Entropy in dynamical systems

By Lai-Sang Young. 2003

263 citations recorded by the source corpus.

In this article, the word entropy is used exclusively to refer to the entropy of a dynamical system, i.e. a map or a flow. It measures the rate of increase in dynamical complexity as the system evolves with time. This is not to be confused with other notions of entropy connected with spatial complexity. I will attempt to give a brief survey of the role of entropy in dynamical systems and especially in smooth ergodic theory. The topics are chosen to give a flavor of this invariant; they are also somewhat biased toward my own interests. This article is aimed at nonexperts. After a review of some basic definitions and ideas, I will focus on one main topic, namely the relation of entropy to Lyapunov exponents and dimension, which I will discuss in some depth. Several other interpretations of entropy, including its relations to volume growth, periodic orbits and horseshoes, large deviations and rates of escape are treated briefly.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/72e68d2510afa78db7de97bb2851c02467e75452

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In Mathematical Physics

1. Yongxia Hua
2. Meng Wu
3. Weisheng Wu
4. Hugo Touchette
5. N. Haydn
6. Huyi Hu
7. Tomasz Downarowicz
8. Lai-Sang Young

In Statistical and Nonlinear Physics

1. Yongxia Hua
2. Meng Wu
3. Weisheng Wu
4. Hugo Touchette
5. N. Haydn
6. Huyi Hu
7. Tomasz Downarowicz
8. U. Seifert

Citations in the corpus

• Entry and return times distribution (2013, background): “Theorem 5 ([12,13]): Let μ be an ergodic invariant measure of positive entropy and A a finite generating partition.”
• A proof of Furstenberg's conjecture on the intersections of ⨯p- and ⨯q-invariant sets (2016, background): “We refer the reader to [3, 40] for more information.”
• Positive topological entropy implies chaos DC2 (2011, methodology): “T,P|F), where µy is the disintegration measure of µ supported by the atom y. This conditional version (with a proof, and in the stronger, almost everywhere version) can be found e.g. in Appendix B of [D]. The following fact is completely standard, we leave the proof as an exercise. Fact 1. Let X′ be”
• The large deviation approach to statistical mechanics (2008, background): “A number of results that establish a direct connection between dynamical systems and large deviation theory can be found in [252,256–260].”
• Unstable Entropies and Variational Principle for Partially Hyperbolic Diffeomorphisms (2017, background): “ Since µ → hu µ(f) is affine and upper semi-continuous by Propositions 2.14 and 2.15, then (5.1) hu µ(f) = Z Me f (M) hu ν(f)dτ(ν) by a classical result in convex analysis (cf. Fact A.2.10 on p. 356 in [1]). Therefore, we only need to prove the proposition for ergodic measures. Suppose µ is ergodic. L”