
Abstract
We study the case of a smooth noninvertible map $f$ with Axiom A, in
higher dimension. In this paper, we look first
at the unstable dimension (i.e the Hausdorff dimension of the intersection between
local unstable manifolds and a basic set $\Lambda$), and prove that it is given
by the zero
of the pressure function of the unstable potential, considered on the natural
extension $\hat\Lambda$ of the basic set $\Lambda$; as a consequence,
the unstable dimension is independent of the prehistory $\hat x$.
Then we take a closer look at the theorem of construction for the local
unstable
manifolds of a perturbation $g$ of $f$, and for the conjugacy $\Phi_g$ defined
on $\hat \Lambda$.
If the map $g$ is holomorphic, one can prove some special estimates of the
Hölder
exponent of $\Phi_g$ on the liftings of the local unstable manifolds.
In this way we obtain a new estimate of the speed of convergence of the unstable
dimension of $g$, when $g \rightarrow f$.
Afterwards we prove the real analyticity of the unstable dimension when
the map $f$ depends on a real analytic parameter.
In the end we show that there exist Gibbs measures on the intersections between
local unstable manifolds and basic sets,
and that they are in fact geometric measures; using this, the unstable dimension
turns out to be equal to the upper box dimension. We notice also that in the
noninvertible case, the Hausdorff dimension of basic sets does not vary continuously with respect to the perturbation $g$
of $f$. In the case of noninvertible Axiom A maps on $\mathbb P^2$,
there can exist an infinite number of local unstable manifolds passing
through the
same point $x$ of the basic set $\Lambda$, thus there is no unstable
lamination. Therefore many of the methods used in the case of diffeomorphisms break down and new phenomena and methods of proof must appear. The results in this paper answer to some questions of Urbanski
([21]) about the extension
of one dimensional theory of Hausdorff dimension of fractals to the
higher dimensional case. They also improve some results and
estimates from [7].
Mathematics Subject Classification: Primary: 37D20, 37A35; Secondary: 37F35.
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