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DCDS

In the first part of the paper we show that a hyperbolic area
preserving Hénon map has a unique Gibbs measure whose Hausdorff
dimension is equal to the Hausdorff dimension of its nonwandering
(Julia) set.
In the second part we introduce the notion of strong hyperbolicity
for diffeomorphisms of compact manifolds.
It is a foliation of the tangent space over a hyperbolic set to one
dimensional contracting and expanding subspaces with different rates
of contractions and expansions.
We show that strong hyperbolicity is structurally stable.
For a Strong Axiom A diffeomorphism $f$ we state a conjectured
variational characterization of the Hausdorff dimension of the
nonwandering set of $f$.
In the third part we study the dynamics of polynomial maps
$f:\mathbb{C}^2 \rightarrow \mathbb{C}^2$ which lift to holomorphic maps
of $\mathbb C\mathbb P^2$.
Let $J(f)$ be the closure of repelling periodic points of $f$.
Using the structural stability results we exhibit open set of
$f$ for which $J(f)$ behaves like the Julia set of one dimensional
polynomial map.

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