Periodic structures for nonlinear piecewise contracting maps

TL;DR

This paper demonstrates that nonlinear monotonic contracting maps with a single discontinuity are conjugate to piecewise linear maps with the same periodicity, and explores the periodic structure in parameter families related to Farey series.

Contribution

It establishes conjugacy between nonlinear and linear contracting maps and analyzes the periodic structure called Arnold tongue in parameter families.

Findings

Nonlinear monotonic contracting maps are conjugate to piecewise linear maps with the same periodic points.

The parameter family exhibits Arnold tongue structure related to Farey series.

Existence of parameter sets with positive measure for arbitrary periods.

Abstract

In this paper, we first show that any nonlinear monotonic increasing contracting maps with one discontinuous point on a unit interval which has an unique periodic point with period $n$ conjugates to a piecewise linear contracting map which has periodic point with same period. Second, we consider one parameter family of monotonic increasing contracting maps, and show that the family has the periodic structure called Arnold tongue for the parameter which is associated with the Farey series. This implies that there exist a parameter set with a positive Lebesgue measure such that the map has a periodic point with an arbitrary period. Moreover, the parameter set with period $(m+n)$ exists between the parameter set with period $m$ and $n$.