Dynamics of piecewise increasing contractions

Abstract

13 pages, 2 figuresLet $0<\lambda<1$ and $I_1=[a_0,a_1),\ldots,I_{k}= [a_{k-1},a_k)$ be a partition of the interval $I=[0,1)$ into $k\ge1$ subintervals. Let $f:I\to I$ be a map where each restriction $f|_{I_i}$ is an increasing $\lambda$-Lipschitz function for $i=1,\ldots,k$. We prove that any piecewise increasing contraction $f$ admits at most $k$ periodic orbits, where the upper bound is sharp. Our second result concerns piecewise $\lambda$-affine maps. Let $b_1,\ldots,b_k$ be real numbers. Let $F_\lambda: I\to \mathbb{R}$ be a family of piecewise $\lambda$-affine functions, where each restriction $F_\lambda|_{I_i}(x)=\lambda x +b_i$. Under a generic assumption on the parameters $a_1,\ldots,a_{k-1},b_1,\ldots,b_k$ which define $F_\lambda$, we prove that, up to a zero Hausdorff dimension set of slopes $0<\lambda<1$, the $\omega$-limit set of the piecewise $\lambda$-affine map $f_\lambda:x\in I \to F_\lambda(x)\pmod{1}$ at every point equals a periodic orbit and there exist at most $k$ periodic orbits