Closure Lemmas for Interval Translation Mappings

Interval (circular arcs) translation mappings, which can be represented as interval exchange transformations with overlap, are studied. It is known that for any mapping of this type there is a Borel probabilistic invariant atomless measure, which is constructed as a weak limit of invariant measures of mappings with periodic parameters. In the latter case, this is simply the normalized Lebesgue measure on some family of subsegments. For such limit measures in the case of shifting arcs of a circle, it is shown that any point of the support of this measure can be made periodic by an arbitrarily small change in the parameters of the system without changing the number of segments. For an arbitrary invariant measure, using the Poincaré recurrence theorem, it is shown that any point can be made periodic with a small change in the parameters of the system, and the number of intervals for mapping increases by no more than two.