Observable invariant measures

TL;DR

This paper introduces observable invariant measures for continuous maps on compact manifolds, generalizing physical measures, and characterizes their properties, existence, and relation to attractors and equilibrium states.

Contribution

It defines observable measures for non-measure-preserving maps, characterizes physical measures via observability, and links observable measures to generalized attractors and equilibrium states.

Findings

Existence of observable measures for all continuous maps.

Characterization of physical measures through observability.

Observable measures form a finite or countably infinite set under certain conditions.

Abstract

For continuous maps on a compact manifold M, particularly for those that do not preserve the Lebesgue measure m, we define the observable invariant probability measures as a generalization of the physical measures. We prove that any continuous map has observable measures, and characterize those that are physical in terms of the observability. We prove that there exist physical measures whose basins cover Lebesgue a.e, if and only if the set of all observable measures is finite or infinite numerable. We define for any continuous map, its generalized attractors using the set of observable invariant measures where there is no physical measure, and prove that any continuous map defines a decomposition of the space in up to infinitely many generalized attractors whose basins cover Lebesgue a.e. We apply the results to the C1 expanding maps f in the circle, proving that the set of observable measures (even if f is not C1 plus Holder, is a subset of the equilibrium states.