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Semiconjugacy to a map of a constant slope

Jozef Bobok (2012)

Studia Mathematica

It is well known that any continuous piecewise monotone interval map f with positive topological entropy h t o p ( f ) is semiconjugate to some piecewise affine map with constant slope e h t o p ( f ) . We prove this result for a class of Markov countably piecewise monotone continuous interval maps.

Semi-étale groupoids and applications

Klaus Thomsen (2010)

Annales de l’institut Fourier

We associate a C * -algebra to a locally compact Hausdorff groupoid with the property that the range map is locally injective. The construction generalizes J. Renault’s reduced groupoid C * -algebra of an étale groupoid and has the advantage that it works for the groupoid arising from a locally injective dynamical system by the method introduced in increasing generality by Renault, Deaconu and Anantharaman-Delaroche. We study the C * -algebras of such groupoids and give necessary and sufficient conditions...

Sequence entropy pairs and complexity pairs for a measure

Wen Huang, Alejandro Maass, Xiangdong Ye (2004)

Annales de l’institut Fourier

In this paper we explore topological factors in between the Kronecker factor and the maximal equicontinuous factor of a system. For this purpose we introduce the concept of sequence entropy n -tuple for a measure and we show that the set of sequence entropy tuples for a measure is contained in the set of topological sequence entropy tuples [H- Y]. The reciprocal is not true. In addition, following topological ideas in [BHM], we introduce a weak notion and a strong notion of complexity pair for a...

Sets of k -recurrence but not ( k + 1 ) -recurrence

Nikos Frantzikinakis, Emmanuel Lesigne, Máté Wierdl (2006)

Annales de l’institut Fourier

For every k , we produce a set of integers which is k -recurrent but not ( k + 1 ) -recurrent. This extends a result of Furstenberg who produced a 1 -recurrent set which is not 2 -recurrent. We discuss a similar result for convergence of multiple ergodic averages. We also point out a combinatorial consequence related to Szemerédi’s theorem.

Shadow trees of Mandelbrot sets

Virpi Kauko (2003)

Fundamenta Mathematicae

The topology and combinatorial structure of the Mandelbrot set d (of degree d ≥ 2) can be studied using symbolic dynamics. Each parameter is mapped to a kneading sequence, or equivalently, an internal address; but not every such sequence is realized by a parameter in d . Thus the abstract Mandelbrot set is a subspace of a larger, partially ordered symbol space, Λ d . In this paper we find an algorithm to construct “visible trees” from symbolic sequences which works whether or not the sequence is realized....

Shadowing and expansivity in subspaces

Andrew D. Barwell, Chris Good, Piotr Oprocha (2012)

Fundamenta Mathematicae

We address various notions of shadowing and expansivity for continuous maps restricted to a proper subset of their domain. We prove new equivalences of shadowing and expansive properties, we demonstrate under what conditions certain expanding maps have shadowing, and generalize some known results in this area. We also investigate the impact of our theory on maps of the interval.

Shadowing and internal chain transitivity

Jonathan Meddaugh, Brian E. Raines (2013)

Fundamenta Mathematicae

The main result of this paper is that a map f: X → X which has shadowing and for which the space of ω-limits sets is closed in the Hausdorff topology has the property that a set A ⊆ X is an ω-limit set if and only if it is closed and internally chain transitive. Moreover, a map which has the property that every closed internally chain transitive set is an ω-limit set must also have the property that the space of ω-limit sets is closed. As consequences of this result, we show that interval maps with...

Shadowing in multi-dimensional shift spaces

Piotr Oprocha (2008)

Colloquium Mathematicae

We show that the class of expansive d actions with P.O.T.P. is wider than the class of actions topologically hyperbolic in some direction ν d . Our main tool is an extension of a result by Walters to the multi-dimensional symbolic dynamics case.

Shape index and other indices of Conley type for local maps on locally compact Hausdorff spaces

Marian Mrozek (1994)

Fundamenta Mathematicae

We present a scheme for constructing various Conley indices for locally defined maps. In particular, we extend the shape index of Robbin and Salamon to the case of a locally defined map in a locally compact Hausdorff space. We compare the shape index with the cohomological Conley index for maps. We also prove the commutativity property of the Conley index, which is analogous to the commutativity property of the fixed point index.

Shape index in metric spaces

Francisco R. Ruiz del Portal, José M. Salazar (2003)

Fundamenta Mathematicae

We extend the shape index, introduced by Robbin and Salamon and Mrozek, to locally defined maps in metric spaces. We show that this index is additive. Thus our construction answers in the affirmative two questions posed by Mrozek in [12]. We also prove that the shape index cannot be arbitrarily complicated: the shapes of q-adic solenoids appear as shape indices in natural modifications of Smale's horseshoes but there is not any compact isolated invariant set for any locally defined map in a locally...

Simple connection matrices

Piotr Bartłomiejczyk (2007)

Annales Polonici Mathematici

We introduce simple connection matrices. We prove the existence of simple connection matrices for filtered differential vector spaces and Morse decompositions of compact metric spaces.

Some dynamical properties of S-unimodal maps

Tomasz Nowicki (1993)

Fundamenta Mathematicae

We study 1) the slopes of central branches of iterates of S-unimodal maps, comparing them to the derivatives on the critical trajectory, 2) the hyperbolic structure of Collet-Eckmann maps estimating the exponents, and under a summability condition 3) the images of the density one under the iterates of the Perron-Frobenius operator, 4) the density of the absolutely continuous invariant measure.

Some generic properties of concentration dimension of measure

Józef Myjak, Tomasz Szarek (2003)

Bollettino dell'Unione Matematica Italiana

Let K be a compact quasi self-similar set in a complete metric space X and let M 1 K denote the space of all probability measures on K , endowed with the Fortet-Mourier metric. We will show that for a typical (in the sense of Baire category) measure in M 1 K the lower concentration dimension is equal to 0 , while the upper concentration dimension is equal to the Hausdorff dimension of K .

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