Rotation number and dynamics of 3-interval piecewise $\lambda$-affine contractions

TL;DR

This paper analyzes a family of 3-interval piecewise affine contractions with a rotation number, characterizing parameter values that produce specific rotation numbers, symbolic dynamics, and types of attractors, extending known results from 2-interval cases.

Contribution

It introduces a detailed parameter characterization for 3-interval contractions with a rotation number, expanding the understanding of their dynamics beyond the well-studied 2-interval case.

Findings

Identifies parameter sets for given rotation numbers.

Determines conditions for periodic orbits and Cantor set attractors.

Connects symbolic dynamics with rotation numbers and parameters.

Abstract

We consider a family of piecewise contractions admitting a rotation number and defined for every $x\in[0,1)$ by $f(x)=\lambda x + \delta + d \theta_a(x) \pmod 1$, where $\lambda\in(0,1)$, $d\in(0,1-\lambda)$, $\delta\in[0,1]$, $a\in[0,1]$ and $\theta_a(x)=1$ if $x\geq a$ and $\theta_a(x)=0$ otherwise. In the special case where $a=1$, the family reduces to the well studied ``contracted rotations" $x\mapsto \lambda x + \delta \pmod 1$, which are 2-interval piecewise $\lambda$-affine contractions when $\delta\in(1-\lambda,1)$. Considering $a\in(0,1)$ allows maps with an additional discontinuity, that is, $3$-interval piecewise $\lambda$-affine contractions. Supposing $\lambda$ and $d$ fixed, for any $\rho\in(0,1)$ and $\alpha\in[0,1]$, we provide the values of the parameters $\delta$ and $a$ for which the corresponding map has rotation number $\rho$, and a symbolic dynamics containing that of the rotation $R_\rho:[0,1)\to[0,1)$ of angle $\rho$ with respect to the partition given by the positions of $1-\rho$ and $\alpha$ in $[0,1)$. This enables in particular to determine the maps that have a given number of periodic orbits of an arbitrary period, or a Cantor set attractor supporting a dynamics of a given complexity.