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Uniformly recurrent sequences and minimal Cantor omega-limit sets

Lori Alvin (2015)

Fundamenta Mathematicae

We investigate the structure of kneading sequences that belong to unimodal maps for which the omega-limit set of the turning point is a minimal Cantor set. We define a scheme that can be used to generate uniformly recurrent and regularly recurrent infinite sequences over a finite alphabet. It is then shown that if the kneading sequence of a unimodal map can be generated from one of these schemes, then the omega-limit set of the turning point must be a minimal Cantor set.

Unimodular Pisot substitutions and their associated tiles

Jörg M. Thuswaldner (2006)

Journal de Théorie des Nombres de Bordeaux

Let σ be a unimodular Pisot substitution over a d letter alphabet and let X 1 , ... , X d be the associated Rauzy fractals. In the present paper we want to investigate the boundaries ∂ X i ( 1 ≤ i ≤ d ) of these fractals. To this matter we define a certain graph, the so-called contact graph 𝒞 of σ . If σ satisfies a combinatorial condition called the super coincidence condition the contact graph can be used to set up a self-affine graph directed system whose attractors are certain pieces of the boundaries ∂ X 1 , ... , ∂ X d . From this graph...

Unique Bernoulli g -measures

Anders Johansson, Anders Öberg, Mark Pollicott (2012)

Journal of the European Mathematical Society

We improve and subsume the conditions of Johansson and Öberg and Berbee for uniqueness of a g -measure, i.e., a stationary distribution for chains with complete connections. In addition, we prove that these unique g -measures have Bernoulli natural extensions. We also conclude that we have convergence in the Wasserstein metric of the iterates of the adjoint transfer operator to the g -measure.

Uniqueness of limit cycles bounded by two invariant parabolas

Eduardo Sáez, Iván Szántó (2012)

Applications of Mathematics

In this paper we consider a class of cubic polynomial systems with two invariant parabolas and prove in the parameter space the existence of neighborhoods such that in one the system has a unique limit cycle and in the other the system has at most three limit cycles, bounded by the invariant parabolas.

Universally finitary symbolic extensions

Jacek Serafin (2009)

Fundamenta Mathematicae

We prove that by considering a finitary (almost continuous) symbolic extension of a topological dynamical system instead of a continuous extension, one cannot achieve any drop of the entropy of the extension.

Unstable Orbits and Milnor Attractors in the Discontinuous Flat Top Tent Map

Viktor Avrutin, Ben Futter, Laura Gardini, Michael Schanz (2012)

ESAIM: Proceedings

In this work we consider the discontinuous flat top tent map which represents an example for discontinuous piecewise-smooth maps, whereby the system function is constant on some interval. Such maps show several characteristics caused by this constant value which are still insufficiently investigated. In this work we demonstrate that in the discontinuous flat top tent map every unstable periodic orbit may become a Milnor attractor. Moreover, it turns...

Upper Estimate of Concentration and Thin Dimensions of Measures

H. Gacki, A. Lasota, J. Myjak (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

We show upper estimates of the concentration and thin dimensions of measures invariant with respect to families of transformations. These estimates are proved under the assumption that the transformations have a squeezing property which is more general than the Lipschitz condition. These results are in the spirit of a paper by A. Lasota and J. Traple [Chaos Solitons Fractals 28 (2006)] and generalize the classical Moran formula.

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