# American Institute of Mathematical Sciences

ISSN:
1930-5311

eISSN:
1930-532X

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## Journal of Modern Dynamics

July & October 2014 , Volume 8 , Issue 3&4

Special Issue: Proceedings of the 2008 and 2011 Summer Institutes at the Mathematics Research and Conference Center of Polish Academy of Sciences at Będlewo, Poland

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2014, 8(3&4): i-i doi: 10.3934/jmd.2014.8.3i +[Abstract](1191) +[PDF](51.8KB)
Abstract:
This special issue presents some of the lecture notes of the courses held in the 2008 and 2011 Summer Institutes at the Mathematics Research and Conference Center of Polish Academy of Sciences at Będlewo, Poland. The school was structured as daily courses with a double lecture each, in two parts of 45-50 minutes with a break in between.

2014, 8(3&4): 271-436 doi: 10.3934/jmd.2014.8.271 +[Abstract](2296) +[PDF](6418.1KB)
Abstract:
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).
2014, 8(3&4): 437-497 doi: 10.3934/jmd.2014.8.437 +[Abstract](1853) +[PDF](442.7KB)
Abstract:
These notes are based on lectures delivered in the summer school Modern Dynamics and its Interaction with Analysis, Geometry and Number Theory'', held in Będlewo, Poland, in the summer of 2011. The course is an exposition of Furstenberg's conjectures on transversality'' of the maps $x\rightarrow ax$mod1 and $x\mapsto bx$mod1 for multiplicatively independent integers $a,b$, and of the associated problems on intersections and sums of invariant sets for these maps. The first part of the course is a short introduction to fractal geometry. The second part develops the theory of Furstenberg's CP-chains and local entropy averages, ending in proofs of the sumset problem and of the known case of the intersections conjecture.
2014, 8(3&4): 499-548 doi: 10.3934/jmd.2014.8.499 +[Abstract](1653) +[PDF](432.7KB)
Abstract:
We prove that the Julia set of a Hénon type automorphism on $\mathbb{C}^2$ is very rigid: it supports a unique positive $dd^c$-closed current of mass 1. A similar property holds for the cohomology class of the Green current associated with an automorphism of positive entropy on a compact Kähler surface. Relations between this phenomenon, several quantitative equidistribution properties and the theory of value distribution will be discussed. We also survey some rigidity properties of Hénon type maps on $\mathbb{C}^k$ and of automorphisms of compact Kähler manifolds.
2014, 8(3&4): 549-576 doi: 10.3934/jmd.2014.8.549 +[Abstract](1738) +[PDF](272.2KB)
Abstract:
In this survey, we discuss the problem of removing zero Lyapunov exponents of smooth invariant measures along the center direction of a partially hyperbolic diffeomorphism and various related questions. In particular, we discuss disintegration of a smooth invariant measure along the center foliation. We also simplify the proofs of some known results and include new questions and conjectures.

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