scholarly journals On topological dynamical systems with discrete spectrum

1985 ◽  
Vol 61 (1) ◽  
pp. 8-10
Author(s):  
Koukichi Sakai
Keyword(s):  
Dynamical Systems ◽  
Discrete Spectrum ◽  
Topological Dynamical Systems

scholarly journals Weak forms of topological and measure-theoretical equicontinuity: relationships with discrete spectrum and sequence entropy

2016 ◽  
Vol 37 (4) ◽  
pp. 1211-1237 ◽  
Author(s):  
FELIPE GARCÍA-RAMOS
Keyword(s):  
Dynamical Systems ◽  
Discrete Spectrum ◽  
Haar Measure ◽  
Topological Category ◽  
Sequence Entropy ◽  
Ergodic Measures ◽  
Factor Map ◽  
Topological Dynamical Systems ◽  
Theoretical Sensitivity ◽  
Topological Sequence Entropy

We define weaker forms of topological and measure-theoretical equicontinuity for topological dynamical systems, and we study their relationships with sequence entropy and systems with discrete spectrum. We show that for topological systems equipped with ergodic measures having discrete spectrum is equivalent to$\unicode[STIX]{x1D707}$-mean equicontinuity. In the purely topological category we show that minimal subshifts with zero topological sequence entropy are strictly contained in diam-mean equicontinuous systems; and that transitive almost automorphic subshifts are diam-mean equicontinuous if and only if they are regular (i.e. the maximal equicontinuous factor map is one–one on a set of full Haar measure). For both categories we find characterizations using stronger versions of the classical notion of sensitivity. As a consequence, we obtain a dichotomy between discrete spectrum and a strong form of measure-theoretical sensitivity.

scholarly journals Q-entropy for general topological dynamical systems

2019 ◽  
Vol 39 (4) ◽  
pp. 2059-2075 ◽  
Author(s):  
Yun Zhao ◽  
◽  
Wen-Chiao Cheng ◽  
Chih-Chang Ho ◽  
◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Topological Dynamical Systems

Almost topological dynamical systems

1979 ◽  
Vol 34 (1-2) ◽  
pp. 139-160 ◽  
Author(s):  
Manfred Denker ◽  
Michael Keane
Keyword(s):  
Dynamical Systems ◽  
Topological Dynamical Systems

scholarly journals A combinatorial approach to products of Pisot substitutions

2015 ◽  
Vol 36 (6) ◽  
pp. 1757-1794 ◽  
Author(s):  
VALÉRIE BERTHÉ ◽  
JÉRÉMIE BOURDON ◽  
TIMO JOLIVET ◽  
ANNE SIEGEL
Keyword(s):  
Dynamical Systems ◽  
Discrete Spectrum ◽  
Continued Fraction ◽  
Combinatorial Approach ◽  
Pisot Numbers ◽  
Algorithmic Framework ◽  
Multidimensional Continued Fraction ◽  
Finite Products ◽  
Dynamical Properties ◽  
Coincidence Conditions

We define a generic algorithmic framework to prove a pure discrete spectrum for the substitutive symbolic dynamical systems associated with some infinite families of Pisot substitutions. We focus on the families obtained as finite products of the three-letter substitutions associated with the multidimensional continued fraction algorithms of Brun and Jacobi–Perron. Our tools consist in a reformulation of some combinatorial criteria (coincidence conditions), in terms of properties of discrete plane generation using multidimensional (dual) substitutions. We also deduce some topological and dynamical properties of the Rauzy fractals, of the underlying symbolic dynamical systems, as well as some number-theoretical properties of the associated Pisot numbers.

On the dynamics of the Tent function - Phase diagrams

2016 ◽  
Vol 7 (4) ◽  
pp. 261
Author(s):  
Prince Amponsah Kwabi ◽  
William Obeng Denteh ◽  
Richard Kena Boadi
Keyword(s):  
Dynamical Systems ◽  
Phase Diagrams ◽  
Initial Conditions ◽  
Asymptotic Properties ◽  
Topological Dynamical System ◽  
Topological Transitivity ◽  
Tent Map ◽  
One Dimensional ◽  
Definition Of ◽  
Topological Dynamical Systems

This paper focuses on the study of a one-dimensional topological dynamical system, the tent function. We give a good background to the theory of dynamical systems while establishing the important asymptotic properties of topological dynamical systems that distinguishes it from other fields by way of an example - the tent function. A precise definition of the tent function is given and iterates are clearly shown using the phase diagrams. By this same method, chaos in the tent map is shown as iterates become higher. We also show that the tent map has infinitely many chaotic orbits and also express some important features of chaos such as topological transitivity, boundedness and sensitivity to change in initial conditions from a topological viewpoint.

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