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Unstable manifolds and Hölder structures associated with noninvertible maps

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  • 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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