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An Introduction to Ergodic Theory
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The first part of the text as discussed by the authors provides an introduction to ergodic theory suitable for readers knowing basic measure theory, including recurrence properties, mixing properties, the Birkhoff Ergodic theorem, isomorphism, and entropy theory.Abstract:
This text provides an introduction to ergodic theory suitable for readers knowing basic measure theory. The mathematical prerequisites are summarized in Chapter 0. It is hoped the reader will be ready to tackle research papers after reading the book. The first part of the text is concerned with measure-preserving transformations of probability spaces; recurrence properties, mixing properties, the Birkhoff ergodic theorem, isomorphism and spectral isomorphism, and entropy theory are discussed. Some examples are described and are studied in detail when new properties are presented. The second part of the text focuses on the ergodic theory of continuous transformations of compact metrizable spaces. The family of invariant probability measures for such a transformation is studied and related to properties of the transformation such as topological traitivity, minimality, the size of the non-wandering set, and existence of periodic points. Topological entropy is introduced and related to measure-theoretic entropy. Topological pressure and equilibrium states are discussed, and a proof is given of the variational principle that relates pressure to measure-theoretic entropies. Several examples are studied in detail. The final chapter outlines significant results and some applications of ergodic theory to other branches of mathematics.read more
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Journal ArticleDOI
On Weakly Hyperbolic Iterated Function Systems
TL;DR: In this paper, the authors studied weakly hyperbolic iterated function systems on compact metric spaces and proved the existence of attractors, both in the topological and measure theoretical viewpoint and the ergodicity of invariant measure.
Posted Content
Fast decay of correlations of equilibrium states of open classes of non-uniformly expanding maps and potentials
Alexander Arbieto,Carlos Matheus +1 more
TL;DR: In this paper, the authors studied the existence, uniqueness and rate of decay of correlation of equilibrium measures associated to robust classes of non-uniformly ex- panding local diffeomorphisms and Holder continuous potentials.
Journal ArticleDOI
Small denominators in complex p-adic dynamics
TL;DR: In this paper, the problem of small denominators in the field of complex p-adic numbers C p was studied, and it was shown that the radius of convergence for conjugate maps for C p -analytic dynamical systems at neutral fixed points (or cycles) can be obtained from a small denominator, which is used in the construction of a congugate map for a dynamical system f having the derivative x = f '(a ) in the fixed point a.
Journal ArticleDOI
The long-run behavior of periodic competitive Kolmogorov systems
Yi Wang,Jifa Jiang +1 more
TL;DR: In this paper, the authors studied persistent trajectories of the Poincare map T generated by the ndimensional periodic system ẋi = xiNi(t, x1, · · ·, xn), xi ≥ 0, under the assumptions that the system is dissipative, competitive and strongly competitive in C = {x : xi > 0}.
Book ChapterDOI
Unique ergodicity and random matrix products
TL;DR: In this article, the authors investigate the question: if T:X → X is a uniquely ergodic homeomorphism of a compact metrizable space, and B: X → GL(k,R) is a continuous map of X into the space of invertible, k × k, real matrices does it converge uniformly to a constant?