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JMD

The editors of the Journal of Modern Dynamics are happy to dedicate this issue to Gregory Margulis, who, over the last four decades, has inﬂuenced dynamical systems as
deeply as few others have, and who has blazed broad trails in the application of dynamical systems to other ﬁelds of core mathematics.

For more information please click the “Full Text” above.

Additional editors: Leonid Polterovich, Ralf Spatzier, Amie Wilkinson and Anton Zorich.

For more information please click the “Full Text” above.

Additional editors: Leonid Polterovich, Ralf Spatzier, Amie Wilkinson and Anton Zorich.

keywords:

JMD

This text is an expanded version of the lecture notes of a minicourse (with the same title of this text) delivered by the authors in the Będlewo school ``Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory'' (from 4 to 16 July, 2011).

In the first part of this text, i.e., from Sections 1 to 5, we discuss the Teichmüller and moduli space of translation surfaces, the Teichmüller flow and the $SL(2,\mathbb{R})$-action on these moduli spaces and the Kontsevich--Zorich cocycle over the Teichmüller geodesic flow. We sketch two applications of the ergodic properties of the Teichmüller flow and Kontsevich--Zorich cocycle, with respect to Masur--Veech measures, to the unique ergodicity, deviation of ergodic averages and weak mixing properties of typical interval exchange transformations and translation flows. These applications are based on the fundamental fact that the Teichmüller flow and the Kontsevich--Zorich cocycle work as

In the second part, i.e., from Sections 6 to 9, we start by pointing out that it is interesting to study the ergodic properties of the Kontsevich--Zorich cocycle with respect to invariant measures other than the Masur--Veech ones, in view of potential applications to the investigation of billiards in rational polygons (for instance). We then study some examples of measures for which the ergodic properties of the Kontsevich--Zorich cocycle are very different from the case of Masur--Veech measures. Finally, we end these notes by constructing some examples of closed $SL(2,\mathbb{R})$-orbits such that the restriction of the Teichmüller flow to them has arbitrary small rate of exponential mixing, or, equivalently, the naturally associated unitary $SL(2,\mathbb{R})$-representation has arbitrarily small spectral gap (and in particular it has complementary series).

In the first part of this text, i.e., from Sections 1 to 5, we discuss the Teichmüller and moduli space of translation surfaces, the Teichmüller flow and the $SL(2,\mathbb{R})$-action on these moduli spaces and the Kontsevich--Zorich cocycle over the Teichmüller geodesic flow. We sketch two applications of the ergodic properties of the Teichmüller flow and Kontsevich--Zorich cocycle, with respect to Masur--Veech measures, to the unique ergodicity, deviation of ergodic averages and weak mixing properties of typical interval exchange transformations and translation flows. These applications are based on the fundamental fact that the Teichmüller flow and the Kontsevich--Zorich cocycle work as

*renormalization dynamics*for interval exchange transformations and translation flows.In the second part, i.e., from Sections 6 to 9, we start by pointing out that it is interesting to study the ergodic properties of the Kontsevich--Zorich cocycle with respect to invariant measures other than the Masur--Veech ones, in view of potential applications to the investigation of billiards in rational polygons (for instance). We then study some examples of measures for which the ergodic properties of the Kontsevich--Zorich cocycle are very different from the case of Masur--Veech measures. Finally, we end these notes by constructing some examples of closed $SL(2,\mathbb{R})$-orbits such that the restriction of the Teichmüller flow to them has arbitrary small rate of exponential mixing, or, equivalently, the naturally associated unitary $SL(2,\mathbb{R})$-representation has arbitrarily small spectral gap (and in particular it has complementary series).

JMD

We review the Brin prize work of Artur Avila on Teichmüller dynamics
and Interval Exchange Transformations. The paper is a nontechnical
self-contained summary that intends to shed some light on Avila's
early approach to the subject and on the significance of his
achievements.

JMD

We establish a geometric criterion on a $SL(2, R)$-invariant ergodic probability measure
on the moduli space of holomorphic abelian differentials on Riemann surfaces
for the nonuniform hyperbolicity of the Kontsevich--Zorich cocycle on the real
Hodge bundle. Applications include measures supported on the $SL(2, R)$-orbits of
all algebraically primitive Veech surfaces (see also [7]) and
of all Prym eigenforms discovered in [34], as well as all canonical
absolutely continuous measures on connected components of strata of the moduli
space of abelian differentials (see also [4, 17]). The argument
simplifies and generalizes our proof for the case of canonical measures
[17]. In the Appendix, Carlos Matheus discusses several relevant
examples which further illustrate the power and the limitations of our
criterion.

JMD

Let $X$ be a vector field on a compact connected manifold $M$. An important question in dynamical systems is to know when a function $g: M\to \mathbb{R}$ is a coboundary for the flow generated by $X$, i.e., when there exists a function $f: M\to \mathbb{R}$ such that $Xf=g$. In this article we investigate this question for nilflows on nilmanifolds. We show that there exists countably many independent Schwartz distributions $D_n$ such that any sufficiently smooth function $g$ is a coboundary iff it belongs to the kernel of all the distributions $D_n$.

ERA-MS

JMD

We consider smooth time-changes of the classical horocycle flows on
the unit tangent bundle of a compact hyperbolic surface and prove
sharp bounds on the rate of equidistribution and the rate of
mixing. We then derive results on the spectrum of smooth time-changes
and show that the spectrum is absolutely continuous with respect to
the Lebesgue measure on the real line and that the maximal spectral
type is equivalent to Lebesgue.

JMD

A cyclic cover of the complex projective line branched at four
appropriate points has a natural structure of a square-tiled surface.
We describe the combinatorics of such a square-tiled surface, the
geometry of the corresponding Teichmüller curve, and compute the
Lyapunov exponents of the determinant bundle over the Teichmüller
curve with respect to the geodesic flow. This paper includes a new
example (announced by G. Forni and C. Matheus in [17]
of a Teichmüller curve of a square-tiled cyclic cover in a stratum
of Abelian differentials in genus four with a maximally degenerate
Kontsevich--Zorich spectrum (the only known example in genus three
found previously by Forni also corresponds to a square-tiled cyclic
cover [15].
We present several new examples of Teichmüller curves in
strata of holomorphic and meromorphic quadratic differentials with
a maximally degenerate Kontsevich--Zorich spectrum.
Presumably, these examples cover all possible Teichmüller curves
with maximally degenerate spectra. We prove that this is indeed the case
within the class of square-tiled cyclic covers.

JMD

We prove that every homogeneous flow on a finite-volume homogeneous manifold has countably many independent invariant distributions unless it is conjugate to a linear flow on a torus. We also prove that the same conclusion holds for every affine transformation of a homogenous space which is not conjugate to a toral translation. As a part of the proof, we have that any smooth partially hyperbolic flow on any compact manifold has countably many distinct minimal sets, hence countably many distinct ergodic probability measures. As a consequence, the Katok and Greenfield-Wallach conjectures hold in all of the above cases.

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