Dynamics of piecewise $\lambda$-affine contractions

In this talk we discuss the dynamics of interval piecewise $\lambda$-affine contractions $f_\lambda$ where $\lambda\in(0,1)$. The map $f_\lambda$ is called asymptotically periodic if the $\omega$-limit set of every point is a periodic orbit and there are at most a finite number of periodic orbits. Under a generic assumption on $f_\lambda$, we show that the set of $\lambda\in(0,1)$ such that $f_\lambda$ is not asymptotically periodic has zero Hausdorff dimension.