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Holomorphic maps for which the unstable manifolds depend on prehistories

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  • For points $x$ belonging to a basic set $\Lambda$ of an Axiom A holomorphic endomorphism of $\mathbb P^2$, one can construct the local stable manifold $W_{\varepsilon_0}^s(x)$ and the local unstable manifolds $W_{\varepsilon_0}^u(\hat x)$. A priori, $W_{\varepsilon_0}^u(\hat x)$ should depend on the entire prehistory $\hat x$ of $x$. However, all known examples have all their local unstable manifolds depending only on the base point $x$. Therefore a natural problem is to give actual examples where, for different prehistories of points in the basic sets of holomorphic Axiom A maps, we obtain different unstable manifolds. We solve this problem by considering the map $(z^4+\varepsilon w^2, w^4)$ and then also show that, by perturbing $(z^2+c, w^2)$, one can get also maps $f_\varepsilon$ which are injective on $\Lambda_\varepsilon$, their corresponding basic sets, hence the cardinality of the set $(f_\varepsilon|_{\Lambda_\varepsilon})^{-1}(x), x \in \Lambda_\varepsilon$, is not stable under perturbation.
    Mathematics Subject Classification: 37D20, 37F45.


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