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2010 Symbolic extensions of smooth interval maps
Tomasz Downarowicz
Probab. Surveys 7: 84-104 (2010). DOI: 10.1214/10-PS164

Abstract

In this course we will present the full proof of the fact that every smooth dynamical system on the interval or circle $X$, constituted by the forward iterates of a function $f : X \rightarrow X$ which is of class $C^r$ with $r > 1$, admits a symbolic extension, i.e., there exists a bilateral subshift $(Y, S)$ with $Y$ a closed shift-invariant subset of $\Lambda^{\mathbb{Z}}$, where $\Lambda$ is a finite alphabet, and a continuous surjection $\pi : Y \rightarrow X$ which intertwines the action of $f$ (on $X$) with that of the shift map $S$ (on $Y$). Moreover, we give a precise estimate (from above) on the entropy of each invariant measure $\upsilon$ supported by $Y$ in an optimized symbolic extension. This estimate depends on the entropy of the underlying measure $\mu$ on $X$, the “Lyapunov exponent” of $\mu$ (the genuine Lyapunov exponent for ergodic $\mu$, otherwise its analog), and the smoothness parameter $r$. This estimate agrees with a conjecture formulated in [15] around 2003 for smooth dynamical systems on manifolds.

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Tomasz Downarowicz. "Symbolic extensions of smooth interval maps." Probab. Surveys 7 84 - 104, 2010. https://doi.org/10.1214/10-PS164

Information

Published: 2010
First available in Project Euclid: 18 May 2010

zbMATH: 1193.37050
MathSciNet: MR2684163
Digital Object Identifier: 10.1214/10-PS164

Subjects:
Primary: 37C40, 37E05
Secondary: 37A35

Rights: Copyright © 2010 The Institute of Mathematical Statistics and the Bernoulli Society

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