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Journal of Modern Dynamics

Open Access Articles

Federico Rodriguez Hertz
2014, 8(3/4): i-i doi: 10.3934/jmd.2014.8.3i +[Abstract](74) +[PDF](51.8KB)
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.

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On the Brin Prize work of Artur Avila in Teichmüller dynamics and interval-exchange transformations
Giovanni Forni
2012, 6(2): 139-182 doi: 10.3934/jmd.2012.6.139 +[Abstract](137) +[PDF](366.9KB)
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.
Forty years of unimodal dynamics: On the occasion of Artur Avila winning the Brin Prize
Mikhail Lyubich
2012, 6(2): 183-203 doi: 10.3934/jmd.2012.6.183 +[Abstract](123) +[PDF](307.2KB)
The field of one-dimensional dynamics, real and complex, emerged from obscurity in the 1970s and has been intensely explored ever since. It combines the depth and complexity of chaotic phenomena with a chance to fully understand it in probabilistic terms: to describe the dynamics of typical orbits for typical maps. It also revealed fascinating universality features that had never been noticed before. The interplay between real and complex worlds illuminated by beautiful pictures of fractal structures adds special charm to the field. By now, we have reached a full probabilistic understanding of real analytic unimodal dynamics, and Artur Avila has been the key player in the final stage of the story (which roughly started with the new century). To put his work into perspective, we will begin with an overview of the main events in the field from the 1970s up to the end of the last century. Then we will describe Avila's work on unimodal dynamics that effectively closed up the field. We will finish by describing his results in the closely related direction, the geometry of Feigenbaum Julia sets, including a recent construction of a new class of Julia sets of positive area.
The 2011 Michael Brin Prize in Dynamical Systems
The Editors
2012, 6(2): i-ii doi: 10.3934/jmd.2012.6.2i +[Abstract](79) +[PDF](4673.4KB)
Professor Michael Brin of the University of Maryland endowed an international prize for outstanding work in the theory of dynamical systems and related areas. The prize is given biennially for specific mathematical achievements that appear as a single publication or a series thereof in refereed journals, proceedings or monographs.

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Counting closed geodesics in moduli space
Alex Eskin and Maryam Mirzakhani
2011, 5(1): 71-105 doi: 10.3934/jmd.2011.5.71 +[Abstract](178) +[PDF](396.8KB)
We compute the asymptotics, as $R$ tends to infinity, of the number $N(R)$ of closed geodesics of length at most $R$ in the moduli space of compact Riemann surfaces of genus $g$. In fact, $N(R)$ is the number of conjugacy classes of pseudo-Anosov elements of the mapping class group of a compact surface of genus $g$ of translation length at most $R$.
Density of positive Lyapunov exponents for quasiperiodic SL(2, R)-cocycles in arbitrary dimension
Artur Avila
2009, 3(4): 631-636 doi: 10.3934/jmd.2009.3.631 +[Abstract](108) +[PDF](87.4KB)
We show that given a fixed irrational rotation of the $d$-dimensional torus, any analytic SL(2, R)-cocycle can be perturbed in such a way that the Lyapunov exponent becomes positive. This result strengthens and generalizes previous results of Krikorian [6] and Fayad-Krikorian [5]. The key technique is the analyticity of $m$-functions (under the hypothesis of stability of zero Lyapunov exponents), first observed and used in the solution of the Ten-Martini Problem [2].
Uniform exponential growth for some SL(2, R) matrix products
Artur Avila and Thomas Roblin
2009, 3(4): 549-554 doi: 10.3934/jmd.2009.3.549 +[Abstract](94) +[PDF](89.0KB)
Given a hyperbolic matrix $H\in SL(2,\R)$, we prove that for almost every $R\in SL(2,\R)$, any product of length $n$ of $H$ and $R$ grows exponentially fast with $n$ provided the matrix $R$ occurs less than $o(\frac{n}{\log n\log\log n})$ times.
On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method
Manfred Einsiedler and Elon Lindenstrauss
2008, 2(1): 83-128 doi: 10.3934/jmd.2008.2.83 +[Abstract](100) +[PDF](523.7KB)
We consider measures on locally homogeneous spaces $\Gamma \backslash G$ which are invariant and have positive entropy with respect to the action of a single diagonalizable element $a \in G$ by translations, and prove a rigidity statement regarding a certain type of measurable factors of this action.
    This rigidity theorem, which is a generalized and more conceptual form of the low entropy method of [14,3] is used to classify positive entropy measures invariant under a one parameter group with an additional recurrence condition for $G=G_1 \times G_2$ with $G_1$ a rank one algebraic group. Further applications of this rigidity statement will appear in forthcoming papers.
Dmitry Dolgopyat, Giovanni Forni, Rostislav Grigorchuk, Boris Hasselblatt, Anatole Katok, Svetlana Katok, Dmitry Kleinbock, Raphaël Krikorian and Jens Marklof
2008, 2(1): i-v doi: 10.3934/jmd.2008.2.1i +[Abstract](85) +[PDF](723.1KB)
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.

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