scholarly journals Dichotomy law for shrinking target problems in a nonautonomous dynamical system: Cantor series expansion

2017 ◽  
Vol 145 (6) ◽  
pp. 2349-2359 ◽  
Author(s):  
Yu Sun ◽  
Chun-Yun Cao
Keyword(s):  
Dynamical System ◽  
Series Expansion ◽  
Nonautonomous Dynamical System ◽  
Cantor Series

scholarly journals SHRINKING TARGETS FOR NONAUTONOMOUS DYNAMICAL SYSTEMS CORRESPONDING TO CANTOR SERIES EXPANSIONS

2015 ◽  
Vol 92 (2) ◽  
pp. 205-213 ◽  
Author(s):  
LIOR FISHMAN ◽  
BILL MANCE ◽  
DAVID SIMMONS ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Hausdorff Dimension ◽  
Diophantine Approximation ◽  
Wide Class ◽  
Series Expansions ◽  
Closed Formula ◽  
Nonautonomous Dynamical Systems ◽  
Nonautonomous Dynamical System ◽  
Cantor Series

We provide a closed formula of Bowen type for the Hausdorff dimension of a very general shrinking target scheme generated by the nonautonomous dynamical system on the interval$[0,1)$, viewed as$\mathbb{R}/\mathbb{Z}$, corresponding to a given method of Cantor series expansion. We also examine a wide class of examples utilising our theorem. In particular, we give a Diophantine approximation interpretation of our scheme.

scholarly journals Metric Entropy of Nonautonomous Dynamical Systems

2014 ◽  
Vol 1 (1) ◽  
Author(s):  
Christoph Kawan
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Metric Entropy ◽  
Nonautonomous Dynamical Systems ◽  
Nonautonomous Dynamical System

AbstractWe introduce the notion of metric entropy for a nonautonomous dynamical system given by a sequence (X

scholarly journals Dynamics of Weakly Mixing Nonautonomous Systems

2019 ◽  
Vol 29 (09) ◽  
pp. 1950123 ◽  
Author(s):  
Mohammad Salman ◽  
Ruchi Das
Keyword(s):  
Dynamical System ◽  
Sufficient Conditions ◽  
Unit Interval ◽  
Nonautonomous System ◽  
Probability Measures ◽  
Topological Transitivity ◽  
Weak Mixing ◽  
Nonautonomous Systems ◽  
Nonautonomous Dynamical System ◽  
Weakly Mixing

For a commutative nonautonomous dynamical system we show that topological transitivity of the nonautonomous system induced on probability measures (hyperspaces) is equivalent to the weak mixing of the induced systems. Several counter examples are given for the results which are true in autonomous but need not be true in nonautonomous systems. Wherever possible sufficient conditions are obtained for the results to hold true. For a commutative periodic nonautonomous system on intervals, it is proved that weak mixing implies Devaney chaos. Given a periodic nonautonomous system, it is shown that sensitivity is equivalent to some stronger forms of sensitivity on a closed unit interval.

Remarks on Topological Entropy of Nonautonomous Dynamical Systems

2015 ◽  
Vol 25 (12) ◽  
pp. 1550158 ◽  
Author(s):  
Zhiming Li
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Topological Entropy ◽  
Nonautonomous Dynamical Systems ◽  
Nonautonomous Dynamical System

In this paper, we give several classical definitions of topological entropy (on a noncompact and noninvariant subset) for nonautonomous dynamical system. Furthermore, their relationships are established.

Directional entropy of ℤ+k-actions

2015 ◽  
Vol 16 (01) ◽  
pp. 1650004 ◽  
Author(s):  
Jinlian Zhang ◽  
Wenda Zhang
Keyword(s):  
Dynamical System ◽  
Variational Principle ◽  
Lyapunov Exponents ◽  
Riemannian Manifolds ◽  
Metric Space ◽  
Compact Metric Space ◽  
Finite Graphs ◽  
Nonautonomous Dynamical System

In this paper, topological and measure-theoretic directional entropies are investigated for [Formula: see text]-actions. Let [Formula: see text] be a [Formula: see text]-action on a compact metric space. For each ray [Formula: see text] in [Formula: see text] we introduce a notion of positive expansivity for [Formula: see text] along [Formula: see text]. We apply the technique of “coding” which was given by Boyle and Lind in [1] to show that these directional entropies are both continuous at positively expansive directions. We relate the directional entropies of a [Formula: see text]-action at a ray [Formula: see text] to the entropies of a nonautonomous dynamical system which induced by the compositions of a sequence of maps along [Formula: see text]. And hence the variational principle relating topological and measure-theoretic directional entropies is given at positively expansive directions. Applying some known results relating entropies and other invariants (such as preimage entropies, degrees and Lyapunov exponents), we obtain the formulas of directional entropies for some classic examples, such as the [Formula: see text]-subshift actions on [Formula: see text], [Formula: see text]-actions on finite graphs and certain smooth [Formula: see text]-actions on Riemannian manifolds.

scholarly journals The Existence of WeakD-Pullback Exponential Attractor for Nonautonomous Dynamical System

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Yongjun Li ◽  
Xiaona Wei ◽  
Yanhong Zhang
Keyword(s):  
Dynamical System ◽  
Diffusion Equation ◽  
External Force ◽  
Exponential Growth ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Exponential Attractor ◽  
Exponential Attractors ◽  
Nonautonomous Dynamical System

First, for a processU(t,τ)∣t≥τ, we introduce a new concept, called the weakD-pullback exponential attractor, which is a family of setsM(t)∣t≤T, for anyT∈R, satisfying the following: (i)M(t)is compact, (ii)M(t)is positively invariant, that is,U(t,τ)M(τ)⊂M(t), and (iii) there existk,l>0such thatdist(U(t,τ)B(τ),M(t))≤ke-(t-τ); that is,M(t)pullback exponential attractsB(τ). Then we give a method to obtain the existence of weakD-pullback exponential attractors for a process. As an application, we obtain the existence of weakD-pullback exponential attractor for reaction diffusion equation inH01with exponential growth of the external force.

scholarly journals BETA CANTOR SERIES EXPANSION AND ADMISSIBLE SEQUENCES

2020 ◽  
Vol 60 (3) ◽  
pp. 214-224
Author(s):  
Jonathan Caalim ◽  
Shiela Demegillo
Keyword(s):  
Real Number ◽  
Series Expansion ◽  
Numeration System ◽  
Integer Sequence ◽  
Cantor Series ◽  
Admissible Sequences

We introduce a numeration system, called the <em>beta Cantor series expansion</em>, that generalizes the classical positive and negative beta expansions by allowing non-integer bases in the Q-Cantor series expansion. In particular, we show that for a fix $\gamma \in \mathbb{R}$ and a sequence $B=\{\beta_i\}$ of real number bases, every element of the interval $x \in [\gamma,\gamma+1)$ has a <em>beta Cantor series expansion</em> with respect to B where the digits are integers in some alphabet $\mathcal{A}(B)$. We give a criterion in determining whether an integer sequence is admissible when $B$ satisfies some condition. We provide a description of the reference strings, namely the expansion of $\gamma$ and $\gamma+1$, used in the admissibility criterion.

scholarly journals On the Hausdorff dimension faithfulness and the Cantor series expansion

2020 ◽  
Vol 26 (4) ◽  
pp. 298-310
Author(s):  
S. Albeverio ◽  
Ganna Ivanenko ◽  
Mykola Lebid ◽  
Grygoriy Torbin
Keyword(s):  
Hausdorff Dimension ◽  
Series Expansion ◽  
Cantor Series
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