In this thesis we prove three results concerning existence of absolutely continuous invariant probability measures (ACIPs) in families of real multimodal maps, focusing on the relation between combinatorial and topological properties of the maps, and these measures.
The first result proves in a novel way that semi-hyperbolic multimodal maps automatically support an ACIP. This is then used to produce a new example of a bicritical map where one of the critical points satisfies $\liminf_{n\to\infty}\lvert Df_n(c)\rvert = 0$. We also investigate the relation of this result with the literature.
Then we shift our attention to families of bicritical polynomials, and we prove an extension of the classical regular and stochastic dichotomy, but for the class of cubic polynomials with a preperiodic critical point. We focus on how to reduce this multimodal problem to a unimodal one.
In the final chapter, inspired by [BRLSvS08], we show how a local derivative condition for bicritical real maps with specific critical combinatorics is su cient to conclude that a map supports an ACIP.