Backward Asymptotics in S-Unimodal Maps

Equilibrium States for S-unimodal Maps

Asymptotics of scaling parameters for period-doubling in unimodal maps with asymmetric critical points

The universal period-doubling scaling of a unimodal map with an asymmetric critical point is governed by a period-2 point of a renormalisation operator. The period-2 point is parametrised by the degree of the critical point and the asymmetry modulus. In this paper we study the asymptotics of period-2 points and their associated scaling parameters in the singular limit of degree tending to 1.

Bifurcations of Unimodal Maps

We review recent results that lead to a very precise understanding of the dynamics of typical unimodal maps from the statistical point of view. We also describe the (generalized) renormalization approach to the study of the statistical properties of typical unimodal maps.

Chaotic Unimodal and Bimodal Maps

We describe up to conjugacy all unimodal and bimodal maps that are chaotic, by giving necessary and sufficient conditions for unimodal and bimodal maps of slopes ±s to be transitive.

Homeomorphic Restrictions of Unimodal Maps

Examples are given of tent maps T for which there exist nontrivial sets B ⊂ [0, 1] such that T : B → B is a homeomorphism.