Topological and statistical attractors for interval maps

TL;DR

This paper investigates topological and statistical attractors for interval maps, including discontinuous ones, establishing finiteness of non-periodic attractors and analyzing the behavior of averages for generic points.

Contribution

It introduces the use of Baire Ergodicity to analyze attractors and proves finiteness and coincidence results for various classes of interval maps.

Findings

Finiteness of non-periodic topological attractors for piecewise C^2 maps with discontinuities.

Coincidence of statistical and topological attractors for C^2 interval maps without discontinuities.

Calculation of upper Birkhoff averages for generic points, even with abundant historical behavior.

Abstract

We use the concept of Baire Ergodicity and Ergodic Formalism introduced to study topological and statistical attractors for interval maps, even with discontinuities. For that we also analyze the {\em wandering intervals attractors}. As a result, we establish the finiteness of the non-periodic topological attractors for piecewise $C^2$ maps with discontinuities. For $C^2$ interval maps without discontinuities, we show the coincidence of the statistical attractors with the topological ones and we calculate the upper Birkhoff averages of continuous functions for generic points, even when the map has abundance of historical behavior.