Symbolic dynamics of piecewise contractions

A map f : [0, 1] → [0, 1] is a piecewise contraction if locally f contracts distance, i.e., if there exist 0 < λ < 1 and a partition of [0,1] into intervals I1,I2,...,In such that |f(x) − f(y)| ≤ λ|x − y| for all x, y ∈ Ii (1 ≤ i ≤ 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 i0 i1i2 . . . over the alphabet A = {1, 2, ..., n} is the natural coding of x∈[0,1] iff k(x)∈Ik for all k≥0. The aim of this talk is to provide a complete classification of the words that appear as natural codings of injective piecewise contractions. 

Date and Venue

Start Date
Room FC1.031


Benito Pires

Speaker's Institution

Universidade de São Paulo, Brasil


Dynamical Systems