Discrete Holomorphic Local Dynamical Systems

Ergodic Theory and Dynamical Systems, 2007

Let f 1 and f 2 be rational maps with Julia sets J 1 and J 2 , and let : J 1 → P 1 be any continuous map such that •f 1 = f 2 • on J 1. We show that if is C-differentiable, with non-vanishing derivative, at some repelling periodic point z 1 ∈ J 1 , then admits an analytic extension to P 1 \E 1 , where E 1 is the exceptional set of f 1. Moreover, this extension is a semiconjugacy. This generalizes a result of Julia (Ann. Sci.École Norm. Sup. (3) 40 (1923), 97-150). Furthermore, if E 1 = ∅ then the extended map is rational, and in this situation (J 1) = J 2 and −1 (J 2) = J 1 , provided that is not constant. On the other hand, if E 1 = ∅ then the extended map may be transcendental: for example, when f 1 is a power map (conjugate to z → z ±d) or a Chebyshev map (conjugate to ±Q d with Q d (z + z −1) = z d + z −d), and when f 2 is an integral Lattès example (a quotient of the multiplication by an integer on a torus). Eremenko (Algebra i Analiz 1(4) (1989), 102-116) proved that these are the only such examples. We present a new proof. 1074 X. Buff and A. L. Epstein whose domain may have empty interior. Consider any map h : Z → P 1 where Z ⊂ P 1 , and let ζ ∈ Z be any accumulation point. If ζ = ∞ = h(ζ) then we say that h is C-differentiable at ζ if lim z→ζ h(z) − h(ζ) z − ζ exists. Note that the existence of a non-zero finite limit is unaffected by precomposition or postcomposition by local analytic isomorphisms, and in particular by Möbius transformations. Thus, it makes sense to speak of C-differentiability with non-vanishing derivative even when ζ or h(ζ) is the point at infinity. In what follows, we use the term analytic in the sense of Riemann surface theory. As the relevant Riemann surfaces will all be open subsets of P 1 , our usage of this term is identical with the classical usage of meromorphic.

The Nepali Mathematical Sciences Report

In this paper, we prove that the escaping set of a transcendental semi group is S-forward invari-ant. We also prove that if a holomorphic semi group is a belian, then the Fatou, Julia, and escaping sets are S-completely invariant. We also investigate certain cases and conditions that the holomorphic semi group dynamics exhibits the similar dynamical behavior just like a classical holomorphic dynamics. Frequently, we also examine certain amount of connections and contrasts between classical holomorphic dynamics and holomorphic semi group dynamics.

American Journal of Mathematics, 2000

We shall describe a canonical procedure to associate to any (germ of) holomorphic self-map F of C n fixing the origin so that dF O is invertible and non-diagonalizable an n-dimensional complex manifold M , a holomorphic map π: M → C n , a point e ∈ M and a (germ of) holomorphic self-mapF of M such that: π restricted to M \ π −1 (O) is a biholomorphism between M \ π −1 (O) and C n \ {O}; π •F = F • π; and e is a fixed point ofF such that dF e is diagonalizable. Furthermore, we shall use this construction to describe the local dynamics of such an F nearby the origin when sp(dF O ) = {1}.

International Mathematics Research Notices, 2013

We prove that the stable manifold of every point in a compact hyperbolic invariant set of a holomorphic automorphism of a complex manifold is biholomorphic to C d , provided that a bunching condition, which is weaker than the classical bunching condition for linearizability, holds.

Contemporary Mathematics, 2011

Two dynamical systems u t (t, •) = (Lu)(t, •), L = A, B, u(0, •) = f , which can serve as toy models for infinite systems of interacting particles in continuum are studied. Here u(t, •) and f are holomorphic functions in some K ⊂ C, and A, B are linear operators in a certain Banach space E of such functions. It is proven that both A and B generate C 0 semigroups and hence the above Cauchy problems have solutions in E. In some particular cases, ergodicity and reversibility are proven.