scholarly journals Expansivity of semi-hyperbolic Lipschitz mappings

1995 ◽  
Vol 51 (2) ◽  
pp. 301-308 ◽  
Author(s):  
P. Diamond ◽  
P. Kloeden ◽  
V. Kozyakin ◽  
A. Pokrovskii
Keyword(s):  
Dynamical Systems ◽  
Classical Systems ◽  
Hyperbolic Diffeomorphisms ◽  
Lipschitz Mappings ◽  
Hyperbolic Dynamical Systems

Semi-hyperbolic dynamical systems generated by Lipschitz mappings are shown to be exponentially expansive, locally at least, and explicit rates of expansion are determined. The result is applicable to nonsmooth noninvertible systems such as those with hysteresis effects as well as to classical systems involving hyperbolic diffeomorphisms.

scholarly journals Large deviations in non-uniformly hyperbolic dynamical systems

2008 ◽  
Vol 28 (2) ◽  
pp. 587-612 ◽  
Author(s):  
LUC REY-BELLET ◽  
LAI-SANG YOUNG
Keyword(s):  
Dynamical Systems ◽  
Generating Functions ◽  
Limit Theorems ◽  
Large Deviation ◽  
Return Times ◽  
Hyperbolic Diffeomorphisms ◽  
Large Deviation Principles ◽  
Hyperbolic Dynamical Systems ◽  
Rank One ◽  
Dispersing Billiards

AbstractWe prove large deviation principles for ergodic averages of dynamical systems admitting Markov tower extensions with exponential return times. Our main technical result from which a number of limit theorems are derived is the analyticity of logarithmic moment generating functions. Among the classes of dynamical systems to which our results apply are piecewise hyperbolic diffeomorphisms, dispersing billiards including Lorentz gases, and strange attractors of rank one including Hénon-type attractors.

scholarly journals Almost all $S$-integer dynamical systems have many periodic points

1998 ◽  
Vol 18 (2) ◽  
pp. 471-486 ◽  
Author(s):  
T. B. WARD
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Cellular Automata ◽  
Zeta Function ◽  
Periodic Points ◽  
Radius Of Convergence ◽  
Dynamical Zeta Function ◽  
Hyperbolic Dynamical Systems ◽  
Almost All

We show that for almost every ergodic $S$-integer dynamical system the radius of convergence of the dynamical zeta function is no larger than $\exp(-\frac{1}{2}h_{\rm top})<1$. In the arithmetic case almost every zeta function is irrational.We conjecture that for almost every ergodic $S$-integer dynamical system the radius of convergence of the zeta function is exactly $\exp(-h_{\rm top})<1$ and the zeta function is irrational.In an important geometric case (the $S$-integer systems corresponding to isometric extensions of the full $p$-shift or, more generally, linear algebraic cellular automata on the full $p$-shift) we show that the conjecture holds with the possible exception of at most two primes $p$.Finally, we explicitly describe the structure of $S$-integer dynamical systems as isometric extensions of (quasi-)hyperbolic dynamical systems.

scholarly journals A spectral approach for quenched limit theorems for random hyperbolic dynamical systems

2019 ◽  
Vol 373 (1) ◽  
pp. 629-664 ◽  
Author(s):  
D. Dragičević ◽  
G. Froyland ◽  
C. González-Tokman ◽  
S. Vaienti
Keyword(s):  
Dynamical Systems ◽  
Limit Theorems ◽  
Spectral Approach ◽  
Hyperbolic Dynamical Systems

New developments in the ergodic theory of nonlinear dynamical systems

1994 ◽  
Vol 346 (1679) ◽  
pp. 145-157 ◽  
Keyword(s):  
Dynamical Systems ◽  
Ergodic Theory ◽  
Nonlinear Dynamical Systems ◽  
Statistical Properties ◽  
Strange Attractors ◽  
New Developments ◽  
Nonlinear Dynamical ◽  
Henon Maps ◽  
Open Questions ◽  
Hyperbolic Dynamical Systems

The purpose of this paper is to give a survey of recent results on non-uniformly hyperbolic dynamical systems. The emphasis is on the existence of strange attractors and Sinai-Ruelle-Bowen measures for Henon maps, but we also describe results about statistical properties of such dynamical systems and state some of the open questions in this area.

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