On the relations between positive Lyapunov exponents, positive  entropy, and sensitivity for interval maps

Alejo Barrio Blaya; Víctor Jiménez López

doi:10.3934/dcds.2012.32.433

Article Contents

2012, Volume 32, Issue 2: 433-466. Doi: 10.3934/dcds.2012.32.433

This issue Previous Article Large solutions of elliptic systems of second order and applications to the biharmonic equation Next Article Asymptotic estimates for unimodular Fourier multipliers on modulation spaces

On the relations between positive Lyapunov exponents, positive entropy, and sensitivity for interval maps

Alejo Barrio Blaya^1, and
Víctor Jiménez López^1,

1.
Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia

Received: August 31, 2010

Revised: June 30, 2011

Published: January 2012

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Abstract

Let $f:I=[0,1]\rightarrow I$ be a Borel measurable map and let $\mu$ be a probability measure on the Borel subsets of $I$. We consider three standard ways to cope with the idea of ``observable chaos'' for $f$ with respect to the measure $\mu$: $h_\mu(f)>0$ ---when $\mu$ is invariant---, $\mu(L^+(f))>0$ ---when $\mu$ is absolutely continuous with respect to the Lebesgue measure---, and $\mu(S^\mu(f))>0$. Here $h_\mu(f)$, $L^+(f)$ and $S^\mu(f)$ denote, respectively, the metric entropy of $f$, the set of points with positive Lyapunov exponent, and the set of sensitive points to initial conditions with respect to $\mu$.
It is well known that if $h_\mu(f)>0$ or $\mu(L^+(f))>0$, then $\mu(S^\mu(f))>0$, and that (when $\mu$ is invariant and absolutely continuous) $h_\mu(f)>0$ and $\mu(L^+(f))>0$ are equivalent properties. However, the available proofs in the literature require substantially stronger hypotheses than those strictly necessary. In this paper we revisit these notions and show that the above-mentioned results remain true in, essentially, the most general (reasonable) settings. In particular, we improve some previous results from [2], [6], and [23].

Keywords:

Absolutely continuous invariant measure,
acip,
invariant measure,
Lyapunov exponents,
metric entropy,
Rohlin's formula,
sensitivity to initial conditions.

Mathematics Subject Classification: Primary: 37E05; Secondary: 37A05, 37A25, 37A35, 37D25, 37D45.

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