Periodic points of post-critically algebraic holomorphic endomorphisms

A holomorphic endomorphism of ${{\mathbb {CP}}}^n$ is post-critically algebraic if its critical hypersurfaces are periodic or preperiodic. This notion generalizes the notion of post-critically finite rational maps in dimension one. We will study the eigenvalues of the differential of such a map along a periodic cycle. When $n=1$ , a well-known fact is that the eigenvalue along a periodic cycle of a post-critically finite rational map is either superattracting or repelling. We prove that, when $n=2$ , the eigenvalues are still either superattracting or repelling. This is an improvement of a result by Mattias Jonsson [Some properties of 2-critically finite holomorphic maps of P2. Ergod. Th. & Dynam. Sys.18(1) (1998), 171–187]. When $n\geq 2$ and the cycle is outside the post-critical set, we prove that the eigenvalues are repelling. This result improves one obtained by Fornæss and Sibony [Complex dynamics in higher dimension. II. Modern Methods in Complex Analysis (Princeton, NJ, 1992) (Annals of Mathematics Studies, 137). Ed. T. Bloom, D. W. Catlin, J. P. D’Angelo and Y.-T. Siu, Princeton University Press, 1995, pp. 135–182] under a hyperbolicity assumption on the complement of the post-critical set.

Chaotically driven sigmoidal maps

We consider skew product dynamical systems [Formula: see text] with a (generalized) baker transformation [Formula: see text] at the base and uniformly bounded increasing [Formula: see text] fibre maps [Formula: see text] with negative Schwarzian derivative. Under a partial hyperbolicity assumption that ensures the existence of strong stable fibres for [Formula: see text], we prove that the presence of these fibres restricts considerably the possible structures of invariant measures — both topologically and measure theoretically, and that this finally allows to provide a “thermodynamic formula” for the Hausdorff dimension of set of those base points over which the dynamics are synchronized, i.e. over which the global attractor consists of just one point.

A NOTE ON THE NEW GLIMM FUNCTIONAL FOR GENERAL SYSTEMS OF HYPERBOLIC CONSERVATION LAWS

For systems of hyperbolic conservation laws, a new Glimm functional was recently constructed when the linearly degenerate manifold in each characteristic field is either the whole space or it consists of a finitely many smooth and transversal manifolds of codimension one. This new functional leads to the neat consistency and convergence rate estimation of the Glimm scheme. In this paper, by the motivation of the result in Ref. 2, we show that the corresponding new Glimm functional can be constructed for general systems only under the strict hyperbolicity assumption.