Our next seminar will be held on Monday, November 10, from 3:30 p.m. to 4:30 p.m. (Rio de Janeiro local time). The meeting will take place at room C116- Bloco C - CT – Instituto de Matemática – UFRJ. There will be no transmission online.
Speaker: Luciana Silva Salgado (IM-UFRJ)
Title: Sensitivity and historic behavior for continuous maps on Baire metric spaces
Abstract:
Given a Baire metric space $(Y, d)$, a continuous map $T : Y \to Y$ and $\varphi$ a continuous bounded real valued function of $Y$, the set of $(T,\varphi)$-irregular points, also points with historic behavior, is formed by those points whose Cesàro average does not converge.
The Birkhoff's ergodic theorem ensures that, for any Borel $T$-invariant probability measure $\mu$ and every $\mu$-integrable observable $\varphi: Y \to \mathbb R$, the sequence of averages converges at $\mu$-almost every point in $Y$.
So, the set of irregular points is negligible with respect to any $T$-invariant probability measure.
In the last decades, though, there has been an intense study concerning the set of points for which Cesàro averages do not converge. Contrary to the previous measure-theoretical description, the set of the irregular points may be Baire generic.
We introduce a notion of sensitivity with respect to a continuous real-valued bounded map which provides a sufficient condition for a continuous transformation, acting on a Baire metric space, to exhibit a Baire generic subset of points with historic behavior.
This is a joint work with M. Carvalho (CMUP), V. Coelho (UFOB) and P. Varandas (UFBA/Univ. de Aveiro).
References:
- Carvalho, V. Coelho, L. Salgado and P. Varandas, Sensitivity and historic behavior for continuous maps on Baire metric spaces. \emph{Ergodic Theory and Dynamical Systems} 2024;44(1):1-30.
More complete information about the seminars can be found at
https://ppge.im.ufrj.br/seminarios-de-probabilidade/
Sincerely,
Organizers: Giulio Iacobelli and Maria Eulalia Vares
