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Dynamic classification of escape time Sierpiński curve Julia sets

Robert L. Devaney, Kevin M. Pilgrim (2009)

Fundamenta Mathematicae

For n ≥ 2, the family of rational maps F λ ( z ) = z ⁿ + λ / z ⁿ contains a countably infinite set of parameter values for which all critical orbits eventually land after some number κ of iterations on the point at infinity. The Julia sets of such maps are Sierpiński curves if κ ≥ 3. We show that two such maps are topologically conjugate on their Julia sets if and only if they are Möbius or anti-Möbius conjugate, and we give a precise count of the number of topological conjugacy classes as a function of n and κ.

Dynamical attraction to stable processes

Albert M. Fisher, Marina Talet (2012)

Annales de l'I.H.P. Probabilités et statistiques

We apply dynamical ideas within probability theory, proving an almost-sure invariance principle in log density for stable processes. The familiar scaling property (self-similarity) of the stable process has a stronger expression, that the scaling flow on Skorokhod path space is a Bernoulli flow. We prove that typical paths of a random walk with i.i.d. increments in the domain of attraction of a stable law can be paired with paths of a stable process so that, after applying a non-random regularly...

Dynamical behavior of two permutable entire functions

Kin-Keung Poon, Chung-Chun Yang (1998)

Annales Polonici Mathematici

We show that two permutable transcendental entire functions may have different dynamical properties, which is very different from the rational functions case.

Dynamical characterization of C-sets and its application

Jian Li (2012)

Fundamenta Mathematicae

We set up a general correspondence between algebraic properties of βℕ and sets defined by dynamical properties. In particular, we obtain a dynamical characterization of C-sets, i.e., sets satisfying the strong Central Sets Theorem. As an application, we show that Rado systems are solvable in C-sets.

Dynamical directions in numeration

Guy Barat, Valérie Berthé, Pierre Liardet, Jörg Thuswaldner (2006)

Annales de l’institut Fourier

This survey aims at giving a consistent presentation of numeration from a dynamical viewpoint: we focus on numeration systems, their associated compactification, and dynamical systems that can be naturally defined on them. The exposition is unified by the fibred numeration system concept. Many examples are discussed. Various numerations on rational integers, real or complex numbers are presented with special attention paid to β -numeration and its generalisations, abstract numeration systems and...

Dynamical entropy of a non-commutative version of the phase doubling

Johan Andries, Mieke De Cock (1998)

Banach Center Publications

A quantum dynamical system, mimicking the classical phase doubling map z ↦ z 2 on the unit circle, is formulated and its ergodic properties are studied. We prove that the quantum dynamical entropy equals the classical value log2 by using compact perturbations of the identity as operational partitions of unity.

Dynamical properties of some classes of entire functions

A. Eremenko, M. Yu Lyubich (1992)

Annales de l'institut Fourier

The paper is concerned with the dynamics of an entire transcendental function whose inverse has only finitely many singularities. It is rpoven that there are no escaping orbits on the Fatou set. Under some extra assumptions the set of escaping orbits has zero Lebesgue measure. If a function depends analytically on parameters then a periodic point as a function of parameters has only algebraic singularities. This yields the Structural Stability Theorem.

Dynamical properties of the automorphism groups of the random poset and random distributive lattice

Alexander S. Kechris, Miodrag Sokić (2012)

Fundamenta Mathematicae

A method is developed for proving non-amenability of certain automorphism groups of countable structures and is used to show that the automorphism groups of the random poset and random distributive lattice are not amenable. The universal minimal flow of the automorphism group of the random distributive lattice is computed as a canonical space of linear orderings but it is also shown that the class of finite distributive lattices does not admit hereditary order expansions with the Amalgamation Property....

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