Abstract
We studied topological and metric properties of the so-called interval translation maps (ITMs). For these maps, we introduced the maximal invariant measure and study its properties. Further, we study how the invariant measures depend on the parameters of the system. These results were illustrated by a simple example or a risk management model where interval translation maps appear naturally.
Highlights
Studying piecewise continuous maps, we can hardly rely on the classical numerical methods of approximating individual trajectories since the solutions are extremely sensitive to the variations of the initial conditions and the parameters of the system.By contrast, invariant measures are more robust objects
We describe the set of invariant measures for an interval translation map and discuss how they affect the topological properties of solutions
In order to be able to discuss the topological minimality of the points of the set M (S), we introduce a symbolic model for our dynamics
Summary
We can hardly rely on the classical numerical methods of approximating individual trajectories since the solutions are extremely sensitive to the variations of the initial conditions and the parameters of the system. As such, they provide a better language for representing the results of numerical models. We describe the set of invariant measures for an interval translation map and discuss how they affect the topological properties of solutions. We introduce the so-called maximal invariant measure and study its properties. Another new result of this part of the paper is a theorem on the continuity of the set of invariant measures with respect to the residual set of parameters. This part of the paper extends and completes the results of [1]. We study the parameter values for which the map is a rotation or a double rotation
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