New Exotic Minimal Sets from Pseudo-Suspensions of Cantor Systems

We develop a technique, pseudo-suspension, that applies to invariant sets of homeomorphisms of a class of annulus homeomorphisms we describe, Handel–Anosov–Katok (HAK) homeomorphisms, that generalize the homeomorphism first described by Handel. Given a HAK homeomorphism and a homeomorphism of the Cantor set, the pseudo-suspension yields a homeomorphism of a new space that combines features of both of the original homeomorphisms. This allows us to answer a well known open question by providing examples of hereditarily indecomposable continua that admit homeomorphisms with positive finite entropy. Additionally, we show that such examples occur as minimal sets of volume preserving smooth diffeomorphisms of 4-dimensional manifolds.We construct an example of a minimal, weakly mixing and uniformly rigid homeomorphism of the pseudo-circle, and by our method we are also able to extend it to other one-dimensional hereditarily indecomposable continua, thereby producing the first examples of minimal, uniformly rigid and weakly mixing homeomorphisms in dimension 1. We also show that the examples we construct can be realized as invariant sets of smooth diffeomorphisms of a 4-manifold. Until now the only known examples of connected spaces that admit minimal, uniformly rigid and weakly mixing homeomorphisms were modifications of those given by Glasner and Maon in dimension at least 2.

Topological Dynamics of Zadeh’s Extension on Upper Semi-Continuous Fuzzy Sets

In this paper, some characterizations are obtained on the transitivity, mildly mixing property, a-transitivity, equicontinuity, uniform rigidity and proximality of Zadeh’s extensions restricted on some invariant closed subsets of all upper semi-continuous fuzzy sets in the level-wise metric. In particular, it is proved that a dynamical system is weakly mixing (resp., mildly mixing, weakly mixing and a-transitive, equicontinuous, uniformly rigid) if and only if the corresponding Zadeh’s extension is transitive (resp., mildly mixing, a-transitive, equicontinuous, uniformly rigid).

Chaos of Transformations Induced Onto the Space of Probability Measures

For a dynamical system [Formula: see text], let [Formula: see text] be its induced dynamical system on the space of Borel probability measures with weak*-topology. It is proved that [Formula: see text] is [Formula: see text]-transitive (resp., exact, uniformly rigid) if and only if [Formula: see text] is weakly mixing and [Formula: see text]-transitive (resp., exact, uniformly rigid), where [Formula: see text] is an [Formula: see text]-vector of integers. Moreover, some analogous results are obtained for the hyperspace.

When are all closed subsets recurrent?

In the paper we study relations of rigidity, equicontinuity and pointwise recurrence between an invertible topological dynamical system (t.d.s.) $(X,T)$ and the t.d.s. $(K(X),T_{K})$ induced on the hyperspace $K(X)$ of all compact subsets of $X$, and provide some characterizations. Among other examples, we construct a minimal, non-equicontinuous, distal and uniformly rigid t.d.s. and a weakly mixing t.d.s. which induces dense periodic points on the hyperspace $K(X)$ but itself does not have dense distal points, solving in that way a few open questions from earlier articles by Dong, and Li, Yan and Ye.

Rigidity in topological dynamics

By analogy with the ergodic theoretical notion, we introduce notions of rigidity for a minimal flow (X, T) according to the various ways a sequence Tni can tend to the identity transformation. The main results obtained are:(i) On a rigid flow there exists a T-invariant, symmetric, closed relation Ñ such that (X, T) is uniformly rigid iff Ñ = Δ, the diagonal relation.(ii) For syndetically distal (hence distal) flows rigidity is equivalent to uniform rigidity.(iii) We construct a family of rigid flows which includes Körner's example, in which Ñ exhibits various kinds of behaviour, e.g. Ñ need not be an equivalence relation.(iv) The structure of flows in the above mentioned family is investigated. It is shown that these flows are almost automorphic.