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DCDS

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].

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].

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