scholarly journals Limit sets of dynamical systems on the space of probability measures

1973 ◽  
Vol 14 (3) ◽  
pp. 559-567 ◽  
Author(s):  
Abraham Boyarsky
Keyword(s):  
Dynamical Systems ◽  
Probability Measures ◽  
Limit Sets

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

scholarly journals A non-Borel special alpha-limit set in the square

2021 ◽  
pp. 1-11
Author(s):  
STEPHEN JACKSON ◽  
BILL MANCE ◽  
SAMUEL ROTH
Keyword(s):  
Dynamical Systems ◽  
Limit Set ◽  
Limit Sets ◽  
Unit Square

Abstract We consider the complexity of special $\alpha $ -limit sets, a kind of backward limit set for non-invertible dynamical systems. We show that these sets are always analytic, but not necessarily Borel, even in the case of a surjective map on the unit square. This answers a question posed by Kolyada, Misiurewicz, and Snoha.

scholarly journals AN INDICATOR FUNCTION LIMIT THEOREM IN DYNAMICAL SYSTEMS

2011 ◽  
Vol 11 (04) ◽  
pp. 681-690
Author(s):  
OLIVIER DURIEU ◽  
DALIBOR VOLNÝ
Keyword(s):  
Dynamical System ◽  
Limit Theorem ◽  
Dynamical Systems ◽  
Indicator Function ◽  
Constructive Proof ◽  
Probability Measures

We give a constructive proof of the following result: in all aperiodic dynamical system, for all sequences (an)n∈ℕ ⊂ ℝ+ such that an ↗ ∞ and [Formula: see text] as n → ∞, there exists a set [Formula: see text] having the property that the sequence of the distributions of [Formula: see text] is dense in the space of all probability measures on ℝ. This extends a result of O. Durieu and D. Volný, Ergod. Th. Dynam. Syst. to the non-ergodic case.

scholarly journals A Characterization of theω-Limit Sets of Planar Continuous Dynamical Systems

1998 ◽  
Vol 145 (2) ◽  
pp. 469-488 ◽  
Author(s):  
Francisco Balibrea ◽  
Víctor Jiménez López
Keyword(s):  
Dynamical Systems ◽  
Limit Sets

scholarly journals On sums of indicator functions in dynamical systems

2009 ◽  
Vol 30 (5) ◽  
pp. 1419-1430 ◽  
Author(s):  
OLIVIER DURIEU ◽  
DALIBOR VOLNÝ
Keyword(s):  
Dynamical System ◽  
Limit Theorem ◽  
Dynamical Systems ◽  
Partial Sums ◽  
Probability Measures ◽  
Ergodic Dynamical System ◽  
Indicator Functions

AbstractIn this paper, we are interested in the limit theorem question for sums of indicator functions. We show that in every invertible ergodic dynamical system, for every increasing sequence (an)n∈ℕ⊂ℝ+ such that an↗∞ and an/n→0 as n→∞, there exists a dense Gδ of measurable sets A such that the sequence of the distributions of the partial sums $(\sfrac {1}{a_n})\sum _{i=0}^{n-1}(\ind _A-\mu (A))\circ T^i$ is dense in the set of the probability measures on ℝ.

scholarly journals A Note on Variational Principle of Subsets for Nonautonomous Dynamical Systems

2021 ◽  
pp. 1-16
Author(s):  
Jiao Yang
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Variational Principles ◽  
Borel Probability Measure ◽  
Jel Classification ◽  
Probability Measures ◽  
Nonautonomous Dynamical Systems ◽  
Nonautonomous Case ◽  
Borel Probability ◽  
Discrete Type

Abstract In this paper, we introduce measure-theoretic for Borel probability measures to characterize upper and lower Katok measure-theoretic entropies in discrete type and the measure-theoretic entropy for arbitrary Borel probability measure in nonautonomous case. Then we establish new variational principles for Bowen topological entropy for nonautonomous dynamical systems. JEL classification numbers: 37A35. Keywords: Nonautonomous, Measure-theoretical entropies, Variational principles.

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