×

Stable manifolds of holomorphic diffeomorphisms. (English) Zbl 1048.37047

Let \(M\) be a complex manifold endowed with a complete Riemann metric and \(f\) a holomorphic diffeomorphism of \(M\). The stable manifold \(W^s_p\) through a point \(p\in M\) with bounded orbit is defined by \[ W^s_p:= \{x\in M\mid \text{dist}(f^N_x, f^N_p)\leq C\rho^N\text{ for }N\geq 0\}, \] where \(\rho= \rho_p< 1\) and \(C= C_p> 0\).
The authors consider a problem raised by E. Bedford [Open problem session of the biholomorphic mappings meeting at the American Institute of Mathematics, Palo Alto, CA, July 2000] to determine the complex structure of the stable manifolds of \(f\). One of their result asserts that there exists a bord set \(K(f)\subset M\) with the property that for every compactly supported invariant probability measure \(\mu\), one has \(\mu(K(f)= 1\), and such that, for every \(p\in K(f)\), \(W^s_p\) is a complex manifold biholomorphic to the complex Euclidean space.

MSC:

37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32H50 Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
PDFBibTeX XMLCite
Full Text: DOI arXiv