Disjoint topological transitivity for weighted translations generated by group actions

In this note, we give a sufficient and necessary condition for weighted translations, generated by group actions, to be disjoint topologically transitive in terms of the weights, the group element and the measure. The characterization of disjoint topological mixing is obtained as well. Moreover, we apply the results to the quotient spaces of locally compact groups and hypergroups.

Topologically mixing tiling of generated by a generalized substitution

This paper presents sufficient conditions for a substitution tiling dynamical system of $\mathbb {R}^2$ , generated by a generalized substitution on three letters, to be topologically mixing. These conditions are shown to hold on a large class of tiling substitutions originally presented by Kenyon in 1996. This problem was suggested by Boris Solomyak, and many of the techniques that are used in this paper are based on the work by Kenyon, Sadun, and Solomyak [Topological mixing for substitutions on two letters. Ergod. Th. & Dynam. Sys.25(6) (2005), 1919–1934]. They studied one-dimensional tiling dynamical systems generated by substitutions on two letters and provided similar conditions sufficient to ensure that one-dimensional substitution tiling dynamical systems are topologically mixing. If a tiling dynamical system of $\mathbb {R}^2$ satisfies our conditions (and thus is topologically mixing), we can construct additional topologically mixing tiling dynamical systems of $\mathbb {R}^2$ . By considering the stepped surface constructed from a tiling $T_\sigma $ , we can get a new tiling of $\mathbb {R}^2$ by projecting the surface orthogonally onto an irrational plane through the origin.

Variants of shadowing properties for iterated function systems on uniform spaces

In this paper, the notions of topological shadowing, topological ergodic shadowing, topological chain transitivity and topological chain mixing are introduced and studied for an iterated function system (IFS) on uniform spaces. It is proved that if an IFS has topological shadowing property and is topological chain mixing, then it has topological ergodic shadowing and it is topological mixing. Moreover, if an IFS has topological shadowing property and is topological chain transitive, then it is topologically ergodic and hence topologically transitive. Also, these notions are studied for the product IFS on uniform spaces.

Topological mixing of Weyl chamber flows

In this paper we study topological properties of the right action by translation of the Weyl chamber flow on the space of Weyl chambers. We obtain a necessary and sufficient condition for topological mixing.

Jℱ-Class Weighted Backward Shifts

In this article [Formula: see text]-class operators are introduced and some basic properties of [Formula: see text]-vectors are given. The [Formula: see text]-class operators include the [Formula: see text]-class operators and [Formula: see text]-class operators introduced by Costakis and Manoussos in 2008. This class also includes the [Formula: see text]-class and [Formula: see text]-class operators defined by Zhang [2012]. Furthermore, for the unilateral weighted backward shifts on a Fréchet sequence space, we establish a criterion under which the shift operators belong to the [Formula: see text]-class. From the criterion it is easy to obtain the existing criteria of hypercyclic backward shifts and of the topological mixing backward shifts. The obtained criterion also reveals the characteristic of [Formula: see text]-class shift operators by the recurrence property. Meanwhile, we obtain infinite topological entropy when the shifts have stronger recurrence property, which generalizes the related results by Brian et al. in 2017.

Electronic Chaotic Oscillator Realization with Potential Uses in Communication Systems

Chaotic systems have some unique properties that can be taken advantage of in some practical systems. These systems have characteristics such as long-term aperiodicity, continuous power spectral density, topological mixing, and sensitivity to initial conditions, all while still having a clearly defined deterministic structure. The property of continuous power spectral density is of particular interest in spread spectrum communication applications. This work looks to maintain these complex properties in a practical custom electronic realization through careful layout and device selection. Included are simulation results demonstrating the system's sensitivity to initial conditions and topological mixing. In addition to this, the electronic simulation maintains a continuous spectral power density up the fundamental frequency of the oscillator. These simulation results are used design the chaotic oscillator in a hardware demonstration. The hardware results exhibit similar dynamics to the original motivation system. Presented here is a relatively simple electronic implementation that closely maintains the complex properties of an ideal chaotic differential equation.