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Dynamics of circle maps with flat spots

100%

Graczyk J.

Fundamenta Mathematicae

2010

tom 209

nr 3

267-290

We study a certain class of weakly order preserving, non-invertible circle maps with irrational rotation numbers and exactly one flat interval. For this class of circle maps we explain the geometric and dynamic structure of orbits. In particular, we formulate the so called upper and lower scaling rules which show an asymmetric and double exponential decay of geometry.

Non-recurrent meromorphic functions

51%

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 Graczyk J.

1 Kotus J.

1 Świątek G.

1 2010

1 2004

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Dynamics of circle maps with flat spots

Non-recurrent meromorphic functions