# Analysis, Geometry, and Dynamical Systems Seminar

### Symbolic dynamics of piecewise contractions

A map $f:[0,1]\to [0,1]$ is a piecewise contraction if locally $f$ contracts distance, i.e., if there exist $0<\lambda<1$ and a partition of $[0,1]$ into intervals $I_1,I_2,\ldots,I_n$ such that $\left\vert f(x)-f(y)\right\vert \le\lambda \vert x-y\vert$ for all $x,y\in I_i$ $(1\le i\le n)$. Piecewise contractions describe the dynamics of many systems such as traffic control systems, queueing systems, outer billiards and Cherry flows. Here I am interested in the symbolic dynamics of such maps. More precisely, we say that an infinite word $i_0 i_1 i_2\ldots$ over the alphabet $\mathcal{A}=\{1,2,\ldots,n\}$ is the natural coding of $x\in [0,1]$ if $f^k(x)\in I_{i_k}$ for all $k\ge 0$. The aim of this talk is to provide a complete classification of the words that appear as natural codings of injective piecewise contractions.