## Journals

- Advances in Mathematics of Communications
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- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
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- Mathematical Foundations of Computing
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- Numerical Algebra, Control & Optimization
- AIMS Mathematics
- Conference Publications
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- Mathematics in Engineering

### Open Access Journals

JMD

We establish stable ergodicity of diffeomorphisms with partially
hyperbolic attractors whose Lyapunov exponents along the central direction
are all negative with respect to invariant SRB-measures.

keywords:
Partial hyperbolicity
,
stable ergodicity
,
accessibility
,
Lyapunov exponents
,
SRB-measures.

JMD

We introduce a class of continuous maps $f$ of a compact topological space $I$ admitting inducing schemes and describe the tower constructions associated with them. We then establish a thermodynamic formalism, \ie describe a class of real-valued potential functions $\varphi$ on $I$, which admit a unique equilibrium measure $\mu_\varphi$ minimizing the free energy for a certain class of invariant measures. We also describe ergodic properties of equilibrium measures, including decay of correlation and the Central Limit Theorem. Our results apply to certain maps of the interval with critical points and/or singularities (including some unimodal and multimodal maps) and to potential functions $\varphi_t=-t\log|df|$ with $t\in(t_0, t_1)$ for some $t_0<1 < t_1$. In the particular case of $S$-unimodal maps we show that one can choose $t_0<0$ and that the class of measures under consideration consists of all invariant Borel probability measures.

keywords:
equilibrium
meassures
,
inducing schemes
,
liftability.
,
towers
,
Thermodynamic formalism

JMD

The paper is a nontechnical survey and is aimed to illustrate Sarig's
profound contributions to statistical physics and in particular,
thermodynamic formalism for countable Markov shifts. I will discuss
some of Sarig's work on characterization of existence of Gibbs
measures, existence and uniqueness of equilibrium states as well as
phase transitions for Markov shifts on a countable set of states.

JMD

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.

keywords:
Brin prize
,
accessibility
,
SRB-measures.
,
partial hyperbolicity
,
stable ergodicity
,
Lyapunov exponents
,
Dolgopyat

DCDS

This is a survey-type article whose goal is to review some recent developments in studying the genericity problem for non-uniformly hyperbolic dynamical systems with discrete time on compact smooth manifolds. We discuss both cases of systems which are conservative (preserve the Riemannian volume) and dissipative (possess hyperbolic attractors). We also consider the problem of coexistence of hyperbolic and regular behaviour.

DCDS

We provide a general mechanism for obtaining uniform information from
pointwise data. For instance, a diffeomorphism of a compact Riemannian
manifold with pointwise expanding and contracting continuous invariant cone
families is an Anosov diffeomorphism, i.e., the entire manifold is uniformly
hyperbolic.

## Year of publication

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## Related Keywords

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