Chapter 10 Symbolic and algebraic dynamical systems

2002 ◽  
pp. 765-812 ◽  
Author(s):  
Douglas Lind ◽  
Klaus Schmidt
Keyword(s):  
Dynamical Systems ◽  
Algebraic Dynamical Systems

scholarly journals Surjunctivity and topological rigidity of algebraic dynamical systems

2017 ◽  
Vol 39 (3) ◽  
pp. 604-619 ◽  
Author(s):  
SIDDHARTHA BHATTACHARYA ◽  
TULLIO CECCHERINI-SILBERSTEIN ◽  
MICHEL COORNAERT
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Sufficient Conditions ◽  
Countable Group ◽  
Continuous Group ◽  
Continuous Map ◽  
Algebraic Dynamical Systems ◽  
Metrizable Group ◽  
Group Automorphisms

Let$X$be a compact metrizable group and let$\unicode[STIX]{x1D6E4}$be a countable group acting on$X$by continuous group automorphisms. We give sufficient conditions under which the dynamical system$(X,\unicode[STIX]{x1D6E4})$is surjunctive, i.e. every injective continuous map$\unicode[STIX]{x1D70F}:X\rightarrow X$commuting with the action of$\unicode[STIX]{x1D6E4}$is surjective.

Sur une notion d'autonomie de systèmes dynamiques, appliquée aux ensembles invariants des flots d' Anosov algébriques

1995 ◽  
Vol 15 (1) ◽  
pp. 175-207 ◽  
Author(s):  
A. Zeghib
Keyword(s):  
Dynamical Systems ◽  
Finite Volume ◽  
Algebraic Dynamical Systems ◽  
Anosov Systems ◽  
Anosov System ◽  
Autonomous Dynamical Systems

AbstractWe introduce a notion of autonomous dynamical systems which generalizes algebraic dynamical systems. We show by giving examples and by describing some properties that this generalization is not a trivial one. We apply the methods then developed to algebraic Anosov systems. We prove that a C1-submanifold of finite volume, which is invariant by an algebraic Anosov system is ‘essentially’ algebraic.

Ensembles invariants des flots géodésiques des variétés localement symétriques

1995 ◽  
Vol 15 (2) ◽  
pp. 379-412 ◽  
Author(s):  
A. Zeghib
Keyword(s):  
Dynamical Systems ◽  
Symmetric Spaces ◽  
Positive Curvature ◽  
Geodesic Flows ◽  
Geometric Description ◽  
Locally Symmetric Spaces ◽  
Algebraic Dynamical Systems

AbstractWe study the rectifiable invariant subsets of algebraic dynamical systems determined by ℝ-semisimple one parameter groups. We show that their ergodic components are algebraic. A more precise geometric description of these components is possible in some cases of geodesic flows of locally symmetric spaces with non-positive curvature.

An almost mixing of all orders property of algebraic dynamical systems

2017 ◽  
Vol 39 (5) ◽  
pp. 1211-1233
Author(s):  
L. ARENAS-CARMONA ◽  
D. BEREND ◽  
V. BERGELSON
Keyword(s):  
Dynamical Systems ◽  
Mild Conditions ◽  
Algebraic Dynamical Systems ◽  
Strongly Mixing

We consider dynamical systems, consisting of $\mathbb{Z}^{2}$-actions by continuous automorphisms on shift-invariant subgroups of $\mathbb{F}_{p}^{\mathbb{Z}^{2}}$, where $\mathbb{F}_{p}$ is the field of order $p$. These systems provide natural generalizations of Ledrappier’s system, which was the first example of a 2-mixing $\mathbb{Z}^{2}$-action that is not 3-mixing. Extending the results from our previous work on Ledrappier’s example, we show that, under quite mild conditions (namely, 2-mixing and that the subgroup defining the system is a principal Markov subgroup), these systems are almost strongly mixing of every order in the following sense: for each order, one just needs to avoid certain effectively computable logarithmically small sets of times at which there is a substantial deviation from mixing of this order.

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