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On enveloping semigroups of nilpotent group actions generated by unipotent affine transformations of the torus

Rafał Pikuła (2010)

Studia Mathematica

Let G be a group generated by a set of affine unipotent transformations T: X → X of the form T(x) = A x + α, where A is a lower triangular unipotent matrix, α is a constant vector, and X is a finite-dimensional torus. We show that the enveloping semigroup E(X,G) of the dynamical system (X,G) is a nilpotent group and find upper and lower bounds on its nilpotency class. Also, we obtain a description of E(X,G) as a quotient space.

On heredity of strongly proximal actions

C. Robinson Edward Raja (2003)

Archivum Mathematicum

We prove that action of a semigroup T on compact metric space X by continuous selfmaps is strongly proximal if and only if T action on 𝒫 ( X ) is strongly proximal. As a consequence we prove that affine actions on certain compact convex subsets of finite-dimensional vector spaces are strongly proximal if and only if the action is proximal.

On local aspects of topological weak mixing in dimension one and beyond

Piotr Oprocha, Guohua Zhang (2011)

Studia Mathematica

We introduce the concept of weakly mixing sets of order n and show that, in contrast to weak mixing of maps, a weakly mixing set of order n does not have to be weakly mixing of order n + 1. Strictly speaking, we construct a minimal invertible dynamical system which contains a non-trivial weakly mixing set of order 2, whereas it does not contain any non-trivial weakly mixing set of order 3. In dimension one this difference is not that much visible, since we prove that every continuous...

On some notions of chaos in dimension zero

Rafał Pikuła (2007)

Colloquium Mathematicae

We compare four different notions of chaos in zero-dimensional systems (subshifts). We provide examples showing that in that case positive topological entropy does not imply strong chaos, strong chaos does not imply complicated dynamics at all, and ω-chaos does not imply Li-Yorke chaos.

On the Lyapunov numbers

Sergiĭ Kolyada, Oleksandr Rybak (2013)

Colloquium Mathematicae

We introduce and study the Lyapunov numbers-quantitative measures of the sensitivity of a dynamical system (X,f) given by a compact metric space X and a continuous map f: X → X. In particular, we prove that for a minimal topologically weakly mixing system all Lyapunov numbers are the same.

Properties of dynamical systems with the asymptotic average shadowing property

Marcin Kulczycki, Piotr Oprocha (2011)

Fundamenta Mathematicae

This article investigates under what conditions nontransitivity can coexist with the asymptotic average shadowing property. We show that there is a large class of maps satisfying both conditions simultaneously and that it is possible to find such examples even among maps on a compact interval. We also study the limit shadowing property and its relation to the asymptotic average shadowing property.

Proximality in Pisot tiling spaces

Marcy Barge, Beverly Diamond (2007)

Fundamenta Mathematicae

A substitution φ is strong Pisot if its abelianization matrix is nonsingular and all eigenvalues except the Perron-Frobenius eigenvalue have modulus less than one. For strong Pisot φ that satisfies a no cycle condition and for which the translation flow on the tiling space φ has pure discrete spectrum, we describe the collection φ P of pairs of proximal tilings in φ in a natural way as a substitution tiling space. We show that if ψ is another such substitution, then φ and ψ are homeomorphic if and...

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...

Some model theory of SL(2,ℝ)

Jakub Gismatullin, Davide Penazzi, Anand Pillay (2015)

Fundamenta Mathematicae

We study the action of G = SL(2,ℝ), viewed as a group definable in the structure M = (ℝ,+,×), on its type space S G ( M ) . We identify a minimal closed G-flow I and an idempotent r ∈ I (with respect to the Ellis semigroup structure * on S G ( M ) ). We also show that the “Ellis group” (r*I,*) is nontrivial, in fact it is the group with two elements, yielding a negative answer to a question of Newelski.

Currently displaying 61 – 80 of 97