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A Case of Monotone Ratio Growth for Quadratic-Like Mappings

Waldemar Pałuba (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

This is a study of the monotone (in parameter) behavior of the ratios of the consecutive intervals in the nested family of intervals delimited by the itinerary of a critical point. We consider a one-parameter power-law family of mappings of the form f a = - | x | α + a . Here we treat the dynamically simplest situation, before the critical point itself becomes strongly attracting; this corresponds to the kneading sequence RRR..., or-in the quadratic family-to the parameters c ∈ [-1,0] in the Mandelbrot set. We allow...

A note on the entropy of a doubly stochastic operator

Brunon Kamiński, José de Sam Lazaro (2000)

Colloquium Mathematicae

We investigate the properties of the entropy and conditional entropy of measurable partitions of unity in the space of essentially bounded functions defined on a Lebesgue probability space.

C¹-maps having hyperbolic periodic points

N. Aoki, Kazumine Moriyasu, N. Sumi (2001)

Fundamenta Mathematicae

We show that the C¹-interior of the set of maps satisfying the following conditions: (i) periodic points are hyperbolic, (ii) singular points belonging to the nonwandering set are sinks, coincides with the set of Axiom A maps having the no cycle property.

Exponential limit shadowing

S. A. Ahmadi, M. R. Molaei (2013)

Annales Polonici Mathematici

We introduce the notion of exponential limit shadowing and show that it is a persistent property near a hyperbolic set of a dynamical system. We show that Ω-stability implies the exponential limit shadowing property.

Heteroclinic solutions for perturbed second order systems

Massimiliano Berti (1997)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The existence of infinitely many heteroclinic orbits implying a chaotic dynamics is proved for a class of perturbed second order Lagrangian systems possessing at least 2 hyperbolic equilibria.

Markov partitions for fibre expanding systems

Manfred Denker, Hajo Holzmann (2008)

Colloquium Mathematicae

Fibre expanding systems have been introduced by Denker and Gordin. Here we show the existence of a finite partition for such systems which is fibrewise a Markov partition. Such partitions have direct applications to the Abramov-Rokhlin formula for relative entropy and certain polynomial endomorphisms of ℂ².

Normal points for generic hyperbolic maps

Mark Pollicott (2009)

Fundamenta Mathematicae

We consider families of hyperbolic maps and describe conditions for a fixed reference point to have its orbit evenly distributed for maps corresponding to generic parameter values.

On the relationship between hyperbolic and cone-hyperbolic structures in metric spaces

Marcin Mazur (2013)

Annales Polonici Mathematici

We give necessary and sufficient conditions for topological hyperbolicity of a homeomorphism of a metric space, restricted to a given compact invariant set. These conditions are related to the existence of an appropriate finite covering of this set and a corresponding cone-hyperbolic graph-directed iterated function system.

Perturbations of flexible Lattès maps

Xavier Buff, Thomas Gauthier (2013)

Bulletin de la Société Mathématique de France

We prove that any Lattès map can be approximated by strictly postcritically finite rational maps which are not Lattès maps.

Robust transitivity in hamiltonian dynamics

Meysam Nassiri, Enrique R. Pujals (2012)

Annales scientifiques de l'École Normale Supérieure

A goal of this work is to study the dynamics in the complement of KAM tori with focus on non-local robust transitivity. We introduce C r open sets ( r = 1 , 2 , ⋯ , ∞ ) of symplectic diffeomorphisms and Hamiltonian systems, exhibitinglargerobustly transitive sets. We show that the C ∞ closure of such open sets contains a variety of systems, including so-calleda priori unstable integrable systems. In addition, the existence of ergodic measures with large support is obtained for all those systems. A main ingredient of...

Some pinching deformations of the Fatou function

Patricia Domínguez, Guillermo Sienra (2015)

Fundamenta Mathematicae

We are interested in deformations of Baker domains by a pinching process in curves. In this paper we deform the Fatou function F ( z ) = z + 1 + e - z , depending on the curves selected, to any map of the form F p / q ( z ) = z + e - z + 2 π i p / q , p/q a rational number. This process deforms a function with a doubly parabolic Baker domain into a function with an infinite number of doubly parabolic periodic Baker domains if p = 0, otherwise to a function with wandering domains. Finally, we show that certain attracting domains can be deformed by a pinching...

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