Equidistribution problems in complex dynamics of higher dimension

Abstract

ABSTRACT. Equidistribution of the orbits of points, subvarieties or of periodic points in complex dynamics is a fundamental problem. It is often related to strong ergodic properties of the dynamical system and to a deep understanding of analytic cycles, or more generally positive closed currents, of arbitrary dimension and degree. The later topic includes the study of the potentials and super-potentials of positive closed currents, their intersection with or without dimension excess. In this paper, we will survey some results and tools developed during the last two decades. Related concepts, new techniques and open problems will be presented.Classification AMS 2010: 32H, 32U, 37F.

Arithmetic and dynamical degrees of self-morphisms of semi-abelian varieties

“…Note that the spectral radius of f * H 2 (X; Z) is equal to the spectral radius of f * H 1,1 (X) (cf. the inequality above Proposition 4.4 in [5]), this proves the theorem. Now, let X be an abelian variety and f : X −→ X a homomorphism, both defined over Q.…”

Finiteness properties of pseudo-hyperbolic varieties

“…, λ d ) is log-concave [65, Theorem 1.1. ( 3)] (see also [18,23,25,60]), so that λ d ≤ λ d 1 . Since deg f > 1 (by assumption), we have that λ d > 1.…”

Growth of the number of periodic points for meromorphic maps

“…We then present some properties of tangent currents, a fundamental notion in theory of densities, that will be used in this work. The reader will find more details in [8,18,19,21,40].…”

“…We expect that our techniques may be extended to study other dynamical questions of algebraic nature. The readers can find in a list of open problems, related results and references.…”