ASYMPTOTICS FOR RETURN TIMES OF RANK-ONE SYSTEMS

2005 ◽  
Vol 05 (01) ◽  
pp. 65-73 ◽  
Author(s):  
VINCENT CHAUMOÎTRE ◽  
MICHAL KUPSA
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Distribution ◽  
Return Times ◽  
Rank One ◽  
Exponential Asymptotics

We give a condition for nonperiodic rank-one systems to have non-exponential asymptotic distribution (equal to 1[1,∞[) of return times along subsequences of cylinders. Applying this result to the staircase transformation, we derive mixing dynamical systems with non-exponential asymptotics. Moreover, we show for two columns rank-one systems unique asymptotic along full sequences of cylinders.

scholarly journals Large deviations in non-uniformly hyperbolic dynamical systems

2008 ◽  
Vol 28 (2) ◽  
pp. 587-612 ◽  
Author(s):  
LUC REY-BELLET ◽  
LAI-SANG YOUNG
Keyword(s):  
Dynamical Systems ◽  
Generating Functions ◽  
Limit Theorems ◽  
Large Deviation ◽  
Return Times ◽  
Hyperbolic Diffeomorphisms ◽  
Large Deviation Principles ◽  
Hyperbolic Dynamical Systems ◽  
Rank One ◽  
Dispersing Billiards

AbstractWe prove large deviation principles for ergodic averages of dynamical systems admitting Markov tower extensions with exponential return times. Our main technical result from which a number of limit theorems are derived is the analyticity of logarithmic moment generating functions. Among the classes of dynamical systems to which our results apply are piecewise hyperbolic diffeomorphisms, dispersing billiards including Lorentz gases, and strange attractors of rank one including Hénon-type attractors.

scholarly journals Potential Well in Poincaré Recurrence

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 379
Author(s):  
Miguel Abadi ◽  
Vitor Amorim ◽  
Sandro Gallo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Recurrence Time ◽  
Potential Well ◽  
Exponential Approximation ◽  
Mixing Processes ◽  
System Perspective ◽  
Return Times ◽  
Error Terms

From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes two new results: (1) For ϕ-mixing and ψ-mixing processes, we give a new exponential approximation for hitting and return times using the potential well as the scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well. Our results apply to processes on countable alphabets and do not assume a complete grammar.

scholarly journals Hitting and return times in ergodic dynamical systems

2005 ◽  
Vol 33 (5) ◽  
pp. 2043-2050 ◽  
Author(s):  
N. Haydn ◽  
Y. Lacroix ◽  
S. Vaienti
Keyword(s):  
Dynamical Systems ◽  
Return Times

The return times and the Wiener—Wintner property for mean-bounded positive operators in Lp

1992 ◽  
Vol 12 (1) ◽  
pp. 1-12
Author(s):  
I. Assani
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Positive Operator ◽  
Positive Operators ◽  
Return Times ◽  
Good Sequence

AbstractWe prove the following two results for mean-bounded positive operators on Lp(µ) (1<p>∞).(1) If (X, , µ, ϕ) is a dynamical system and f ∈ L∞ (X) then the sequence f(ϕn x) is a.e. a universal good sequence for mean-bounded positive operators in Lp. (Return times property.)(2) If T is a mean-bounded positive operator on LP(X, , µ) and f ∈ Lp (µ) then the sequence Tnf)(x) is a.e. a universal good sequence for all dynamical systems (Y, , v,S) in L∞(v). A corollary of (2) is a Wiener-Wintner property for mean-bounded positive operators on Lp.

scholarly journals Inducing schemes for multi-dimensional piecewise expanding maps

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Peyman Eslami
Keyword(s):  
Dynamical Systems ◽  
Statistical Properties ◽  
Expanding Maps ◽  
Return Times ◽  
Return Map ◽  
Exponential Tails ◽  
Inducing Schemes ◽  
Piecewise Expanding Maps

<p style='text-indent:20px;'>We construct inducing schemes for general multi-dimensional piecewise expanding maps where the base transformation is Gibbs-Markov and the return times have exponential tails. Such structures are a crucial tool in proving statistical properties of dynamical systems with some hyperbolicity. As an application we check the conditions for the first return map of a class of multi-dimensional non-Markov, non-conformal intermittent maps.</p>

scholarly journals Asymptotic Information-Theoretic Detection of Dynamical Organization in Complex Systems

Entropy ◽  
2021 ◽  
Vol 23 (4) ◽  
pp. 398
Author(s):  
Gianluca D’Addese ◽  
Laura Sani ◽  
Luca La Rocca ◽  
Roberto Serra ◽  
Marco Villani
Keyword(s):  
Dynamical Systems ◽  
Complex Systems ◽  
Origin Of Life ◽  
Asymptotic Distribution ◽  
Random Variables ◽  
Computationally Efficient ◽  
Information Theoretic ◽  
Simple Index ◽  
Formidable Challenge ◽  
The Origin Of Life

The identification of emergent structures in complex dynamical systems is a formidable challenge. We propose a computationally efficient methodology to address such a challenge, based on modeling the state of the system as a set of random variables. Specifically, we present a sieving algorithm to navigate the huge space of all subsets of variables and compare them in terms of a simple index that can be computed without resorting to simulations. We obtain such a simple index by studying the asymptotic distribution of an information-theoretic measure of coordination among variables, when there is no coordination at all, which allows us to fairly compare subsets of variables having different cardinalities. We show that increasing the number of observations allows the identification of larger and larger subsets. As an example of relevant application, we make use of a paradigmatic case regarding the identification of groups in autocatalytic sets of reactions, a chemical situation related to the origin of life problem.

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