Abstract. We give a brief survey on the entropy of holomorphic self maps f of compact Kähler manifolds and rational dominating self maps f of smooth projective varieties. We emphasize the connection between the entropy and the spectral radii of the induced action of f on the homology of the compact manifold. The main conjecture for the rational maps states that modulo birational isomorphism all various notions of entropy and the spectral radii are equal.

Introduction

The subject of the dynamics of a map f : X → X has been studied by hundreds, or perhaps thousands, of mathematicians, physicists and other scientists in the last 150 years. One way to classify the complexity of the map f is to assign to it a number h(f) ∈ [0, ∞], which called the entropy of f. The entropy of f should be an invariant with respect to certain automorphisms of X. The complexity of the dynamics of f should be reflected by h(f): the larger h(f) the more complex is its dynamics.

The subject of this short survey paper is mostly concerned with the entropy of a holomorphic f : X → X, where X is a compact Kähler manifold, and the entropy of a rational map of f : Y ⇢ Y, where Y is a smooth projective variety. In the holomorphic case the author showed that entropy of f is equal to the logarithm of the spectral radius of the finite dimensional f* on the total homology group H*(X) over ℝ.