Dmitry Dolgopyat Giovanni Forni Rostislav Grigorchuk Boris Hasselblatt Anatole Katok Svetlana Katok Dmitry Kleinbock Raphaël Krikorian Jens Marklof
Journal of Modern Dynamics 2008, 2(1): i-v doi: 10.3934/jmd.2008.2.1i
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.

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Additional editors: Leonid Polterovich, Ralf Spatzier, Amie Wilkinson and Anton Zorich.
Introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards
Giovanni Forni Carlos Matheus
Journal of Modern Dynamics 2014, 8(3&4): 271-436 doi: 10.3934/jmd.2014.8.271
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 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).
keywords: \mathbb{R})$-action on moduli spaces Moduli spaces Kontsevich–Zorich cocycle Abelian differentials translation surfaces Teichmüller flow $SL(2 Lyapunov exponents.
On the Brin Prize work of Artur Avila in Teichmüller dynamics and interval-exchange transformations
Giovanni Forni
Journal of Modern Dynamics 2012, 6(2): 139-182 doi: 10.3934/jmd.2012.6.139
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.
keywords: Brin prize Avila.
A geometric criterion for the nonuniform hyperbolicity of the Kontsevich--Zorich cocycle
Giovanni Forni
Journal of Modern Dynamics 2011, 5(2): 355-395 doi: 10.3934/jmd.2011.5.355
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.
keywords: Teichmüller geodesic flow Kontsevich--Zorich cocycle non-uniform hyperbolicity. moduli space of abelian differentials
On the cohomological equation for nilflows
Livio Flaminio Giovanni Forni
Journal of Modern Dynamics 2007, 1(1): 37-60 doi: 10.3934/jmd.2007.1.37
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 $D_n$.
keywords: Nilflows Cohomological Equations.
The cohomological equation for area-preserving flows on compact surfaces
Giovanni Forni
Electronic Research Announcements 1995, 1(0): 114-123
keywords: higher genus surfaces. Cohomological equation area-preserving flows
Time-changes of horocycle flows
Giovanni Forni Corinna Ulcigrai
Journal of Modern Dynamics 2012, 6(2): 251-273 doi: 10.3934/jmd.2012.6.251
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.
keywords: Time-changes horocycle flows quantitative mixing spectral theory. quantitative equidistribution
Square-tiled cyclic covers
Giovanni Forni Carlos Matheus Anton Zorich
Journal of Modern Dynamics 2011, 5(2): 285-318 doi: 10.3934/jmd.2011.5.285
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.
keywords: Kontsevich--Zorich cocycle Teichmüller geodesic flow square-tiled surfaces.
Invariant distributions for homogeneous flows and affine transformations
Livio Flaminio Giovanni Forni Federico Rodriguez Hertz
Journal of Modern Dynamics 2016, 10(02): 33-79 doi: 10.3934/jmd.2016.10.33
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.
keywords: Cohomological equations homogeneous flows.

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