Continuous Piecewise Linear Iterated Function Systems on the Line

We study the dimension theory of Continuous Piecewise Linear Iterated Function Systems (CPLIFS) on the line. A CPLIFS $\mathcal{F}$ consists of finitely many continuous contracting self mappings $\left\{ f_1,\dots ,f_m \right\}$ of $\mathbf{R}$ such that for all $i=1,\dots ,m$ the mapping $f_i$ is strictly contractive, piecewise linear and all of its slopes are different from zero. However, we do not assume that these maps are injective. Our goal is to compute the dimension (Hausdorff and box) of the attractor $\Lambda $ of the CPLIFS $\mathcal{F}$. The new results of the talk are joint with D\'aniel Prokaj (Budapest) and Peter Raith (Vienna).