scholarly journals A return time invariant for finitary isomorphisms

1984 ◽  
Vol 4 (2) ◽  
pp. 225-231 ◽  
Author(s):  
U. R. Fiebig
Keyword(s):  
Finite Measure ◽  
Measurable Subset ◽  
Return Time ◽  
Time Function ◽  
Markov Shifts ◽  
Measure Preserving Transformation ◽  
Recurrence Theorem ◽  
First Return Time ◽  
Time Invariant ◽  
Finitary Isomorphisms

AbstractPoincare's recurrence theorem says that, given a measurable subset of a space on which a finite measure-preserving transformation acts, almost every point of the subset returns to the subset after a finite number of applications of the transformation. Moreover, Kac's recurrence theorem refines this result by showing that the average of the first return times to the subset over the subset is at most one, with equality in the ergodic case. In particular, the first return time function to any measurable set is integrable. By considering the supremum over all p ≥ 1 for which the first return time function is p-integrable for all open sets, we obtain a number for each almost-topological dynamical system, which we call the return time invariant. It is easy to show that this invariant is non-decreasing under finitary homomorphism. We use the invariant to construct a continuum number of countable state Markov shifts with a given entropy (and hence measure-theoretically isomorphic) which are pairwise non-finitarily isomorphic.

scholarly journals ON YOUNG TOWERS ASSOCIATED WITH INFINITE MEASURE PRESERVING TRANSFORMATIONS

2009 ◽  
Vol 09 (04) ◽  
pp. 635-655 ◽  
Author(s):  
H. BRUIN ◽  
M. NICOL ◽  
D. TERHESIU
Keyword(s):  
Dynamical System ◽  
Common Factor ◽  
Sufficient Conditions ◽  
Finite Measure ◽  
Original System ◽  
Return Time ◽  
Tail Behavior ◽  
Necessary And Sufficient ◽  
Young Towers ◽  
Measure Preserving Transformations

For a σ-finite measure preserving dynamical system (X, μ, T), we formulate necessary and sufficient conditions for a Young tower (Δ, ν, F) to be a (measure theoretic) extension of the original system. Because F is pointwise dual ergodic by construction, one immediate consequence of these conditions is that the Darling–Kac theorem carries over from F to T. One advantage of the Darling–Kac theorem in terms of Young towers is that sufficient conditions can be read off from the tail behavior and we illustrate this with relevant examples. Furthermore, any two Young towers with a common factor T, have return time distributions with tails of the same order.

Scalings of first-return time for random walks on generalized and weighted transfractal networks

2019 ◽  
Vol 33 (26) ◽  
pp. 1950306
Author(s):  
Qin Liu ◽  
Weigang Sun ◽  
Suyu Liu
Keyword(s):  
Random Walks ◽  
Large Scale ◽  
Probability Generating Function ◽  
Return Time ◽  
Network Size ◽  
Large Network ◽  
Large Scale Network ◽  
Scale Network ◽  
The Mean ◽  
First Return Time

The first-return time (FRT) is an effective measurement of random walks. Presently, it has attracted considerable attention with a focus on its scalings with regard to network size. In this paper, we propose a family of generalized and weighted transfractal networks and obtain the scalings of the FRT for a prescribed initial hub node. By employing the self-similarity of our networks, we calculate the first and second moments of FRT by the probability generating function and obtain the scalings of the mean and variance of FRT with regard to network size. For a large network, the mean FRT scales with the network size at the sublinear rate. Further, the efficiency of random walks relates strongly with the weight factor. The smaller the weight, the better the efficiency bears. Finally, we show that the variance of FRT decreases with more number of initial nodes, implying that our method is more effective for large-scale network size and the estimation of the mean FRT is more reliable.

scholarly journals Optimal networks revealed by global mean first return time

2021 ◽  
Author(s):  
JunHao Peng ◽  
Renxiang Shao ◽  
Huoyun Wang
Keyword(s):  
Return Time ◽  
First Return Time ◽  
Global Mean ◽  
Optimal Networks

scholarly journals Regular variation and rates of mixing for infinite measure preserving almost Anosov diffeomorphisms

2018 ◽  
Vol 40 (3) ◽  
pp. 663-698 ◽  
Author(s):  
HENK BRUIN ◽  
DALIA TERHESIU
Keyword(s):  
Regular Variation ◽  
Return Time ◽  
Main Task ◽  
Two Dimensional ◽  
Dimensional Torus ◽  
Anosov Diffeomorphisms ◽  
Infinite Measure ◽  
Mixing Rates ◽  
First Return Time ◽  
Almost Anosov

The purpose of this paper is to establish mixing rates for infinite measure preserving almost Anosov diffeomorphisms on the two-dimensional torus. The main task is to establish regular variation of the tails of the first return time to the complement of a neighbourhood of the neutral fixed point.

On the Ergodic Averages and the Ergodic Hilbert Transform

1995 ◽  
Vol 47 (2) ◽  
pp. 330-343
Author(s):  
L. M. Fernández-Cabrera ◽  
F. J. Martín-Reyes ◽  
J. L. Torrea
Keyword(s):  
Hilbert Transform ◽  
Measure Space ◽  
Dual Problem ◽  
Finite Measure ◽  
Positive Function ◽  
Ergodic Averages ◽  
Unified Approach ◽  
Measure Preserving Transformation ◽  
Finite Measure Space ◽  
Abstract Method

AbstractLet T be an invertible measure-preserving transformation on a σ-finite measure space (X, μ) and let 1 < p < ∞. This paper uses an abstract method developed by José Luis Rubio de Francia which allows us to give a unified approach to the problems of characterizing the positive measurable functions v such that the limit of the ergodic averages or the ergodic Hilbert transform exist for all f ∈ Lp(νdμ). As a corollary, we obtain that both problems are equivalent, extending to this setting some results of R. Jajte, I. Berkson, J. Bourgain and A. Gillespie. We do not assume the boundedness of the operator Tf(x) = f(Tx) on Lp(νdμ). However, the method of Rubio de Francia shows that the problems of convergence are equivalent to the existence of some measurable positive function u such that the ergodic maximal operator and the ergodic Hilbert transform are bounded from LP(νdμ) into LP(udμ). We also study and solve the dual problem.

On the critical points of the flight return time function of perturbed closed orbits

2019 ◽  
Vol 266 (12) ◽  
pp. 8344-8369 ◽  
Author(s):  
Adriana Buică ◽  
Jaume Giné ◽  
Maite Grau
Keyword(s):  
Critical Points ◽  
Return Time ◽  
Time Function ◽  
Closed Orbits

scholarly journals On the return time function around monodromic polycycles

2006 ◽  
Vol 228 (1) ◽  
pp. 226-258 ◽  
Author(s):  
D. Marín ◽  
J. Villadelprat
Keyword(s):  
Return Time ◽  
Time Function

scholarly journals Matrix models, Toeplitz determinants and recurrence times for powers of random unitary matrices

2015 ◽  
Vol 04 (03) ◽  
pp. 1550011 ◽  
Author(s):  
O. Marchal
Keyword(s):  
Return Time ◽  
Unitary Matrix ◽  
Unitary Matrices ◽  
Toeplitz Determinants ◽  
Recurrence Times ◽  
Toeplitz Determinant ◽  
Large N ◽  
Single Arc ◽  
Random Unitary Matrices ◽  
First Return Time

The purpose of this paper is to study the eigenvalues [Formula: see text] of Ut where U is a large N×N random unitary matrix and t > 0. In particular we are interested in the typical times t for which all the eigenvalues are simultaneously close to 1 in different ways thus corresponding to recurrence times in the issue of quantum measurements. Our strategy consists in rewriting the problem as a random matrix integral and use loop equations techniques to compute the first-orders of the large N asymptotic. We also connect the problem to the computation of a large Toeplitz determinant whose symbol is the characteristic function of several arc segments of the unit circle. In particular in the case of a single arc segment we recover Widom's formula. Eventually we explain why the first return time is expected to converge toward an exponential distribution when N is large. Numerical simulations are provided along the paper to illustrate the results.

Strong Poincaré recurrence theorem in MV-algebras

2010 ◽  
Vol 60 (5) ◽  
Author(s):  
Beloslav Riečan
Keyword(s):  
Probability Space ◽  
Mv Algebra ◽  
Measure Preserving Transformation ◽  
Poincaré Recurrence Theorem ◽  
Recurrence Theorem ◽  
Poincare Recurrence ◽  
Poincaré Recurrence ◽  
Almost All ◽  
Poincare Theorem ◽  
Mv Algebras

AbstractThe classical Poincaré strong recurrence theorem states that for any probability space (Ω, ℒ, P), any P-measure preserving transformation T, and any A ∈ ℒ, almost all points of A return to A infinitely many times. In the present paper the Poincaré theorem is proved when the σ-algebra ℒ is substituted by an MV-algebra of a special type. Another approach is used in [RIEČAN, B.: Poincaré recurrence theorem in MV-algebras. In: Proc. IFSA-EUSFLAT 2009 (To appear)], where the weak variant of the theorem is proved, of course, for arbitrary MV-algebras. Such generalizations were already done in the literature, e.g. for quantum logic, see [DVUREČENSKIJ, A.: On some properties of transformations of a logic, Math. Slovaca 26 (1976), 131–137.

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