Ergodic properties of families of Hénon maps

Abstract

Let {H λ } be a continuous family of Hénon maps parametrized by λ inM , where M ⊂ C k is compact. The purpose of this paper is to understand some aspects of the random dynamical system obtained by iterating maps from this family. As an application, we study skew products of Hénon maps and obtain lower bounds for their entropy.

Rigidity of Julia sets of families of biholomorphic mappings in higher dimension

“…for (λ, x, y) ∈ M × C 2 where σ H acts as an homeomorphism on M . Recently, the dynamics of these maps has been studied in [13] and [14]. For each λ ∈ M and for each n ≥ 1, let…”

“…for each n ≥ 1 and for all (x, y) ∈ C 2 . Recall form [14] that the sequence of functions defined in (5.2) converge uniformly in compact to the plurisubharmonic functions G ± H,λ in C 2 . Further, note that (see [14]) a uniform filtration defined as follow:…”