scholarly journals Local stable and unstable sets for positive entropy C1 dynamical systems

2021 ◽  
Author(s):  
Shilin Feng ◽  
Rui Gao ◽  
Wen Huang ◽  
Zeng Lian
Keyword(s):  
Dynamical Systems ◽  
Positive Entropy ◽  
Stable And Unstable Sets

Entropy of $C^\ast$-dynamical systems defined by bitstreams

1998 ◽  
Vol 18 (4) ◽  
pp. 859-874 ◽  
Author(s):  
V. YA. GOLODETS ◽  
ERLING ST&\Oslash;RMER
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Positive Entropy ◽  
Completely Positive ◽  
Car Algebra

We study automorphisms of the CAR-algebra obtained from binary shifts. We consider cases when the $C^\ast$-dynamical system is asymptotically abelian, is proximally asymptotically abelian, is an entropic $K$-system or has completely positive entropy. The entropy is computed in several cases.

ON FUNCTIONS ATTRACTING POSITIVE ENTROPY

2017 ◽  
Vol 97 (1) ◽  
pp. 69-79 ◽  
Author(s):  
ANNA LORANTY ◽  
RYSZARD J. PAWLAK
Keyword(s):  
Fixed Point ◽  
Dynamical Systems ◽  
Lebesgue Measure ◽  
Positive Entropy ◽  
Density Point

We examine dynamical systems which are ‘nonchaotic’ on a big (in the sense of Lebesgue measure) set in each neighbourhood of a fixed point $x_{0}$, that is, the entropy of this system is zero on a set for which $x_{0}$ is a density point. Considerations connected with this family of functions are linked with functions attracting positive entropy at $x_{0}$, that is, each mapping sufficiently close to the function has positive entropy on each neighbourhood of $x_{0}$.

scholarly journals THE PRIME DECOMPOSITION OF KNOTTED PERIODIC ORBITS IN DYNAMICAL SYSTEMS

1994 ◽  
Vol 03 (01) ◽  
pp. 83-120 ◽  
Author(s):  
MICHAEL C. SULLIVAN
Keyword(s):  
Dynamical Systems ◽  
Periodic Orbits ◽  
Positive Entropy ◽  
Prime Decomposition ◽  
Prime Factors

Templates are used to capture the knotting and linking patterns of periodic orbits of positive entropy flows in 3 dimensions. Here, we study the properties of various templates, especially whether or not there is a bound on the number of prime factors of the knot types of the periodic orbits. We will also see that determining whether two templates are different is highly nontrivial.

Co-induction in dynamical systems

2011 ◽  
Vol 32 (3) ◽  
pp. 919-940 ◽  
Author(s):  
ANTHONY H. DOOLEY ◽  
GUOHUA ZHANG
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Amenable Group ◽  
Positive Entropy ◽  
Completely Positive ◽  
Infinite Subgroup

AbstractIf a countable amenable group G contains an infinite subgroup Γ, one may define, from a measurable action of Γ, the so-called co-induced measurable action of G. These actions were defined and studied by Dooley, Golodets, Rudolph and Sinelsh’chikov. In this paper, starting from a topological action of Γ, we define the co-induced topological action of G. We establish a number of properties of this construction, notably, that the G-action has the topological entropy of the Γ-action and has uniformly positive entropy (completely positive entropy, respectively) if and only if the Γ-action has uniformly positive entropy (completely positive entropy, respectively). We also study the Pinsker algebra of the co-induced action.

Dimensions of stable sets and scrambled sets in positive finite entropy systems

2011 ◽  
Vol 32 (2) ◽  
pp. 599-628 ◽  
Author(s):  
CHUN FANG ◽  
WEN HUANG ◽  
YINGFEI YI ◽  
PENGFEI ZHANG
Keyword(s):  
Dynamical System ◽  
Riemannian Manifold ◽  
Continuum Hypothesis ◽  
Large Set ◽  
Metric Entropy ◽  
Positive Entropy ◽  
Stable Sets ◽  
Stable And Unstable Sets ◽  
The Continuum ◽  
Hyperbolic Points

AbstractWe study the dimensions of stable sets and scrambled sets of a dynamical system with positive finite entropy. We show that there is a measure-theoretically ‘large’ set containing points whose sets of ‘hyperbolic points’ (i.e. points lying in the intersections of the closures of the stable and unstable sets) admit positive Bowen dimension entropies; under the continuum hypothesis, this set also contains a scrambled set with positive Bowen dimension entropies. For several kinds of specific invertible dynamical systems, the lower bounds of the Hausdorff dimension of these sets are estimated. In particular, for a diffeomorphism on a smooth Riemannian manifold with positive entropy, such a lower bound is given in terms of the metric entropy and Lyapunov exponent.

scholarly journals Example of a non-standard extreme-value law

2014 ◽  
Vol 35 (6) ◽  
pp. 1902-1912
Author(s):  
NICOLAI HAYDN ◽  
MICHAL KUPSA
Keyword(s):  
Dynamical Systems ◽  
Extreme Values ◽  
Short Note ◽  
Step Function ◽  
Extreme Value ◽  
Limiting Distribution ◽  
Positive Entropy ◽  
Irrational Numbers ◽  
Distributed Processes ◽  
Suitable Sequence

It has been shown that sufficiently well mixing dynamical systems with positive entropy have extreme-value laws which in the limit converge to one of the three standard distributions known for independently and identically distributed processes, namely Gumbel, Fréchet and Weibull distributions. In this short note, we give an example which has a non-standard limiting distribution for its extreme values. Rotations of the circle by irrational numbers are used and it will be shown that the limiting distribution is a step function where the limit has to be taken along a suitable sequence given by the convergents.

ENTROPY OF EQUILIBRIUM MEASURES OF CONTINUOUS PIECEWISE MONOTONIC MAPS

2004 ◽  
Vol 04 (01) ◽  
pp. 85-94 ◽  
Author(s):  
JÉRÔME BUZZI
Keyword(s):  
Dynamical Systems ◽  
Positive Entropy ◽  
Unique Equilibrium ◽  
Mixing Properties ◽  
Equilibrium Measures ◽  
Thermodynamical Formalism ◽  
Transfer Operators ◽  
Piecewise Monotonic

Considering the thermodynamical formalism of dynamical systems, P. Walters showed that for β-transformations all Lipschitz weights define quasi-compact transfer operators and therefore unique equilibrium measures which additionally have positive entropy and good mixing properties. In this note we generalize this to continuous piecewise monotonic maps of the interval. The case of piecewise monotonic maps with discontinuities remains open.

DISORDERED GROUND STATES OF CLASSICAL LATTICE MODELS

1991 ◽  
Vol 03 (02) ◽  
pp. 125-135 ◽  
Author(s):  
CHARLES RADIN
Keyword(s):  
Dynamical Systems ◽  
Statistical Mechanics ◽  
Continuous Spectrum ◽  
Ground State ◽  
Short Range ◽  
Ground States ◽  
Lattice Models ◽  
Positive Entropy ◽  
Classical Lattice ◽  
Classical Statistical Mechanics

We use strictly ergodic dynamical systems to describe two methods for constructing short range interactions of classical statistical mechanics models with unique ground states and unusual properties of disorder; in particular, these ground states can be mixing under translations (and therefore have purely continuous spectrum), and can have positive entropy. Because of the uniqueness of the ground state the disorder is not of the usual type associated with local degeneracy.

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