Descriptive set theory for expansive systems

Abstract

Kato [5] and Artigue [3] merged the theory of expansive systems [10] and foliations with the continuum theory [14]. Here we merge the expansive systems but with the descriptive set theory [6] instead. More precisely, we define meagre-expansivity for both homeomorphisms and measures by requiring the interior of the dynamical balls up to some prefixed radio to be either empty or with zero measure respectively. We first prove that every cw-expansive homeomorphism of a locally connected metric space without isolated points is meagre-expansive (but not conversely). Second that a homeomorphism of a metric space is meagre-expansive if and only if every Borel probability measure is meagre-expansive. Next that the space of meagre-expansive measures of a homeomorphism of a compact metric space X is an Fσ subset of the space of Borel probability measures equipped with the weak* topology. In the sequel we prove that every homeomorphism with a meagre-expansive measure of a compact metric space has an invariant meagre-expansive measure. Also that the set of periodic points of every meagre-expansive homeomorphism of a compact metric space has empty interior. In the circle or the interval we prove that there are no meagre-expansive homeomorphisms of the circle or the interval. Moreover, the meagre-expansive measures of an interval homeomorphism or a circle homeomorphism with rational rotation number are precisely the finite convex combinations of Dirac measures supported on isolated periodic points. A circle homeomorphism with irrational rotation number has a meagre-expansive measure if and only if it is a Denjoy map. In such a case the meagre-expansive measures are precisely those measures supported on the unique minimal set of the map. To obtain some of our results we will consider a measurable version of the classical Baire Category.