# Piecewise contractions defined by iterated function systems – Arnaldo Nogueira (Universita’ di Aix Marseille )

#### Venue

Sala Conferenze (Puteano, Centro De Giorgi).

#### Abstract

We are interested in the asymptotical behavior of piecewise contractions of the interval (PCs). A map $f:[0,1)\to [0,1)$ is an {\it $n$ interval PC} if there exists a partition of the interval $[0,1)$ into subintervals $J_1,\ldots,J_n$ such that each restriction $f\vert_{J_\ell}:J_\ell \to [0,1)$ is a Lipschitz-continuous contraction. Let $\{\phi_1,\ldots,\phi_n\}$ be an Iterated Function System (IFS), where each $\phi_\ell:[0,1]\to (0,1)$ is a Lipschitz-continuous contraction. Setting $x_0=0$ and $x_n=1$, we prove that for Lebesgue almost every $(n-1)$-dimensional point $(x_1,… ,x_{n-1})$ with \$0

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