Piecewise contracting maps on the interval: Hausdorff dimension, entropy and attractors

We consider the attractor $\mathrm{\Lambda }$ of a piecewise contracting map $f$ defined on a compact interval. If $f$ is injective, we show that it is possible to estimate the topological entropy of $f$ (according to Bowen’s formula) and the Hausdorff dimension of $\mathrm{\Lambda }$ via the complexity associated with the orbits of the system. Specifically, we prove that both numbers are zero.