Lecture 18. Markov Partitions. H-Theorem for Dynamical Systems. Elements of Thermodynamic Formalism

2017 ◽  
pp. 204-216
Keyword(s):  
Dynamical Systems ◽  
Thermodynamic Formalism ◽  
Markov Partitions ◽  
H Theorem

AN INTRODUCTION TO QUANTITATIVE POINCARÉ RECURRENCE IN DYNAMICAL SYSTEMS

2009 ◽  
Vol 21 (08) ◽  
pp. 949-979 ◽  
Author(s):  
BENOIT SAUSSOL
Keyword(s):  
Dynamical Systems ◽  
Thermodynamic Formalism ◽  
Dimension Theory ◽  
Recurrence Rates ◽  
One Dimensional ◽  
Expanding Map ◽  
System A ◽  
Higher Dimensional ◽  
Technical Details ◽  
Hyperbolic Behavior

We present some recurrence results in the context of ergodic theory and dynamical systems. The main focus will be on smooth dynamical systems, in particular, those with some chaotic/hyperbolic behavior. The aim is to compute recurrence rates, limiting distributions of return times, and short returns. We choose to give the full proofs of the results directly related to recurrence, avoiding as much as possible to hide the ideas behind technical details. This drove us to consider as our basic dynamical system a one-dimensional expanding map of the interval. We note, however, that most of the arguments still apply to higher dimensional or less uniform situations, so that most of the statements continue to hold. Some basic notions from the thermodynamic formalism and the dimension theory of dynamical systems will be recalled.

scholarly journals Existence and convergence properties of physical measures for certain dynamical systems with holes

2009 ◽  
Vol 30 (3) ◽  
pp. 687-728 ◽  
Author(s):  
HENK BRUIN ◽  
MARK DEMERS ◽  
IAN MELBOURNE
Keyword(s):  
Dynamical Systems ◽  
Invariant Measure ◽  
Thermodynamic Formalism ◽  
Physical Property ◽  
Continuous Functions ◽  
Hölder Continuous ◽  
Exponential Decay Of Correlations ◽  
Hölder Continuous Functions ◽  
Young Towers ◽  
Set Of Points

AbstractWe study two classes of dynamical systems with holes: expanding maps of the interval and Collet–Eckmann maps with singularities. In both cases, we prove that there is a natural absolutely continuous conditionally invariant measure μ (a.c.c.i.m.) with the physical property that strictly positive Hölder continuous functions converge to the density of μ under the renormalized dynamics of the system. In addition, we construct an invariant measure ν, supported on the Cantor set of points that never escape from the system, that is ergodic and enjoys exponential decay of correlations for Hölder observables. We show that ν satisfies an equilibrium principle which implies that the escape rate formula, familiar to the thermodynamic formalism, holds outside the usual setting. In particular, it holds for Collet–Eckmann maps with holes, which are not uniformly hyperbolic and do not admit a finite Markov partition. We use a general framework of Young towers with holes and first prove results about the a.c.c.i.m. and the invariant measure on the tower. Then we show how to transfer results to the original dynamical system. This approach can be expected to generalize to other dynamical systems than the two above classes.

scholarly journals Finitely presented dynamical systems

1987 ◽  
Vol 7 (4) ◽  
pp. 489-507 ◽  
Author(s):  
David Fried
Keyword(s):  
Dynamical Systems ◽  
Zeta Function ◽  
Symbolic Dynamics ◽  
Periodic Points ◽  
Finite Cover ◽  
Markov Partitions ◽  
Finitely Presented

AbstractWe extend results of Bowen and Manning on systems with good symbolic dynamics. In particular we identify the class of dynamical systems that admit Markov partitions. For these systems the Manning-Bowen method of counting periodic points is explained in terms of topological coincidence numbers. We show, in particular, that an expansive system with a finite cover by rectangles has a rational zeta function.

scholarly journals Countable Markov partitions suitable for thermodynamic formalism

2018 ◽  
Vol 13 (1) ◽  
pp. 199-219
Author(s):  
Michael Jakobson ◽  
◽  
Lucia D. Simonelli ◽  
Keyword(s):  
Thermodynamic Formalism ◽  
Markov Partitions

scholarly journals Thermodynamic Formalism for Coarse Expanding Dynamical Systems

2021 ◽  
Author(s):  
Tushar Das ◽  
Feliks Przytycki ◽  
Giulio Tiozzo ◽  
Mariusz Urbański ◽  
Anna Zdunik
Keyword(s):  
Dynamical Systems ◽  
Thermodynamic Formalism
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