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Diffusion to infinity for periodic orbits in meromorphic dynamics

100%

Kotus J., Świątek G.

Fundamenta Mathematicae

2002

tom 174

nr 3

263-269

A small perturbation of a rational function causes only a small perturbation of its periodic orbits. We show that the situation is different for transcendental maps. Namely, orbits may escape to infinity under small perturbations of parameters. We show examples where this "diffusion to infinity" occurs and prove certain conditions under which it does not.

Non-recurrent meromorphic functions

81%

Graczyk J., Kotus J., Świątek G.

Fundamenta Mathematicae

2004

tom 182

nr 3

269-281

We consider a transcendental meromorphic function f belonging to the class ℬ (with bounded set of singular values). We show that if the Julia set J(f) is the whole complex plane ℂ, and the closure of the postcritical set P(f) is contained in B(0,R) ∪ {∞} and is disjoint from the set Crit(f) of critical points, then every compact and forward invariant set is hyperbolic, provided that it is disjoint from Crit(f). It is further shown, under general additional hypotheses, that f admits no measurable invariant line-field.

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2 Fundamenta Mathematicae

2 Kotus J.

2 Świątek G.

1 Graczyk J.

1 2004

1 2002

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Diffusion to infinity for periodic orbits in meromorphic dynamics

Non-recurrent meromorphic functions