Journal of Modern Dynamics
April 2010 , Volume 4 , Issue 2
A special issue
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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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We present some of the results and techniques due to Dolgopyat. Thepresentation avoids technicalities as much as possible while trying to focus onthe basic ideas. We also try to present Dolgopyat's work in the context of aresearch program aimed at enlightening the relations between dynamical systemsand nonequilibrium statistical mechanics.
This paper is a nontechnical survey and aims to illustrate Dolgopyat'sprofound contributions to smooth ergodic theory. I will discuss some ofDolgopyat's work on partial hyperbolicity and nonuniform hyperbolicity withemphasis on the interaction between the two-the class of dynamical systemswith mixed hyperbolicity. On one hand, this includes uniformlypartially hyperbolic diffeomorphisms with nonzero Lyapunov exponents in thecenter direction. The study of their ergodic properties has provided analternative approach to the Pugh-Shub stable ergodicity theory for bothconservative and dissipative systems. On the other hand, ideas of mixedhyperbolicity have been used in constructing volume-preservingdiffeomorphisms with nonzero Lyapunov exponents on any manifold.
D. Dolgopyat is the winner of the second Brin Prize in Dynamical Systems(2009). This article overviews his remarkable achievements in anontechnical manner. It complements two other surveys of Dolgopyat's workwritten by Y. Pesin and C. Liverani and published in this issue. Thissurvey covers Dolgopyat's work on various physical models, including theLorentz gas, Galton board, and some systems of hard disks.
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.
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 were introduced in Hamiltonian Floer homology by Viterbo , Oh [20, 21], and Schwarz . 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.
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.
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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