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We studied topological and metric properties of the so-called intervaltranslation maps (ITMs). For these maps, we introduced the maximal invariantmeasure and study its properties. Further, we study how the invariant measuresdepend on the parameters of the system. These results were illustrated by asimple example or a risk management model where interval translation mapsappear~naturally.

Dynamics of Systems with a Discontinuous Hysteresis Operator and Interval Translation Maps