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

Abstract

We consider a family of piecewise contractions admitting a rotation number and defined for every $x\in [0,1)$ by $f(x)=\lambda x+\delta +d{\theta }_a(x)(\mathrm{mod}1)$, where $\lambda \in (0,1)$, $d\in (0,1-\lambda )$, $\delta \in [0,1]$, $a\in [0,1]$ and ${\theta }_a(x)=1$ if $x\ge a$ and ${\theta }_a(x)=0$ otherwise. In the special case where $a=1$, the family reduces to the well studied ``contracted rotations" $x\mapsto \lambda x+\delta (\mathrm{mod}1)$, 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.

45 pages, 7 figures