DCDS
On hyperbolic measures and periodic orbits
Ilie Ugarcovici
We prove that if a diffeomorphism on a compact manifold preserves a nonatomic ergodic hyperbolic Borel probability measure, then there exists a hyperbolic periodic point such that the closure of its unstable manifold has positive measure. Moreover, the support of the measure is contained in the closure of all such hyperbolic periodic points. We also show that if an ergodic hyperbolic probability measure does not locally maximize entropy in the space of invariant ergodic hyperbolic measures, then there exist hyperbolic periodic points that satisfy a multiplicative asymptotic growth and are uniformly distributed with respect to this measure.
keywords: periodic orbits Closing Lemma. Hyperbolic measures
JMD
Structure of attractors for $(a,b)$-continued fraction transformations
Svetlana Katok Ilie Ugarcovici
We study a two-parameter family of one-dimensional maps and related $(a,b)$-continued fractions suggested for consideration by Don Zagier. We prove that the associated natural extension maps have attractors with finite rectangular structure for the entire parameter set except for a Cantor-like set of one-dimensional Lebesgue zero measure that we completely describe. We show that the structure of these attractors can be "computed'' from the data $(a,b)$, and that for a dense open set of parameters the Reduction theory conjecture holds, i.e., every point is mapped to the attractor after finitely many iterations. We also show how this theory can be applied to the study of invariant measures and ergodic properties of the associated Gauss-like maps.
keywords: Continued fractions attractor invariant measure. natural extension
ERA-MS
Theory of $(a,b)$-continued fraction transformations and applications
Svetlana Katok Ilie Ugarcovici
We study a two-parameter family of one-dimensional maps and the related $(a,b)$-continued fractions suggested for consideration by Don Zagier and announce the following results and outline their proofs: (i) the associated natural extension maps have attractors with finite rectangular structure for the entire parameter set except for a Cantor-like set of one-dimensional zero measure that we completely describe; (ii) for a dense open set of parameters the Reduction theory conjecture holds, i.e. every point is mapped to the attractor after finitely many iterations. We also give an application of this theory to coding geodesics on the modular surface and outline the computation of the smooth invariant measures associated with these transformations.
keywords: invariant measure. Continued fractions modular surface attractors
DCDS-S
Preface
Ayşe Şahin Ilie Ugarcovici Marian Gidea
In October, 2007 the AMS Central Sectional Meeting was held at DePaul University in Chicago, IL. At that time, the guest editors of this issue of DCDS-S organized two special sessions dedicated to dynamical systems: “Smooth dynamical systems” and “Ergodic theory and symbolic dynamical systems”. The meeting was preceded by a workshop titled “Applications of measurable and smooth dynamical systems to number theory”, hosted jointly by DePaul and Northeastern Illinois universities, where M. Einsiedler, A. Katok, and A. Venkatesh gave expository lectures. The goal of the workshop was to disseminate the tools and ideas of their work on the Littlewood conjecture and related topics to the larger dynamical systems community. This confluence of events encouraged us to put together this volume which gives a small snapshot of research conducted in dynamical systems around the time of the workshop and conference.
   The volume does not represent the entire scope of what is a very large and active field but does span a variety of distinct areas in dynamical systems. It is worth noting, however, that there are natural groupings around some central ideas which cut across sub-disciplines.

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