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Journal of Modern Dynamics

2010 , Volume 4 , Issue 2

A special issue
CHiLE 2009

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The 2009 Michael Brin Prize in Dynamical Systems
The Editors
2010, 4(2): i-ii doi: 10.3934/jmd.2010.4.2i +[Abstract](30) +[PDF](740.0KB)
Professor Michael Brin of the University of Maryland endowed an international prize for outstandingwork 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.
   The prize recognizes mathematicians who have made substantial impact in the field at an early stage of their careers.
   The prize is awarded by an international committee of experts chaired by Anatole Katok. Its members are Jean Bourgain, John N. Mather, Yakov Pesin, Marina Ratner, Marcelo Viana and BenjaminWeiss.

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On the work and vision of Dmitry Dolgopyat
Carlangelo Liverani
2010, 4(2): 211-225 doi: 10.3934/jmd.2010.4.211 +[Abstract](51) +[PDF](200.6KB)
We present some of the results and techniques due to Dolgopyat. The presentation avoids technicalities as much as possible while trying to focus on the basic ideas. We also try to present Dolgopyat's work in the context of a research program aimed at enlightening the relations between dynamical systems and nonequilibrium statistical mechanics.
On the work of Dolgopyat on partial and nonuniform hyperbolicity
Yakov Pesin
2010, 4(2): 227-241 doi: 10.3934/jmd.2010.4.227 +[Abstract](51) +[PDF](187.5KB)
This paper is a nontechnical survey and aims to illustrate Dolgopyat's profound contributions to smooth ergodic theory. I will discuss some of Dolgopyat's work on partial hyperbolicity and nonuniform hyperbolicity with emphasis on the interaction between the two-the class of dynamical systems with mixed hyperbolicity. On one hand, this includes uniformly partially hyperbolic diffeomorphisms with nonzero Lyapunov exponents in the center direction. The study of their ergodic properties has provided an alternative approach to the Pugh-Shub stable ergodicity theory for both conservative and dissipative systems. On the other hand, ideas of mixed hyperbolicity have been used in constructing volume-preserving diffeomorphisms with nonzero Lyapunov exponents on any manifold.
The work of Dmitry Dolgopyat on physical models with moving particles
Nikolai Chernov
2010, 4(2): 243-255 doi: 10.3934/jmd.2010.4.243 +[Abstract](31) +[PDF](157.2KB)
D. Dolgopyat is the winner of the second Brin Prize in Dynamical Systems (2009). This article overviews his remarkable achievements in a nontechnical manner. It complements two other surveys of Dolgopyat's work written by Y. Pesin and C. Liverani and published in this issue. This survey covers Dolgopyat's work on various physical models, including the Lorentz gas, Galton board, and some systems of hard disks.
Generating product systems
Nir Avni and  Benjamin Weiss
2010, 4(2): 257-270 doi: 10.3934/jmd.2010.4.257 +[Abstract](39) +[PDF](170.7KB)
Generalizing Krieger's finite generation theorem, we give conditions for an ergodic system to be generated by a pair of partitions, each required to be measurable with respect to a given subalgebra, and also required to have a fixed size.
Local rigidity of partially hyperbolic actions
Zhenqi Jenny Wang
2010, 4(2): 271-327 doi: 10.3934/jmd.2010.4.271 +[Abstract](35) +[PDF](547.4KB)
We consider partially hyperbolic abelian algebraic high-rank actions on compact homogeneous spaces obtained from simple indefinite orthogonal and unitary groups. In the first part of the paper, we show local differentiable rigidity for such actions. The conclusions are based on progress toward computations of the Schur multipliers of these non-split groups, which is the main aim of the second part.
Spectral invariants in Rabinowitz-Floer homology and global Hamiltonian perturbations
Peter Albers and  Urs Frauenfelder
2010, 4(2): 329-357 doi: 10.3934/jmd.2010.4.329 +[Abstract](880) +[PDF](292.1KB)
Spectral invariants were introduced in Hamiltonian Floer homology by Viterbo [26], Oh [20, 21], and Schwarz [24]. We extend this concept to Rabinowitz--Floer homology. As an application we derive new quantitative existence results for leafwise intersections. The importance of spectral invariants for this application is that spectral invariants allow us to derive existence of critical points of the Rabinowitz action functional even in degenerate situations where the functional is not Morse.
Fractal trees for irreducible automorphisms of free groups
Thierry Coulbois
2010, 4(2): 359-391 doi: 10.3934/jmd.2010.4.359 +[Abstract](47) +[PDF](358.8KB)
The self-similar structure of the attracting subshift of a primitive substitution is carried over to the limit set of the repelling tree in the boundary of outer space of the corresponding irreducible outer automorphism of a free group. Thus, this repelling tree is self-similar (in the sense of graph directed constructions). Its Hausdorff dimension is computed. This reveals the fractal nature of the attracting tree in the boundary of outer space of an irreducible outer automorphism of a free group.
Dynamics of the Teichmüller flow on compact invariant sets
Ursula Hamenstädt
2010, 4(2): 393-418 doi: 10.3934/jmd.2010.4.393 +[Abstract](62) +[PDF](310.7KB)
Let $S$ be an oriented surface of genus $g\geq 0$ with $m\geq 0$ punctures and $3g-3+m\geq 2$. For a compact subset $K$ of the moduli space of area-one holomorphic quadratic differentials for $S$, let $\delta(K)$ be the asymptotic growth rate of the number of periodic orbits for the Teichmüller flow $\Phi^t$ which are contained in $K$. We relate $\delta(K)$ to the topological entropy of the restriction of $\Phi^t$ to $K$. Moreover, we show that sup$_K\delta(K)=6g-6+2m$.

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