Journal ArticleDOI
Sets of "non-typical" points have full topological entropy and full hausdorff dimension
Luis Barreira,Jörg Schmeling +1 more
TLDR
For subshifts of finite type, conformal repellers, and conformal horseshoes, this article showed that the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages do not exist simultaneously, carries full topological entropy and full Hausdorff dimension.Abstract:
For subshifts of finite type, conformal repellers, and conformal horseshoes, we prove that the set of points where the pointwise dimensions, local entropies, Lyapunov exponents, and Birkhoff averages do not exist simultaneously, carries full topological entropy and full Hausdorff dimension. This follows from a much stronger statement formulated for a class of symbolic dynamical systems which includes subshifts with the specification property. Our proofs strongly rely on the multifractal analysis of dynamical systems and constitute a non-trivial mathematical application of this theory.read more
Citations
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Journal ArticleDOI
Limit theorems for partially hyperbolic systems
TL;DR: In this article, a large class of partially hyperbolic systems containing affine maps, frame flows on negatively curved manifolds, and mostly contracting diffeomorphisms is considered.
Book
Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents
Luis Barreira,Yakov Pesin +1 more
TL;DR: In this paper, the authors present a self-contained and comprehensive account of modern smooth ergodic theory, which provides a rigorous mathematical foundation for the phenomenon known as deterministic chaos -the appearance of 'chaotic' motions in pure deterministic dynamical systems.
Journal ArticleDOI
On the topological entropy of saturated sets
TL;DR: In this paper, the authors introduced two conditions for the set of orbits, called respectively the texttt{g}$-almost product property and the uniform separation property, for dynamical systems with the specification property, but also for many others.
Journal ArticleDOI
Multifractal Analysis of Lyapunov Exponent for Continued Fraction and Manneville–Pomeau Transformations and Applications to Diophantine Approximation
Mark Pollicott,Howard Weiss +1 more
TL;DR: In this paper, the theory of multifractal analysis for conformal expanding systems is extended to two new cases: the non-uniformly hyperbolic example of the Manneville-Pomeau equation and the continued fraction transformation.
Journal ArticleDOI
Variational principles and mixed multifractal spectra
Luis Barreira,Benoît Saussol +1 more
TL;DR: In this article, the conditional variational principle is used to study the regularity of the spectra and the full dimensionality of their irregular sets for several classes of dynamical systems, including the class of maps with upper semi-continuous metric entropy.
References
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Ergodic theory of chaos and strange attractors
Jean-Pierre Eckmann,David Ruelle +1 more
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Lyapunov exponents, entropy and periodic orbits for diffeomorphisms
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Dimension Theory in Dynamical Systems: Contemporary Views and Applications
TL;DR: Pesin this article provides a comprehensive and systematic treatment of modern dimension theory in dynamical systems, summarizes the current state of research, and describes the most important accomplishments of this field, focusing on invariant fractals and their influence on stochastic properties of systems.
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