Holomorphic motions of hyperbolic sets
A holomorphic family \(\{f_a\}_{a\in D}\) of diffeomorphisms of a Hermitian manifold \(M\), parametrised by the unit disk \(D\) is considered. It is shown that if \(f_0\) has a hyperbolic invariant set \(K\), then there exists a small disk \(D_r\) and a holomorphic map \(h:D_r\times M\to M\) such that \(K_a:= h(a,k)\) is a hyperbolic invariant set of \(f_a\), for all \(a\in D_r\), and \(h_a:K \to K_a\) is a homeomorphism verifying \(f_a\circ h_a=h_a\circ f_a\). The map \(h\) is called a holomorphic motion. Sometimes a hyperbolic set \(K\) moves holomorphically with the parameter if \(f_a\) are only endomorphisms (this is the case when \(K\) is a repellor). Most often this is not so and examples are given. The proof of the main theorem above is a kind of complexification of the real case treated in the book of \textit{D. Ruelle} [Elements of differentiable dynamics and bifurcation theory, Boston, Academic Press (1989; Zbl 0684.58001)]. An idea of another proof is given, applying Sullivan's telescope construction. The general setting is very well done and the paper is exceptionally clear.
scientific article; zbMATH DE number 1548273