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By an emph{assignment} we mean a mapping from a Choquet simplex $K$ to probability measure-preserving systems, obeying some natural restrictions. We prove that if $Phi$ is an aperiodic assignment on a Choquet simplex $K$ such that the set of extreme points $mathsf{ex}K$ is a countable union $bigcup_n E_n$, where each set $E_n$ is compact, zero-dimensional, and the restriction of $Phi$ to the Bauer simplex $K_n$ spanned by $E_n$ can be `embedded in some topological dynamical system, then $Phi$ can be `realized in a zero-dimensional system.
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Hindman and Leader first introduced the notion of Central sets near zero for dense subsemigroups of $((0,infty),+)$ and proved a powerful combinatorial theorem about such sets. Using the algebraic structure of the Stone-$breve{C}$ech compactification, Bayatmanesh and Tootkabani generalized and extended this combinatorial theorem to the central theorem near zero. Algebraically one can define quasi-central set near zero for dense subsemigroup of $((0,infty),+)$, and they also satisfy the conclusion of central sets theorem near zero. In a dense subsemigroup of $((0,infty),+)$, C-sets near zero are the sets, which satisfies the conclusions of the central sets theorem near zero. Like discrete case, we shall produce dynamical characterizations of these combinatorically rich sets near zero.
A zero-dimensional (resp. symbolic) flow is a suspension flow over a zero-dimensional system (resp. a subshift). We show that any topological flow admits a principal extension by a zero-dimensional flow. Following [Bur19] we deduce that any topological flow admits an extension by a symbolic flow if and only if its time-$t$ map admits an extension by a subshift for any $t eq 0$. Moreover the existence of such an extension is preserved under orbit equivalence for regular topological flows, but this property does not hold more true for singular flows. Finally we investigate symbolic extensions for singular suspension flows. In particular, the suspension flow over the full shift on ${0,1}^{mathbb Z}$ with a roof function $f$ vanishing at the zero sequence $0^infty$ admits a principal symbolic extension or not depending on the smoothness of $f$ at $0^infty$.
Since its inception, control of data congestion on the Internet has been based on stochastic models. One of the first such models was Random Early Detection. Later, this model was reformulated as a dynamical system, with the average queue sizes at a routers buffer being the states. Recently, the dynamical model has been generalized to improve global stability. In this paper we review the original stochastic model and both nonlinear models of Random Early Detection with a two-fold objective: (i) illustrate how a random model can be smoothed out to a deterministic one through data aggregation, and (ii) how this translation can shed light into complex processes such as the Internet data traffic. Furthermore, this paper contains new materials concerning the occurrence of chaos, bifurcation diagrams, Lyapunov exponents and global stability robustness with respect to control parameters. The results reviewed and reported here are expected to help design an active queue management algorithm in real conditions, that is, when system parameters such as the number of users and the round-trip time of the data packets change over time. The topic also illustrates the much-needed synergy of a theoretical approach, practical intuition and numerical simulations in engineering.
This paper has been withdrawn by the authors due to an error in the main theorem.
This paper is concerned with the dynamics of an infinite-dimensional gradient system under small almost periodic perturbations. Under the assumption that the original autonomous system has a global attractor given as the union of unstable manifolds of a finite number of hyperbolic equilibrium solutions, we prove that the perturbed non-autonomous system has exactly the same number of almost periodic solutions. As a consequence, the pullback attractor of the perturbed system is given by the union of unstable manifolds of these finitely many almost periodic solutions. An application of the result to the Chafee-Infante equation is discussed.
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