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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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