On the relations between positive Lyapunov exponents, positive
entropy, and sensitivity for interval maps
Alejo Barrio Blaya - Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain (email)
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 $\mu$.
Keywords: Absolutely continuous invariant measure, acip, invariant
measure, Lyapunov exponents, metric entropy, Rohlin's formula,
sensitivity to initial conditions.
Received: September 2010; Revised: July 2011; Available Online: September 2011.
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