Closing lemmas for interval translation maps

Abstract

A interval translation mapping (or a circle translation mapping) is studied. Such maps can be regarded as interval exchange maps with overlaps. It is known that for any mapping of that type admits a Borel probability invariant non-atomic measure. This measure can be constructed as a weak limit of invariant measures of maps with periodic parameters. Those measures, are just normalized Lebesgue ones on a family of sub-sectors. For such limit measures, in the case of a shift of the arcs of the circle, it is shown that any point of their supports can be made periodic by arbitrarily small change of the parameters of the system without changing the number of segments. For any invariant measure, it is deduced from Poincar´es Recurrence Theorem shows every point can be made periodic by a small change in the parameters of the system, with the number of intervals increasing by two at most.

Keywords:

interval translation maps, invariant measures, Pugh lemma, Poincar´e recurrence theorem