Abstract

Let $f:I=[0,1]\to$ I be a map and let $\mu$ be a probability measure on the Borel subsets of $I$. Classically, there are three possible ways to cope with the idea of "observable" chaos for $f$ with respect to the measure $\mu$. The analytical approach requires that the set $L$₊(f) of points with positive Lyapunov exponent has positive $\mu$-measure; here we assume that $\mu$ is absolutely continuous with respect to the Lebesgue measure. In the ergodic/probabilistic framework, the measure-theoretic entropy $h$_μ(f) of $f$ must be positive (now $\mu$ must be invariant with respect to $f$). Finally, chaos is observable from a topological point of view whenever the set $S$_f of sensitive points to the initial conditions has positive $\mu$-measure. It is well known that each of the properties $\mu (L$₊(f))>0 and $h$_μ(f)>0 implies $\mu (S\_f)>0$, and that (when $\mu$ is invariant and absolutely continuous) $\mu (L$₊(f))>0 and $h$_μ(f)>0 are equivalent properties. However, the available proofs in the literature require substantially stronger hypotheses than those strictly necessary. In this talk we revisit these notions and show that the above-mentioned results remain true in, essentially, the most general (reasonable) settings.

This is an account of a joint work with Alejo Barrio Blaya, Universidad de Murcia, Spain.