restriction of the roof function to each homogeneity strip is uniformly Hölder continuous (in the sense that, for some , the Hölder norms are bounded by a constant *independent* of ).

Coming back to the one-step growth and distortion bounds, let us content ourselves to formulate simpler *versions* of them (while referring to Section 4 and 5 of Chernov-Zhang paper for precise definitions): indeed, the actual definitions of these notions involve the properties of the derivative along unstable manifolds, and, in our current setting, we have just a *partially defined* map , so that we can not talk about future iterates and unstable manifolds unless we “complete” the definition of .

Nevertheless, even if is only partially defined, we still can give crude analogs to unstable directions for by noticing that the vector field on (whose leaves are ) morally works like an unstable direction: in fact, this vector field is transverse to the singular set which is a sort of “stable set” because all trajectories of the geodesic flow starting at converge in the future to the same point, namely, the cusp at . In terms of the “unstable direction” , we define the *expansion factor* of at a point as , that is, the amount of expansion of the “unstable” vector field under . Note that, from the definitions, the expansion factor depends only on the -coordinate of . So, from now on, we will think of expansion factors as a function of .

In terms of expansion factors, the (variant of the) distortion bound condition is

where satisfies , and the (variant of the) one-step growth condition is

Remark 9The one-step growth condition above is very close to the original version in Chernov-Zhang paper (compare (3) with Equation (5.5) in Chernov-Zhang article). On the other hand, the distortion bound condition (2) differs slightly from its original version in Equation (4.1) in Chernov-Zhang paper. Nevertheless, they can be related as follows. The original distortion condition essentially amounts to give estimates (where is a smooth function such that as ) whenever and belong to the samehomogenous unstable manifold(i.e., a piece of unstable manifold such that never intersects the boundaries of the homogeneity strips for all and ; the existence of homogenous unstable manifolds through almost every point is guaranteed by a Borel-Cantelli type argument described in Appendix 2 of this paper of Bunimovich-Chernov-Sinai here). Here, one sees that

for some . Using the facts that decays exponentially fast (as and are in the same unstable manifold ) and is always contained in a homogeneity strip (as is a homogenous unstable manifold), one can check that the estimate in (2) implies the desired uniform bound on the previous expression in terms of a smooth function such that as . In other words, the estimate (2) can be shown to imply the original version of distortion bounds, so that we can safely concentrate on the proof of (2).

At this point, we can summarize the discussion so far as follows. By Melbourne’s criterion for rapid mixing for contact suspension flows and Chernov-Zhang criterion for the existence of Young towers with exponential tails for the map , we have “reduced” the proof of Theorem 5 to the following statements:

Proposition 6Given and , one has the following “uniform Hölder estimate”

whenever is sufficiently small (depending on , and ).

Proposition 7The expansion factor function satisfies:

- given , we can choose large (and sufficiently small) so that
where ;

- given , we can choose and such that and
for some (sufficiently large) constant and for all .

The proofs of these two propositions are given in the next two subsections and they are based on the study of perpendicular unstable Jacobi fields related to the variations of geodesics of the form , .

** 2.3. The derivative of the roof function **

From now on, we fix (e.g., ) and, for the sake of simplicity, we will denote a geodesic corresponding to an initial vector by . Of course, there is no loss of generality here because of the rotational symmetry of the surface . Also, we will suppose that as the case is symmetric.

Note that the roof function is defined by the condition , or, equivalently,

where denotes the distance from a point to the cusp at and is the distance from to . By taking the derivative with respect to at and by recalling that , we obtain that

where . Since where , we have , and, *a fortiori*,

Let us compute the two inner products above. By definition of the parameter and the symmetry of the revolution surface , we have . Also, if we denote by the perpendicular (“unstable”) Jacobi field along the geodesic associated to the variation of (cf. Section 2 of this previous post here) with initial conditions and , then

From the computation of the inner products above and the fact that they add up to zero, we deduce that , that is,

In other terms, the previous equation says that the derivative can be controlled via the quantity measuring the growth of the perpendincular Jacobi field at the return time . Here, it is worth to recall that Jacobi fields are driven by Jacobi’s equation:

where is the Gaussian curvature of the surface of revolution at the point . Also, it is useful to keep in mind that Jacobi’s equation implies that the quantity satisfies Riccati’s equation

where .

In the context of the surface of revolution , these equations are important tools because we have the following explicit formula for the Gaussian curvature at a point :

In particular, verifies .

Next, we take and we consider the following auxiliary function:

By definition, . Furthermore,

Since the equation (describing the motion of geodesic on ) implies that , we deduce from the previous inequality that

This estimate allows to control the solution of Riccati’s equation along the following lines. The initial data of the Jacobi field is and . Hence,

In particular, there exists a well-defined maximal interval where for all . By plugging this estimate into Jacobi’s equation, we get that

for each .

By integrating this inequality (and using the initial condition ), we obtain that

Therefore,

If , we deduce that (as ). Otherwise, and . Since satisfies Riccati’s equation, we deduce from (5) that

at each time where . It follows that for all . Hence,

and, *a fortiori*,

In other words, we proved that

Now, the quantity can be estimated as follows. By deriving Clairaut’s relation , we get

Since (as we are interested in small angles , large) and (thanks to the relation and the fact that and, thus, for small), we conclude that

for . Here, we used the fact that for . Therefore,

since . Also, the symmetry of the surface implies and, hence,

In summary, we have shown that , i.e.,

By putting together (4), (6) and (8), we conclude that

for some constant depending on and .

At this stage, we are ready to complete the proof of Proposition 6.

*Proof:* Let us estimate the Hölder constant . For this sake, we fix and we write

for some between and . Since and , it follows from (9) that

Because and are arbitrary points in , we have that

where is an appropriate constant.

Now, our assumption implies that we can choose sufficiently small so that . By doing so, we see from the previous estimate that

whenever , i.e., , is sufficently small. This proves Proposition 6.

** 2.4. Some estimates for the expansion factors **

Similarly to the previous subsection, the proof of Proposition 7 uses the properties of Jacobi’s and Riccati’s equation to study

where is the scalar function (with and ) measuring the size of the perpendicular “unstable” Jacobi field along .

We begin by giving a lower bound on . Given , let us choose small so that

Of course, this choice of is possible because . Next, we consider the auxiliary function:

By definition, . Furthermore,

In particular,

Since (cf. the paragraph before (5)), we deduce from the previous estimate that

This inequality implies that the solution of Riccati’s equation satisfies for all . Indeed, the initial condition , says that and the inequality above tells us that

at any time where .

By integrating the estimate over the interval , we obtain that

i.e.,

For sake of concreteness, let us set and let us restrict our attention to geodesics whose initial angle with the meridians of are sufficiently small so that . In this way, we have that (thanks to Jacobi’s equation and our initial conditions and ). In this way, the inequality above becomes

Next, we observe that can be bounded from below in a similar way to our derivation of a bound from above to in the previous subsection: in fact, by repeating the arguments appearing after (7) above, one can show that

and

where is an adequate (small) constant depending on , and .

By putting together the estimates above, we deduce that

where .

This inequality shows that

Thus, if , then we can choose small (with ) and large so that (our variant of) the one-step growth condition (3) holds. This proves the first part of Proposition 7.

Finally, we give an indication of the proof of the second part of Proposition 7 (i.e., the distortion bound (2)). We start by writing

and by noticing that

Next, we take the derivative with respect to of the previous expression. Here, we obtain several terms involving some quantities already estimated above via Jacobi’s and Riccati’s equation (such as , , etc.), but also a new quantity appears, namely, , i.e., the derivative with respect to of the family of solutions of Riccati’s equation along . Here, the “trick” to give bounds on is to derive Riccati’s equation

with respect to in order to get an ODE (in the time variable ) satisfied by . In this way, it is possible to see that one has reasonable bounds on as soon as the derivative of the square root of the absolute value of the Gaussian curvature. Here, can be bounded by recalling that we have an explicit formula

for the Gaussian curvature. By following these lines, one can prove that, for a given , the distortion bound

holds whenever is taken sufficiently small. In other words, by taking , we have .

Note that the estimate in the previous paragraph gives the desired distortion bounds (2) once we show that can be selected such that . In order to check this, it suffices to recall that can be taken arbitrarily small (cf. the proof of the first part of Proposition 7), i.e., . So,

and

Since for , it follows that for adequate choices of and . This completes our sketch of proof of the second part of Proposition 7.

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