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Although Sarnak's conjecture holds for compact group rotations (irrational rotations, odometers), it is not even known whether it holds for all Jewett-Krieger models of such rotations. In this paper we show that it does, as long as the model is at the same a topological extension, via the same map that establishes the isomorphism, of an equicontinuous model. In particular, we recover (after [AKL]) that regular Toeplitz systems satisfy Sarnak's conjecture, and, as another consequence, so do...
We study relations between the almost specification property, the asymptotic average shadowing property and the average shadowing property for dynamical systems on compact metric spaces. We show implications between these properties and relate them to other important notions such as shadowing, transitivity, invariant measures, etc. We provide examples showing that compactness is a necessary condition for these implications to hold. As a consequence, we also obtain a proof that limit shadowing in...
We continue the study of topological properties of the group Homeo(X) of all homeomorphisms of a Cantor set X with respect to the uniform topology τ, which was started by Bezuglyi, Dooley, Kwiatkowski and Medynets. We prove that the set of periodic homeomorphisms is τ-dense in Homeo(X) and deduce from this result that the topological group (Homeo(X),τ) has the Rokhlin property, i.e., there exists a homeomorphism whose conjugacy class is τ-dense in Homeo(X). We also show that for any homeomorphism...
In the paper the fundamental properties of discrete dynamical systems generated by an -condensing mapping ( is the Kuratowski measure of noncompactness) are studied. The results extend and deepen those obtained by M. A. Krasnosel’skij and A. V. Lusnikov in [21]. They are also applied to study a mathematical model for spreading of an infectious disease investigated by P. Takac in [35], [36].
We discuss some results about derivations and crossed homomorphisms arising in the context of locally compact groups and their group algebras, in particular, L¹(G), the von Neumann algebra VN(G) and actions of G on related algebras. We answer a question of Dales, Ghahramani, Grønbæk, showing that L¹(G) is always permanently weakly amenable. Then we show that for some classes of groups (e.g. IN-groups) the homology of L¹(G) with coefficients in VN(G) is trivial. But this is no longer true, in general,...
We consider a class of symbolic systems over a finite alphabet which are minimal almost one-to-one extensions of rotations of compact metric monothetic groups and provide computations of their enveloping semigroups that highlight their algebraic structure. We describe the set of idempotents of these semigroups and introduce a classification that can help distinguish between certain such systems having zero topological entropy.
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.
A theory of essential values of cocycles over minimal rotations with values in locally compact Abelian groups, especially , is developed. Criteria for such a cocycle to be conservative are given. The group of essential values of a cocycle is described.
We prove that action of a semigroup on compact metric space by continuous selfmaps is strongly proximal if and only if action on 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.
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...
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.
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.
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