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Holomorphic maps for which the unstable manifolds depend on prehistories. (English) Zbl 1032.37017
Axiom A holomorphic maps $$f: P^2 \to P^2$$ on the 2-dimensional complex projective space are considered. For points $$x$$ belonging to a basic set $$\Lambda$$ of $$f$$ one can construct the local stable and unstable manifolds for every $$0<\varepsilon_0\ll 1: W_{\varepsilon_0}^s:=\{ y \in P^2,d(f^n x,f^n y)\leq \varepsilon_0$$, $$n \geq 0 \}$$, $$W_{\varepsilon_0}^u:=\{ y \in P^2$$, $$y$$ has a prehistory $$\widehat{y}=(y_n)_{n\leq 0}$$, $$d(x_{-n},y_{-n})\leq \varepsilon_0$$, $$n \geq 0\}$$. $$W_{\varepsilon_0}^s$$, $$W_{\varepsilon_0}^u$$ are complex disks. $$W_{\varepsilon_0}^u$$ depends a priory on the entire prehistory $$\widehat{x}$$, but all known examples have all their local unstable manifolds depending only on the base point $$x$$. The example is constructed where for different prehistories of points in the basic sets different unstable manifolds are obtained. To solve this problem a nontrivial example of a map which is noninjective on its basic set is found. Examples are obtained as perturbations of maps of the form $$(z^4,w^4)$$.
MSC:
 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 37F45 Holomorphic families of dynamical systems; the Mandelbrot set; bifurcations (MSC2010)
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