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Applications of thermodynamic formalism in complex dynamics on $\mathbb{P}^2$

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  • In this paper, we review several notions from thermodynamic formalism, like topological pressure and entropy and show how they can be employed, in order to obtain information about stable and unstable sets of holomorphic endomorphisms of $\mathbb P^2$ with Axiom A.
    In particular, we will consider the non-wandering set of such a mapping and its "saddle" part $S_1$, i.e the subset of points with both stable and unstable directions. Under a derivative condition, the stable manifolds of points in S1 will have a very "thin" intersection with $S_1$, from the point of view of Hausdorff dimension. While for diffeomorphisms there is in fact an equality between $HD(W^s_\varepsilon(x)\cap S_1)$ and the unique zero of $P(t \cdot \phi^s)$ (Verjovsky-Wu [VW]) in the case of endomorphisms this will not be true anymore; counterexamples in this direction will be provided. We also prove that the unstable manifolds of an endomorphism depend Hölder continuously on the corresponding prehistory of their base point and employ this in the end to give an estimate of the Hausdorff dimension of the global unstable set of $S_1$. This set could be à priori very large, since, unlike in the case of H´enon maps, there is an uncountable collection of local unstable manifolds passing through each point of $S_1$.
    Mathematics Subject Classification: 37D35, 37D20, 37C45.


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