`a`
Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

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

Pages: 433 - 466, Volume 32, Issue 2, February 2012

doi:10.3934/dcds.2012.32.433       Abstract        References        Full Text (574.0K)       Related Articles

Alejo Barrio Blaya - Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain (email)
Víctor Jiménez López - Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain (email)

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.

Received: September 2010;      Revised: July 2011;      Published: September 2011.

 References