Dynamics of piecewise increasing contractions

TL;DR

This paper investigates the dynamics of piecewise increasing contractions, establishing upper bounds on periodic orbits and analyzing the typical behavior of piecewise linear maps with respect to periodicity and parameter sets.

Contribution

It provides sharp bounds on the number of periodic orbits for piecewise increasing contractions and characterizes the measure and dimension of parameter sets leading to non-asymptotic periodicity.

Findings

Any such map has at most k periodic orbits, which is sharp.

For generic parameters, the omega-limit set is a periodic orbit, with at most k such orbits.

The set of parameters leading to non-asymptotic periodicity has Lebesgue measure zero and large Hausdorff dimension.

Abstract

Let $I_1=[a_0,a_1),\ldots,I_{k}= [a_{k-1},a_k)$ be a partition of the interval $I=[0,1)$ into $k$ subintervals. Let $f:I\to I$ be a map such that each restriction $f|_{I_i}$ is an increasing Lipschitz contraction. We prove that any $f$ admits at most $k$ periodic orbits, where the upper bound is sharp. We are also interested in the dynamics of piecewise linear $\lambda$-affine maps, where $0<\lambda<1$. Let $b_1,\ldots,b_k$ be real numbers and let $F_\lambda: I\to \mathbb{R}$ be a function such that each restriction $F_\lambda|_{I_i}(x)=\lambda x +b_i$. Under a generic assumption on the parameters $a_1,\ldots,a_{k-1},b_1,\ldots,b_k$, we prove that, up to a zero Hausdorff dimension set of slopes $\lambda$, the $\omega$-limit set of the piecewise $\lambda$-affine maps $f_\lambda:x\in I \mapsto F_\lambda(x)\pmod{1}$ at every point equals a periodic orbit and there exist at most $k$ periodic orbits. Moreover, let $\mathfrak{E}^{(k)}$ be the exceptional set of parameters $\lambda,a_1,\ldots,a_{k-1},b_1,\ldots,b_k$ which define non-asymptotically periodic $f$, we prove that $\mathfrak{E}^{(k)}$ is a Lebesgue null measure set whose Hausdorff dimension is large or equal to $k$.