scholarly journals The limiting distribution and error terms for return times of dynamical systems

2004 ◽  
Vol 10 (3) ◽  
pp. 589-616 ◽  
Author(s):  
Nicolai Haydn ◽  
◽  
Sandro Vaienti ◽  
Keyword(s):  
Dynamical Systems ◽  
Limiting Distribution ◽  
Return Times ◽  
Error Terms

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 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 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

A REMARK CONCERNING THE THERMODYNAMICAL LIMIT OF THE LYAPUNOV SPECTRUM

1996 ◽  
Vol 06 (06) ◽  
pp. 1137-1142 ◽  
Author(s):  
Ya. G. SINAI
Keyword(s):  
Dynamical Systems ◽  
Short Range ◽  
Limiting Distribution ◽  
Lyapunov Spectrum ◽  
Short Range Interaction ◽  
Range Interaction ◽  
Limit Transition ◽  
Thermodynamical Limit ◽  
Definition Of ◽  
General Conditions

We consider dynamical systems of N particles confined in domains of volume V with pair-wise short range interaction. We propose a new definition of the limiting distribution of Lyapunov Spectrum according to which the thermodynamical limit transition is taken before the limit t→∞. The main result proven under rather general conditions gives the existence of this modified limiting distribution of Lyapunov Spectrum.

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 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.

Scaling spectra and return times of dynamical systems

1987 ◽  
Vol 46 (5-6) ◽  
pp. 925-932 ◽  
Author(s):  
Mitchell J. Feigenbaum
Keyword(s):  
Dynamical Systems ◽  
Return Times
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