Recent advances and new perspectives

**Organized by:** G. Forni, K. Fraczek, C. Ulcigrai, B. Weiss

Renormalization of one-frequency cocycles with values on compact Lie groups

Spectral disjointness of rescaling of some surface flows

We are interested in a relation of some surface flows with their rational rescalings. In a recent paper, Kanigowski, Lemanczyk and Ulcigrai proved that different real rescalings of a special flow built over rotation and under roof function with asymmetric logarithmic singularities are disjoint in the sense of Furstenberg. We proof analogous result for roof function with symmetric singularities and only for rational rescalings, we replace however Furstenberg disjointness by spectral disjointness. While Kanigowski-Lemanczyk-Ulcigrai rely on properties of Ratner type, we exploit a different approach. Moreover, by using similar methods, we show that special flows built over IETs and under piecewise linear functions (with both zero and non-zero slope) satisfy identical property. It is partially generalization of results by Frączek and Lemanczyk. All mentioned special flows arise naturally as special representations of flows on surfaces. This is joint work with Adam Kanigowski.A prime transformation with many and big self-joinings.

**Abstract:** Let (X,mu,T) be a measure preserving system. A factor is a system (Y,nu,S) so
that there exists F with SF=FT and so that F pushes mu forward to nu. A measurable
dynamical system is prime if it has no non-trivial factors. A classical way to prove a system
is prime is to show it has few self-joinings, that is, few T x T invariant measures that on X
x X that project to mu. We show that there exists a prime transformation that has many
self-joinings which are also large. In particular, its ergodic self-joinings are dense in its self-
joinings and it has a self-joining that is not a distal extension of itself. As a consequence
we show that being quasi-distal is a meager property in the set of measure preserving
transformations, which answers a question of Danilenko. This is joint work with Bryna Kra.

Weak-mixing for translation flows

**Abstract:** Translation flows are examples of (singular) zero entropy flow on surfaces. I will
review the known results about weak-mixing (Avila-Forni for the generic surfaces and Avila-Delecroix
for Veech surfaces) and explain our more recent work on the so-called "rank one loci" (Aulicino-Avila-
Delecroix).

Countable Lebesgue spectrum for conservative surface flows and other zero entropy non-algebraic dynamical system

**Abstract:** We study the spectral measures of conservative mixing flows on the two torus
having one degenerate singularity. We show that, for a sufficiently strong singularity, the
spectrum of these flows is typically Lebesgue with infinite multiplicity.
For this, we use two main ingredients : 1) a proof of absolute continuity of the maximal
spectral type for this class of non-uniformly stretching flows that have an irregular decay of
correlations, 2) a geometric criterion that yields infinite Lebesgue multiplicity of the
spectrum and that is well adapted to rapidly mixing flows.

Joinings of smooth time changes of horocycle flows

**Abstract:** Recently Kanigowski, Lemańczyk and Ulcigrai proved that if distinct powers of a
time-change of the horocycle flow have a non-trivial joining, then the time change function
is cohomologous to a constant. Their results holds for a class of time-changes of infinite
codimension in the space of smooth time changes. In a joint work with Giovanni Forni we
show the same conclusion holds for all sufficiently smooth time changes the horocycle
flow.

Bernoulli property for some partially hyperbolic systems with parabolic center.

**Abstract:** We will discuss the Bernoulli property for some partially hyperbolic systems with
(non-trivial) parabolic center.
Two main examples are positive entropy homogeneous systems and skew-products with
hyperbolic base and parabolic fiber.

Lengths spectrum of mapping class groups.

**Abstract:** The mapping class group of surfaces is a very rich object, directly related to the
geometry of the moduli space. Pseudo-Anosov homeomorphisms give a way to
understand this geometry. We will explain different constructions permitting to obtain (all)
pseudo-Anosov homeomorphisms. As an application, this allows us to determine the
systole of the hyperelliptic connected components (joint work with C. Boissy), as well as
the systole of any Teichmueller curves in genus two (joint work with V. Delecroix).

On Ratner’s property of special flows over irrational rotations

**Abstract:** A special way of divergence of orbits of nearby points, now called Ratner’s
property, was discovered by Marina Ratner in the 1980th in case of horocycle flows. Two
decades later Ratner's property turned out to be present also in dimension 2, for some
special flows over irrational rotations. I will focus on some examples and measure-
theoretic consequences of this property: mixing properties, centralizer, (possible)
disjointness of time automorphisms and some consequences. Talk is based on my joint
works with J.-P. Conze, K. Fraczek, A. Kanigowski, E. Lesigne and C. Ulcigrai.

On measures invariant under the horospherical flow for geometrically infinite manifolds

**Abstract:** We consider a locally finite (Radon) measure mu on SO(n,1)/Gamma invariant
under a horospherical subgroup of SO(n,1) where Gamma is a discrete, but not
necessarily geometrically finite, subgroup. Following Ledrappier and Sarig, who
considered the case of SO(2,1)=PGL(2,R), we show that such a measure is either highly
degenerate, or has additional invariance properties. We relate these invariance properties
to the behaviour of the trajectory of a mu-typical point under the corresponding contracting
1-parameter diagonalizable flow. One ingredient in our proof is Hochman's ratio ergodic
thoerem for actions of R^d. Joint work with Or Landesberg.

Dynamics of unipotent frame flows on hyperbolic manifolds.

**Abstract:** Joint work with B. Schapira. We give another proof of a theorem of A.
Mohammadi and H. Oh about the ergodicity of the Burger-Roblin measure for kleinian
groups of high enough critical exponents, and relate it with a topological counterpart.

Asymptotic formulas on infinite periodic translation surfaces.

**Abstract:** The Gauss circle problem consists in counting the number of integer points
of bounded length in the plane. This problem is equivalent to counting the number of
closed geodesics of bounded length on a flat two dimensional torus.

Many counting problems in dynamical systems have been inspired by this problem. For 30 years, the experts try to understand the asymptotic behavior of closed geodesics in translation surfaces and periodic trajectories on rational billiards. (Polygonal billiards yield translation surfaces naturally through an unfolding procedure.) H. Masur proved that this number has quadratic growth rate.

In this talk, we will study the counting problem on infinite periodic rational billiards and translation surfaces.

The first example and motivation is the wind-tree model, a Z^2-periodic billiard model. In the classical setting, we place identical rectangular obstacles in the plane at each integer point; we play billiard on the complement.

It is possible to give quite precise results on the counting problem for this model, thanks to the many symmetries it presents. These results, however, do not extend to more general contexts.

I will present a general result on the counting problem for infinite periodic translation surfaces that uses new ideas: a dynamical analogous, for the algebraic hull of a cocycle, to strong and super-strong approximation on algebraic groups. Under these approximation hypothesis I will exhibit asymptotic formulas for the number of closed geodesics of bounded length on infinite periodic translation surfaces. And will present some applications and discuss why I think these hypothesis hold in general (work in progress).

On the proof of the Tree Conjecture for triangle tiling billiards

**Abstract:** Tiling billiards are a class of dynamical systems that model movement of light in
heterogeneous media. The rule for a tiling billiard is the following: in a given tiling on a
plane, a billiard trajectory moves in a straight segment in each tile and each time when it
crosses a border between two tiles, it refracts with a refraction coefficient equal to -1.

This definition may seem quite extravagant from the point of view of a physicist. Although, tiling billiards exhibit rich combinatorial behaviour from the point of view of a mathematician. In this talk, we concentrate ourselves on the dynamics of a tiling billiard in a periodic triangle tiling obtained from a standard equilateral triangle tiling by a linear transformation. In such a dynamical system, called a triangle tiling billiard, almost any trajectory is either a simple closed curve, either linearly escapes. There exists a zero measure set of exceptional trajectories of triangle tiling billiards that escape to infinity in a non-linear manner. Although the exceptional trajectories of such billiards could be a hero of our talk (because of their connection to Arnoux-Rauzy family of interval exchange transformations, and to the Rauzy gasket), the main heroes this time would be ordinary trajectories, and their common feature which is reflected in the Tree Conjecture.

The Tree Conjecture, formulated by Baird-Smith, Davis, Fromm and Iyer, states that any periodic trajectory of a triangle tiling billiard doesn't contour triangles. It means that any periodic trajectory contours a graph in a tiling which is a tree. We will share the proof of this Conjecture that we have found recently and the objects that we discovered along the way.

Degenerating directions on infinite translation surfaces

**Abstract:** This talk is intended to be an advertisement for an open question on infinite translation
surfaces. Whereas a finite translation surface can be described by finitely many polygons
that are glued along edges which are parallel and have the same length, an infinite
translation surface is glued from infinitely many polygons.

One of the open questions about infinite translation surfaces is the following: When we apply the Teichmüller flow (stretching in one direction and shrinking in a transversal direction) to an infinite translation surface, can it happen that its diameter goes to 0? In my talk, I will introduce first some basics about infinite translation surfaces, then I will explain how the question comes up, what wild singularities have to do with it, and why the answer should be "yes". In doing this, I will mention works with W. Patrick Hooper, Howard Masur, and Kasra Rafi.

Parabolic perturbations of unipotent flows

**Abstract:** A perturbation of a unipotent flow produces, in general, a hyperbolic flow, hence
the question of describing which perturbations of a unipotent flow exhibit a parabolic
behaviour appears to be delicate. The simplest family of smooth perturbations which
preserve parabolicity consists of time-changes: they are obtained by adding a (non-
constant) component in the same direction of the original vector field.

In this talk, we consider a new class of smooth perturbations of any unipotent flow on compact quotients of SL(3, R); they are obtained by adding a non-constant component in a transverse commuting direction. We prove that these perturbations are parabo, ergodic, and, in fact, mixing. Moreover, we relate non-trivial perturbations to the first cohomology of a certain parabolic Abelian action.

Expanding translates of shrinking curves on homogenous spaces

Via proving a result on equidistribution of limits of expanding translates of analytic curves on the space of unimodular lattices in R^{n+1}, it was shown that for any analytic curve on R^n which is not contained in a proper affine subspace, the Dirichlet-Minkowski theorem on simultaneous approximation cannot be improved for almost all points on this curve. Recently, in a joint work with Pengyu Yang, we proved a new result on equidistribution of uniformly, as well as non-uniformly, expanding translates of optimally shrinking pieces of curves in homogenous spaces of SL(n,R). Using this we prove the Dirichlet-Minknowski non-improvability for almost all points on ‘regular’ smooth curves. At the heart of such results lies Ratner’s description of invariant measures for unipotent flows, and the new ingredients involve observations on interactive linear dynamics of various SL(2,R)-actions.

Cohomological equations for linear involutions

**Abstract:** The famous Roth theorem about diophantine approximations states that a given
algebraic number may not have too many rational number approximations, that are "very
good". More precisely, Roth first defined a class of numbers that are not very easy to
approximate by rationals (they are called Roth numbers) and then showed that almost all
algebraic irrationals are of Roth type, and that they form a set of a full measure which is
invariant under the natural action of the modular group SL(2,Z).

In addition to their interesting arithmetical properties, Roth type irrationals appear in a study of the cohomological equation associated with a rotation R_a : R_a(x) = x+a of the circle T=R/Z: a is of Roth type if and only iff for all r,s : r>s+1>1 and for all functions g of class C^r on T with zero mean there exists a unique function f in C^s(T) with zero mean such that f-f R_a = f.

In 2005 Marmi, Moussa and Yoccoz established an analogue of Roth theorem for interval exchange transformations (IETs). In particular, they defined the notion of Roth type IETs and proved existence of the solution of cohomological equation for this class; they also showed that IET of Roth type form a full measure set in the parameter space of IETs. In a fresh joint work with Erwan Lanneau and Stefano Marmi we get a certain generalization of this result for linear involutions that can be considered as a natural extension of IETs to non-orientable case.

Dynamics on strata versus dynamics on homogeneous spaces.

**Abstract:** A fundamental challenge in ``modular dynamics” is to compare the behavior
of the horocycle flow on strata to the behavior of unipotent flows on homogeneous
spaces. We would like to compare nearby orbit behavior in both cases and
ask how this behavior influences the topological dynamics and the ergodic theory
of the corresponding flows. In the homogeneous case this problem was solved by
Ratner but in the stratum case many questions remain. I will describe some recent
joint work with Jon Chaika and Barak Weiss and some older work with Barak Weiss
and explain how it is connected to this question.

An effective equidistribution result for SL(2,R)⋉(R^{2})^{⊕k}

**Abstract:** Let G=SL(2,R)⋉(R^{2})^{⊕k} . We prove a polynomially effective asymptotic
equidistribution result for special types of unipotent orbits in Gamma\G which project to
pieces of closed horocycles in SL(2,Z)\ SL(2,R). As an application, we prove an effective
quantitative Oppenheim type result for the quadratic form (m1-a)^2+(m2-b)^2-(m3-a)^2-(m4-b)^2 for (a,b) in R^2 of
Diophantine type, following an approach by Jens Marklof (Ann of Math 158 (2003), 419-471)
using theta sums. This is joint work with Pankaj Vishe.

Effective equidistribution for multiplicative Diophantine approximation on planar lines.

**Abstract:** We will study multiplicative Diophantine approximation property of typical points
on planar lines. We will show that for any planar line, a strengthening of Littlewood
conjecture holds for almost every point on the line. The strengthening is sharp for typical
points on the whole space. This is done by establishing an effective equidistribution result
for certain one-parameter unipotent orbits in SL(3,R)/SL(3,Z). This is a joint work with Sam
Chow.

Kakutani equivalence of unipotent flows

**Abstract:** We study Kakutani equivalence in the class of unipotent flows acting on finite
volume quotients of semisimple Lie groups. For every such ow we compute the Kakutani
invariant of M. Ratner, the value of which being explicitly given by the Jordan block
structure of the unipotent element generating the flow. This, in particular, answers a
question of M. Ratner. Moreover, it follows that the only standard unipotent ows are given by (1 t & 0 1) × id acting on (SL(2, R)×G’))/Γ’, where Γ’ is an irreducible lattice in SL(2,
R)×G’ (with the possibility that G’ = {e}). This is a joint work with Adam Kanigowski and
Kurt Vinhage.