Parabolic dynamics
1. Survey
1.1. What does hyperbolic, elliptic mean in dynamics?
Let be a dynamical system. Even if deterministic, it can exhibit a chaotic behaviour. This has several characteristics. One of them is the Butterfly effect: sensitive dependence on initial conditions (SDIC).
Definition 1 A flow has SDIC if there exists such that , , such that and such that .
A quantitative measurement of SDIC is provided by the dependence of on .
Definition 2 Let be a nondecreasing function. A flow has SDIC of order if in the above definition, one can take such that .
This leads us to a rough division of dynamical systems:
- Elliptic: no SDIC, or if any, subpolynomial.
- Hyperbolic: SDIC is fast, exponential.
- Parabolic: SDIC is slow, subexponential.
This trichotomy is advertised in Katok-Hasselblatt’s book.
For instance, entropy is a measure of chaos. Elliptic or parabolic dynamical systems have zero entropy. Hyperbolic dynamical systems have positive entropy.
1.2. Examples of elliptic dynamical systems
- Circle diffeomorphisms.
- Linear flows on the torus (these two examples are related, one is the suspension of the other).
- Billiards in convex domains. Usually, there are many periodic orbits, trapping regions, caustics.
This the realm of Hamiltonian dynamics and KAM theory.
1.3. Examples of hyperbolic dynamical systems
- Automorphisms of a torus which are Anosov, i.e. all eigenvalues have absolute values . Then orbits diverge exponentially: if , , set . Then mod and reaches in time such that for , an exponential.
- Geodesic flows on constant curvature surfaces.
- Sinai’s billiard: a rectangle with a circular obstacle. Somewhat equivalent to the motion of two hard spheres on a torus. Scattering occurs after hitting the obstacle, due to its strict convexity.
This is the realm of Anosov-Sinai and others’ dynamics. Structural stability occurs: hyperbolic systems form an open set.
1.4. Examples of parabolic dynamical systems
- Horocycle flows on constant curvature surfaces. They were introduced by Hedlund, followed by Dani, Furstenberg, Marcus, Ratner.
- Nilflows on nilmanifolds. If is a cocompact lattive in a nilpotent Lie group, let be a -parameter subgroup of acting by right translations on . It descends to . Both examples are algebraic, this provides us with tools to study them. They are a bit too special to illustrate parabolic dynamics.
- Smooth area-preserving flows on higher genus surfaces.
- Ehrenfest’s billiard: rectangular, with a rectangular obstacle. Here, SDIC is only caused by discontinuities due to corners. More generally, billiards in rational polygons (angles belong to ), or equivalently linear flows on translation surfaces. This field is known as Teichmüller dynamics. Forni considers them as elliptic systems with singularities, I prefer to stress their parabolic character.
1.5. Uniformity
Within hyperbolic dynamics, there is a subdivision in uniformly hyperbolic, nonuniformly hyperbolic and partially hyperbolic.
In the same manner, we see horocycle flows as uniformly parabolic, and nilflows as partially parabolic, with both elliptic and parabolic directions. Area-preserving flows on surfaces have fixed points which introduce partially parabolic behaviour: shearing is uniform or not. However, there is no formal definition.
1.6. More examples of parabolic behaviours
Parabolicity is not stable. However, Ravotti has discovered a -parameter perturbation of unipotent flows in .
A flow is a time-change of a given flow if there exists a function such that
For to be a flow, it is necessary that be a cocycle.
Both flows have the same trajectories. A feature of parabolic dynamics is that a typical time-change is not isomorphic to and has new chaotic features. Indeed, an isomorphism would solve the cohomology equation, and there are obstructions.
1.7. Program
Study smooth time-changes of algebraic flows.
Goes back to Marcus in the 1970’s. Algebraic tools break down, softer methods are required: geometric mechanisms. Also, we expect the features exhibited by time changes to be more typical.
2. Chaotic properties
2.1. Definitions
Definition 3 Let be a measure space with finite measure. Let be a measure preserving flow. The trajectory of a point is equidistributed with respect to if for every smooth observable ,
tends to as tends to .
is ergodic if almost every has equidistributed orbit with respect to .
This is Boltzmann hypothesis.
Definition 4 Say is mixing if for all , the correlation
tends to as tends to .
This means decorrelation of functions. This implies ergodicity.
The speed at which decorrelation occurs is a significative feature too.
Definition 5 The speed of mixing is a function such that for all smooth observables , the correlation decays at speed , i.e.
as tends to .
2.2. Relation to the trichotomy
Elliptic systems often are not ergodic, but even when they are, they are not mixing.
Hyperbolic and parabolic systems can be mixing, but at different speeds:
- hyperbolic exponential decay of correlations.
- in parabolic systems, we expect that, if mixing occurs, the decay of correlations is slower: polynomial or subpolynomial.
A related concept is that of polynomial deviations of ergodic averages: if is smooth and has vanishing integral, s ergodic and is an equidistribution point,
and no faster. This phenomenon was first discovered on horocycle flows, then Teichmüller flows (Zorich, experimentally, Kontsevitch-Zorich for a proof).
2.3. Other features
Spectral properties. Let be the operator on . Then is unitary. What is its spectrum?
Disjointness of rescalings. Rescaling means linear time change .
3. Results
Horocycle flow is mixing (Ratner). The spectrum is Lebesgue absolutely continuous.
Time changes of horocycle flows are mixing (Marcus, by shearing). This can be made quantitative (Forni-Ulcigrai).
Disjointness of rescalings fails for horocycle flows, but hold for nontrivial time changes (Kanigowski-Ulcigrai and Flaminio-Forni). The spectrum is Lebesgue absolutely continuous as well.
Nilflows themselves are not mixing, but typical time changes of nilflows are mixing (Avila-Forni-Ravotti-Ulcigrai).
We shall see geometric mechanisms at work:
- Mixing via shearing.
- Ratner property of shearing.
- Renormizable parabolic flows.
- Deviations of ergodic integrals.
4. Horocycle flows
Today’s goal is to explain the technique of mixing by shearing.
4.1. Algebraic viewpoint
Consider and its subgroups
Every matrix can be uniquely written , hence . The factors do not commute, because of the key relation
The key relation can be interpreted as a selfsimilarity property: is a fixed point of renormalization by .
Take a discrete and cocompact subgroup . Then and act on by left multiplication. The key relation implies that the rescaled flow $latex {h_{\mathbb R}^k=(h_{ks})_{s\in{\mathbb R}}}&fg=000000$ is conjugated to $latex {h_{\mathbb R}}&fg=000000$. This fails for other (nonrescaling) time changes, as I proved recently with Fraczek and Kanigowski.
4.2. Geometric viewpoint
Let denote the upper half plane, with metric .
Fact. acts isometrically and transitively on , and this yields a diffeomorphism of with the unit tangent bundle .
Indeed, the action is by Möbius transformations
on , and by their derivatives on .
With this identification, orbits of are curves which project to geodesics of , and coincide with their lifts by their unit speed vector. On the other hand, orbits of are curves which projects to horocycles of , and coincide with their lifts by their unit normal outward pointing vectors. Lifts by inward pointing normal vectors are orbits of .
is a hyperbolic flow: it contracts in the direction of -orbits, it dilates in the direction of -orbits,
4.3. Classical results
Let denote Haar measure on . It maps via to hyperbolic volume. Consider the induced measure on (still denoted by ). Then and acts on by measure preserving transformations.
Then
- is ergodic (Hopf). It is far from being unique ergodic (plenty of periodic orbits).
- is mixing.
- is uniquely ergodic (Furstenberg). This means that every orbit is equidistributed with respect to .
5. Shearing
This is an alternate way to prove mixing. The idea goes back to Marcus (Annals of Math. 1977). Marcus covered a more general situation, and proved mixing of all orders (i.e. for multicorrelations, integrals involving an arbitrarily large number of functions).
Let us shift viewpoint on the key relation. Let be a piece of -orbit of length . Let
Then is sheared or tilted in the direction of the geodesic flow.
Key idea in parabolic dynamics: In several parabolic systems, the Butterfly effect happens in a special way, e.g. shearing. Points nearby move parallel, but with different speeds. This implies that transverse arcs shear.
5.1. Recipe for mixing in parabolic dynamics
Here are the ingredients:
- A uniquely ergodic flow .
- A transverse direction which is sheared in the direction of the flow.
By assumption, for every , the trajectory equidistributes with respect to . We want to upgrade it to mixing, which is a property of sets: indeed
is equivalent to
i.e equidistributes as .
The idea is to cover by short arcs in the transverse direction. We prove that each such arc equidistributes, and apply Fubini. I.e. if , apply . Then .
Each becomes close to a long piece of orbit of . By unique ergodicity, that piece equidistributes, and this implies equidistribution for .
5.2. Time change
Let be a smooth time change, which is a cocycle with respect to a given smooth flow , i.e.
We are interested in the flow defined by
The generator of is
It is a smooth nonnegative function on . We assume that . We denote by .
Remark. If is generated by a smooth vectorfield , then is generated by the vectorfield .
Theorem 6 (Forni-Ulcigrai) Let be the horocyclic flow of a compact constant curvature surface. For any smooth function , the flow is mixing, with quantitative estimates which imply that the spectrum is absolutely continuous with respect to Lebesgue measure.
Lemma 7 Let denote the generator of and the generator of . Take a segment of -orbit
Let Then where
Remark. , hence . Thus, as tends to , tends to a finite limit, the shear rate.
Proof of Lemma. It relies on , which implies that .
We see that we need compute integrals over sheared arcs . Let be a smooth function. Then
The second term is an ergodic integral which is easy to handle. The main term is the first term, which can be rewritten
where , .
To deduce mixing, one must estimate inner products . We integrate by parts
The second term is again an ergodic integral that tends to .
5.3. Quantitative equidistribution estimates
We are interested in ergodic integrals of the form
Flaminio-Forni treat the un-time-changed case and show that
for some . One can adapt their arguments, using estimates by Bufetov-Forni, to the time-changed case, and get similar estimates.
5.4. Additional references
Marcus original technique already proved mixing. His setting was Anosov flows, with their stable and unstable foliations. From these, a flow can be defined, which satisfies
where $latex {s^*}&fg=000000$ has a continuous mixed partial second derivative $latex {\frac{\partial s^*}{\partial t \partial s}}&fg=000000$. So we see that time-changes were already in the picture.
Kushnirenko was able to prove mixing for smooth time-changes, assuming
Thus small time-changes are mixing. What about larger ones? This is still open.
Tiedra de Aldecoa uses a different method to prove absolute continuity of the spectrum for time changes satisfying (KC).
Generalizations. The setting is algebraic dynamics: a unipotent -parameter subgroup acting on , semisimple Lie group.
Lucia Simonelli (Forni’s student) could prove absolute continuity of the spectrum for time changes satisfying (KC).
Davide Ravotti (my student) could prove quantitative mixing.
Kanigowski and Ravotti could prove quantitative -mixing.
6. Heisenberg nilfows
Let denote the Lie group of unipotent matrices, with as standard generators of its Lie algebra, . Let be a discrete cocompact lattice (for instance, unipotent matrices with integer entries). We call the Heisenberg nilmanifold. acts on by right multiplication. The action of a -parameter subgroup is called a nilflow.
6.1. Classical results
Auslander-Green-Hahn (1963) studied unique ergodicity of nilflows. They showed that is can be written , for the nilflow defined by ,
unique ergodicity ergodicity minimality and are rationally independent.
Rational independence means that no linear relation with nonzero integral coefficients can hold.
In other words, if we project the situation to the -torus where , , then the flow projects to a flow on , and
unique ergodicity for unique ergodicity for .
Here, we have used a theorem of Furstenberg on skew-products of rotations of the circle.
Definition 8 For real numbers , let be the diffeomorphism of the -torus defined by
In general, a skew-product over a map is a map which is fiber-preserving (with respect to the projection ) and the permutation of fibers is given by . In the example at hand, is isometric on fibers.
6.2. First return map
Lemma 9 Assume that the Heisenberg nilflow is uniquely ergodic. There is a transverse submanifold , diffeomorphic to a torus, such that the Poincaré return map , given by , the first return time to , is one of the Furstenberg skew-products .
Proof of the Lemma. lifts to a vertical plane in . Since and commute, is diffeomorphic to a torus. If is uniquely ergodic, , so is transverse to the flow.
We show that is a return time. We use the fact that . So using the Campbell-Hausdorff-Dynkin formula, we compute
This point belongs to , so .
6.3. Lack of mixing
The above Lemma shows th at we can now focus on Furstenberg skew-products. We shall see that the parameter plays no role, so we focus on .
Definition 10 Given a map and a function (called the roof function), we define the special flow over under as follows: it is a flow on an -bundle over the circle, the quotient of the vertical unit speed flow on under the identification
Fact. If a flow admits a global Poincaré section with first return time , then is isomorphic to the special flow of the first return map with roof function .
In the case at hand, the roof function is constant. Therefore, the special flow is not mixing: if , a short interval, so do its images by the vertical unit speed flow, and so do their projections to , which are the images of a set and its images by the special flow.
This shows that Heisenberg nilflows are never mixing.
7. Mixing time-changes
The following contents can be found in Avila-Forni-Ulcigrai. A recent generalization to all step 2 nilflows can be found in Avila-Forni-Ravotti-Ulcigrai. Ravotti has treated filiform nilflows.
Let be a Heisenberg nilflow and a smooth positive function on . Let
Then the time-change is given by
7.1. Time-changes versus special flows
We have seen that Heisenberg nilflows are special flows with constant roof function. The time-change is again a special flow, over the same skew-product, but with roof function
7.2. Trivial time-changes
Beware that there exist smooth time-changes which are trivial, i.e. smoothly conjugate to the original nilflow.
In general, adding a coboundary to the roof function of a special flow produces an isomorphic flow. This leads us to the following problem: understand cohomology of nilflows. Here are our ultimate results.
Theorem 11 Let be a There exists a dense set in of roof functions, and a vectorspace of countable dimension and codimension, such that if a roof function is chosen in , the corresponding special flow is mixing.
Moreover, for ,
In fact, Katok has found a nice characterization of which are smoothly trivial. I will come back to this next week. Today, I merely give one example.
Example. is a smoothly trivial roof function.
Under the assumption that has bounded type, Kanigowski and Forni have proved quantitative mixing.
7.3. Idea of proof
We start from a Furstenberg skew-product and play with roof functions. We use again mixing by shearing. We consider intervals in the fiber (i.e. in the direction). We shall see that many of them shear in the flow () direction. But there are intervals which do not shear or shear in the other direction.
7.4. Special flow dynamics
Given a point in the torus and , we compute . When is large, we join bottom to roof several times. Let
and
Then
In order to exhibit shearing, we want to see how this changes in . Since is an isometry in the -direction, the -derivative of the sum is the sum of -derivatives, i.e.
Take trigonometric polynomials on the torus, which are positive.
Given , let minus its average on the -fiber. Define as the set of such that is not a measurable coboundary.
Step 1. Since is not a measurable coboundary, the sums must grow,
This relies on a result by Gottschalk-Hedlund, plus decoupling.
Step 2. The sums are trigonometric polynomials of bounded degree. It follows that
There can be flat intervals where no stretch occur, so one must throw them away. But the larger , the shorter these intervals are.
Step 3. We use a polynomial bound on level sets of trigonometric polynomials.
The next class will deal with renormalization, deviations of ergodic averages. Only later shall we get back to Katok’s characterization of smoothly trivial special flows and to nilflows.
8. Short recap
Parabolicity is (a bit heuristically) defined by slow butterfly effect. This can be formalized for smooth flows, in terms of growth of derivatives under iteration. It is not that easy to build examples.
Presently, parabolic flows is the following list of examples,
- Horocycle flows of compact constant curvature surfaces.
- Unipotent flows (a generalization of the above).
- Nilflows and their time-changes.
- Smooth area-preserving flows.
- Linear flows on flat surfaces with conical singularites.
The two first are uniformly parabolic. The next is partially parabolic. The fourth is nonuniformly parabolic, the last is elliptic with singularities.
We have proven mixing via the technique of shearing.
Here is a further example where this technique works, due to B. Fayad, of an elliptic flavour. Start with a linear flow on the -torus. If , Fayad has been able to construct anaytic time-changes which are mixing.
Such examples are very rare (they rely on parameters being very Liouville numbers).
9. Renormalization
The word here is taken in a meaning which differs from its use in holomorphic dynamics (not to speak of quantum mechanics).
The idea is to analyze systems which are approximately self-similar, and exhibit several time scales.
We introduce the renormalization flow which rescales: long trajectories become short. Given a map , one way to zoom in is to restrict to a subspace and replace with the first return map to . Eventually rescale space afterwards. But there are other means.
9.1. A series of examples
Example. Start with the horocycle flow . The key relation is
Applying the geodesic flow to a length trajectory , we get a trajectory of length . So the geodesic flow achieves renormalization, on the same space, with no effort.
Example. The cat map associated with the matrix on the -torus. Let , denote the eigenvalues, the eigenvectors. Let be the linear flow in direction , with unit speed. This is an elliptic flow.
Put . Let
Put . Then maps to a trajectory of of length .
Express as the composition of two Dehn twists,
Rotate coordinates so that eigenvector becomes vertical. Then acts by a diagonal matrix in , which was denoted by , earlier.
Example. Higher genus toy model.
View a genus surface as a regular octagon in the Euclidean plane with edge identifications. Fix a direction , consider the linear flow in direction . It descends to a flow on the surface which is not well-defined at the vertex
This generalizes to translation surfaces, made of a disjoint union of polygons, with identifications given by translations. The notion of a direction is well-defined on the surface (except at finally many singularities), whence a flow with singularities.
Consider matrix
It applies a shear on the original regular octagon. Since , This linear map induces a homeomorphism of the surface . Let us repeat in different direction (rotation by 45 degrees), get matrix . Let , this is a hyperbolic matrix with eigenvalues . We get again an affine automorphism of the surface. Let denote the linear flow in the direction of the eigenvector . Then
B. Veech has shown that the affine group of the surface is generated by and the order rotation. This is as large as the affine group of a translation surface can be. Therefore, is clled the Veech surface.
For almost every direction (the condition is that ), there exists a sequence of hyperbolic automorphisms with unstable direction that converge to . can be used to renormalize .
Example.
For a slightly deformed octagon , the affine group of the corresponding surface is trivial. Nevertheless, almost every linear flow is still renormalizable: there is a sequence of surfaces and affine hyperbolic morphisms with expanding direction converging to .
Indeed, consider the space of linear flows on translation surfaces of genus . The flow of diagonal matrices acts on this space, generating a flow on , known as the Teichmüller flow.
Theorem 12 (Masur-Veech) is recurrent.
Therefore for almost every linear flow, there exists a sequance such that tends to .
9.2. What is renormalization good for?
It is used to put diophantine conditions on linear flows in higher genus. See my ICM 2022 talk (watch it on line on july 11th). I do not pursue this topic further.
It is used to study deviations of ergodic averages. Assume a flow is uniquely ergodic. Ergodic integrals take the form
By the ergodic theorem, for every . Let us focus on our favourite example, the Veech surface. The invariant measure is area in the plane. Unique ergodicity is a theorem of Masur, Kerckhoff-Masur-Smillie
We show that for functions with vanishing average,
for some (discovered by Zorich, conjectured by Kontsevitch-Zorich, proven by Forni).
Fix a basis of . Fix a section of the linear flow. Take trajectories from and to their first return to . Closing them by segments of sigma gives representatives of the basis of homology. Let
Let be the matrix expressing the homology basis is the initial basis. Let denote the eigenvalues of . By continuity of , up to a small error,
Let us start from a large instead of ,
For simplicity, let us assume that is the eigenvector . Then
a contradiction. Therefore must be a combination of . Thus
On the other hand, grows like . Let satisfy
Then , as announced.
Of course, I cheated a bit, some more regularity of is needed.
In that example, the fact that can be explicitly. Tomorrow, I will mention results on this for general translation surfaces.
And then, back to nilflows.
9.3. Area preserving flows on surfaces
compact connected oriented surface of genus . Let be a flow which preserves a smooth measure . Note that has fixed points, which are all of saddle type. In the next theorem, the number and types of fixed points are fixed.
Theorem 13 (Zorich, Forni, Avila-Viana) There exists distinct positive exponents such that almost every choice of (Katok’s fundamental class, described in terms of periods) is uniquely ergodic, and for all smooth functions ,
for all .
In this statement, means a function such that
Furthermore, is a distribution on , and .
This type of behavior is called a power deviation spectrum. It was conjectured by Kontsevitch and Zorich, then proved by Zorich in 1997 for special functions , with only one term . Then Forni obtained a proof in 2002 for functions with support away from fixed points, up to distinctness of exponents which was proven by Avila-Viana. Bufetov gave a more precise version, transforming the result into an asymptotic expansion. With Fraczek, we gave a different proof based on Marmi-Moussa-Yoccoz). Finally, Fraczek-Kim could handle generic saddles. The expansion then involves extra terms depending on saddles (and not on ).
9.4. Idea of proof
Choose coordinates such that appears as a time-change of a linear flow on a translation surface. The time change is smooth only away from fixed points. If function has support away from fixed points, one is reduced to study deviation for linear flows. For a general linear flow, a renormalization is given by a sequence of matrices , the Kontsevitch-Zorich cocycle. Eigenvalues are replaced with ratios of Lyapunov exponents.
10. More on renormalization
10.1. Back to nilflows
Let denote the stabilizer of vector is . Defined by
Then is recurrent, its has been used as a renormalization by Flaminio-Forni, in order to prove polynomial deviations of ergodic averages.
10.2. Renormalizable parabolic flows
Horocycle flows: yes.
Unipotent flows: unknown.
Heisenberg nilflows: yes.
Higher step nilflows: unknown in general. Special case (special flows over skew products) studied by Flaminio-Forni. In that case, the maps diverge.
Linear flows over higher genus surfaces (they are smooth and area-preserving).
More general smooth flows on surfaces (not necessarily area preserving). Then typically diverges. This is related to generalized interval exchange transformations.
11. Isomorphisms between time-changes
Recall that a time-change of a flow is
When do such a change lead to a genuinely different, nonisomorphic flow?
11.1. Setting of special flows
Let be a map. Given a roof function $latex {\Phi:Y\rightarrow{\mathbb R}_{>0}}&fg=000000$, the flow of translations on descends to a flow $latex {\psi^{f,\Phi}_{\mathbb R}}&fg=000000$ on the quotient space
Lemma 14 Let be special flows over the same map , under roofs and . If there exists a function such that
then the two special flows are isomorphic.
Indeed, look for a conjugating homeomorphism of the form
It commutes with translations and maps one equivalence relation to the other.
11.2. Cohomological equations
The operator is called the coboundary operator. Hence the equation with unknown is called a cohomological equation. It sometimes appears with a twist: , for some .
There are obvious obstructions.
If is a periodic point, i.e. , then
So every periodic orbit gives an obstruction. If is hyperbolic, it is essentially the only one.
More generally, if is an invariant measure, then
So every invariant measure gives an obstruction.
11.3. An elliptic example
If is elliptic, this is sometimes sufficient, e.g. for circle rotations : if is a trigonometric polynomial and , then there exists a solution . If is smooth, a Diophantine condition on is required in addition for the solution to be smooth. Indeed, in Fourier, if
then
For to decay superpolynomially, one needs that
for some , which amounts to being badly approximable by rationals.
11.4. Parabolic case
In the parabolic world, in addition to invariant measures, invariant distributions provide further obstructions.
Remember that Heisenberg nilflows are special flows with constant roof over Furstenberg skew-products of the form
We need study the corresponding cohomological equation.
Proposition 15 In Fourier series, if
then the cohomological equation has a formal solution if and only if all where is the distribution such that
Indeed, let , and
Then
The matrix acts on . It has a family of orbits and orbits
, for each . splits accordingly, so the cohomological equation can be solved independently in each summand
and on
For , acts like the circle rotation , so we already understand the necessary condition, given by invariant measures.
For fixed and , let us denote
The cohomological equation reads
where
Recursively, one gets
Then
Similarly, the equation
yields
Combining both leads to
i.e. .
Conversely, one can see that this conditions are sufficient for existence of a formal solution, and for existence of smooth solutions under Diophantine conditions.
11.5. More general results on cohomological equations in parabolic dynamics
- The case of Heisenberg nilflows, which we just treated, is due to Katok.
- More general nilflows have been studied by Flaminio-Forni, as well as horocycle flows.
- Linear flows on higher genus surfaces are due to Forni. In this case, one gets distributions . If a smooth function is killed by all of them, ergodic integrals stay bounded. This is equivalent to being a coboundary, according to Gottschalk-Hedlund.
11.6. Cocycle effectiveness
Sometimes, the cohomological equation with measurable data is needed. It is much more difficult, but some miracle occurs in the parabolic setting.
Definition 16 Given a map , say a function on is a measurable coboundary of the exists a measurable such that .
The theorem on mixing smooth time-changes of Heisenberg nilflows (Avila-Forni-Ulcigrai) required the roof not to be a measurable coboundary. It turns out that here, this is equivalent to not being a smooth coboundary.
Proposition 17 Let us study the Furstenberg skew-product on the -torus. Let be a smooth function on . Then
is not a smooth coboundary is not a measurable coboundary.
Indeed, let
denote the Birkhoff sums. Then Flaminio-Forni establish quadratic upper bounds: there exists a sequence such that
Matching lower bounds exist: if is not a smooth coboundary, there exists a nonvanishing , and
If is a measurable coboundary, , then
stays bounded on a set of almost full measure, and grows at most quadratically on the complement. This contradicts the quadratic lower bound.
11.7. More general nilflows
Theorem 18 (Avila-Forni-Ravotti-Ulcigrai) For general nilflows of step , there exists a dense (in ) class of generators of time-changes, which are
- either measurably trivial (i.e. measurably conjugate to the nilflow);
- or mixing.
Unfortunately, the set is not explicitly describable like in the Heisenberg case.
The proof is an induction on central extensions. It uses mixing by shearing.
12. More examples of parabolic dynamics
12.1. Parabolic perturbations which are not time-changes
These were discovered by Ravotti during his PhD at Princeton.
Here, parabolic means that the derivative of the flow grows polynomially. It implies that smooth time-changes are still parabolic.
Start with , a cocompact lattice, . Let be the flow generated by a unipotent element . Let belong to the center of the minimal unipotent and such that .
Let where is a function on such that . Let denote the corresponding flow.
Theorem 19 If preserves a smooth measure with density, then is
- parabolic: ;
- ergodic;
- mixing.
Remark. Existence of there exists a time-change of the -flow which commutes with .
Remark. There exists such which are not smoothly isomorphic to . This follows from the failure of cocycle rigidity for parabolic actions, due to Wang, following many people.
Remark. Ergodicity needs be proven, it does not follow from general principles.
The proof relies on mixing by shearing, although in a setting different from what we have already met. Consider arcs of orbits of and push them by . Since
where is an ergodic integral for .
12.2. What else can shearing be used for?
A strong, quantitative, shearing can be used to establish spectral results. Here, I mean the spectrum of the Koopman operator
To each , there corresponds a spectral measure on . Its Fourier coefficients are given by selfcorrelations of , i.e.
The spectrum of is absolutely continuous for all , is absolutely continuous for all , .
If is ergodic, it is enough to study functions which are smooth coboundaries.
Theorem 20 (Forni-Ulcigrai) Smooth time-changes of a horocycle flow has absolutely continuous spectrum.
The proof uses quantitative bounds
Since , this implies absolute continuity.
Fayad-Forni-Kanigowski consider smooth area-preserving flows on the -torus with a stopping point.
12.3. Ratner property
M. Ratner uses a quantitative form of shearing for unipotent flows.
Shearing takes some time. Ratner requires the following:
For all , for all large enough , there exists a set of measure , for all pairs not in the same orbit, but such that , there exists such that
and
for all .
For a long time, this was used only in algebraic dynamics, until a more flexible variant, called switchability, was introduced. It means that one can switch past and future.
Fayad-Kanigowski and Kanigowski-Kulaga-Ulcigrai established this variant for typical smooth area-preserving flows on surfaces. This implies mixing of all orders.
Here is another application if these ideas:
Theorem 21 (Kanigowski-Lemanczyk-Ulcigrai) For all smooth time-changes of the horocycle flow, the rescaled flow is not isomorphic to .
They are actually disjoint in Furstenberg’s sense. Recall that the horocycle flow itself is isomorphic to its rescalings.
We use a disjointness criterion based on this switchable variant of Ratner’s property.
12.4. Summary
- Parabolic means slow butterfly effect.
- Typically slow mixing.
- Slow equidistribution.
- Disjointness of rescalings.
- Obstructions to cohomological equation.
Tools:
- Shearing.
- Ratner property and switchability.
- Renormalization.