SELF-SIMILAR CORRECTIONS TO THE ERGODIC THEOREM FOR THE PASCAL-ADIC TRANSFORMATION

Let T be the Pascal-adic transformation. For any measurable function g, we consider the corrections to the ergodic theorem [Formula: see text] When seen as graphs of functions defined on {0,…,ℓ - 1}, we show for a suitable class of functions g that these quantities, once properly renormalized, converge to (part of) the graph of a self-affine function. The latter only depends on the ergodic component of x, and is a deformation of the so-called Blancmange function. We also briefly describe the links with a series of works on Conway recursive $10,000 sequence.

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On Krylov’s estimates for optional semimartingales

Random Operators and Stochastic Equations ◽

10.1515/rose-2021-2059 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

M. Abdelghani ◽

A. Melnikov ◽

A. Pak

Keyword(s):

Sobolev Space ◽

Measurable Function ◽

Diffusion Processes ◽

Stochastic Integrals ◽

Controlled Diffusion ◽

Convergence Of Solutions ◽

Change Of Variables ◽

General Class Of Functions ◽

Class Of Functions ◽

Controlled Diffusion Processes

Abstract The estimates of N. V. Krylov for distributions of stochastic integrals by means of the L d {L_{d}} -norm of a measurable function are well-known and are widely used in the theory of stochastic differential equations and controlled diffusion processes. We generalize estimates of this type for optional semimartingales, then apply these estimates to prove the change of variables formula for a general class of functions from the Sobolev space W d 2 {W^{2}_{d}} . We also show how to use these estimates for the investigation of L 2 {L^{2}} -convergence of solutions of optional SDE’s.

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Self-similar extrapolation of nonlinear problems from small-variable to large-variable limit

International Journal of Modern Physics B ◽

10.1142/s0217979220502082 ◽

2020 ◽

Vol 34 (21) ◽

pp. 2050208

Author(s):

V. I. Yukalov ◽

E. P. Yukalova

Keyword(s):

Asymptotic Series ◽

Nonlinear Problems ◽

Numerical Convergence ◽

Physical Problems ◽

Variable Limit ◽

Small Parameters ◽

Self Similar ◽

Similar Approximation ◽

Similar Factor ◽

Class Of Functions

Complicated physical problems are usually solved by resorting to perturbation theory leading to solutions in the form of asymptotic series in powers of small parameters. However, finite, and even large values of the parameters, are often of main physical interest. A method is described for predicting the large-variable behavior of solutions to nonlinear problems from the knowledge of only their small-variable expansions. The method is based on self-similar approximation theory resulting in self-similar factor approximants. The latter can well approximate a large class of functions, rational, irrational, and transcendental. The method is illustrated by several examples from statistical and condensed matter physics, where the self-similar predictions can be compared with the available large-variable behavior. It is shown that the method allows for finding the behavior of solutions at large variables when knowing just a few terms of small-variable expansions. Numerical convergence of approximants is demonstrated.

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V.—Les transformations asymptotiquement presque périodiques discontinues et le lemme ergodique. (Première Note.)

Proceedings of the Royal Society of Edinburgh Section A Mathematical and Physical Sciences ◽

10.1017/s0080454100006981 ◽

1950 ◽

Vol 63 (1) ◽

pp. 61-68

Author(s):

Maurice Fréchet

Keyword(s):

Ergodic Theorem ◽

Continuous Functions ◽

Almost Periodic Functions ◽

Almost Periodic ◽

Periodic Functions ◽

Discontinuous Functions ◽

Extended Class ◽

Class Of Functions ◽

Asymptotically Almost Periodic

SynopsisWith the aim of establishing, under wide conditions, the ergodic theorem of G. D. Birkhoff, the author extends the class of asymptotically almost-periodic functions, considering now not only continuous functions, as he had already done in 1943, but discontinuous functions. Definitions and properties of the extended class of functions are set out, some comparisons being made with almost-periodic functions in the sense of Bohr, Stepanoff, Weyl and Besicovitch. Applications to the ergodic theorem are adumbrated.

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Nonlinear Equations of Infinite Order Defined by an Elliptic Symbol

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2014/656959 ◽

2014 ◽

Vol 2014 ◽

pp. 1-7

Author(s):

Mauricio Bravo Vera

Keyword(s):

Hilbert Space ◽

Laplace Operator ◽

Nonlinear Equations ◽

Measurable Function ◽

Infinite Order ◽

Formal Definition ◽

Regularity Properties ◽

Definition Of ◽

The Laplace Operator ◽

Class Of Functions

The aim of this work is to show existence and regularity properties of equations of the formf(Δ)u=U(x,u(x))onℝn, in whichfis a measurable function that satisfies some conditions of ellipticity andΔstands for the Laplace operator onℝn. Here, we define the class of functions to whichfbelongs and the Hilbert space in which we will find the solution to this equation. We also give the formal definition off(Δ)explaining how to understand this operator.

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Self-Affine Processes and the Ergodic Theorem

Canadian Mathematical Bulletin ◽

10.4153/cmb-1994-037-2 ◽

1994 ◽

Vol 37 (2) ◽

pp. 254-262 ◽

Cited By ~ 2

Author(s):

Wim Vervaat

Keyword(s):

Ergodic Theorem ◽

Sample Path ◽

Stationary Increments ◽

Affine Processes ◽

Self Similar ◽

The One

AbstractKnown results for strictly stable motions as finiteness of moments and local boundednessof sample-path variation are generalized to self-affine processes, i.e., self-similar processes with stationary increments. The proofs are new, even for stable motions, and are obtained by applying the ergodic theorem to powers of the (one-sided) increments.

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Powers of sequences and convergence of ergodic averages

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385709000571 ◽

2009 ◽

Vol 30 (5) ◽

pp. 1431-1456 ◽

Cited By ~ 1

Author(s):

N. FRANTZIKINAKIS ◽

M. JOHNSON ◽

E. LESIGNE ◽

M. WIERDL

Keyword(s):

Ergodic Theorem ◽

Measurable Function ◽

Pointwise Convergence ◽

Ergodic Averages ◽

Bounded Measurable Function ◽

Mean Ergodic Theorem ◽

The Mean ◽

Good For

AbstractA sequence (sn) of integers is good for the mean ergodic theorem if for each invertible measure-preserving system (X,ℬ,μ,T) and any bounded measurable function f, the averages (1/N)∑ Nn=1f(Tsnx) converge in the L2(μ) norm. We construct a sequence (sn) which is good for the mean ergodic theorem but such that the sequence (s2n) is not. Furthermore, we show that for any set of bad exponents B, there is a sequence (sn) where (skn) is good for the mean ergodic theorem exactly when k is not in B. We then extend this result to multiple ergodic averages of the form (1/N)∑ Nn=1f1(Tsnx)f2(T2snx)⋯fℓ(Tℓsnx). We also prove a similar result for pointwise convergence of single ergodic averages.

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Second-order ergodic theorem for self-similar tiling systems

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2013.27 ◽

2013 ◽

Vol 34 (6) ◽

pp. 2018-2053

Author(s):

KONSTANTIN MEDYNETS ◽

BORIS SOLOMYAK

Keyword(s):

Hausdorff Dimension ◽

Euclidean Space ◽

Ergodic Theorem ◽

Second Order ◽

Substitution Rule ◽

Infinite Measure ◽

Tiling Systems ◽

Self Similar

AbstractWe consider infinite measure-preserving non-primitive self-similar tiling systems in Euclidean space ${ \mathbb{R} }^{d} $. We establish the second-order ergodic theorem for such systems, with exponent equal to the Hausdorff dimension of a graph-directed self-similar set associated with the substitution rule.

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ABOUT ONE CLASS OF FUNCTIONS WITH FRACTAL PROPERTIES

Bukovinian Mathematical Journal ◽

10.31861/bmj2021.01.23 ◽

2021 ◽

Vol 9 (1) ◽

pp. 273-283

Author(s):

Ya. Goncharenko ◽

M. Pratsiovytyi ◽

S. Dmytrenko ◽

I. Lysenko ◽

S. Ratushniak

Keyword(s):

Random Variables ◽

Variational Integral ◽

Numerical Series ◽

Fractal Properties ◽

Fractal Functions ◽

Self Similar ◽

Object Of Study ◽

Class Of Functions

We consider one generalization of functions, which are called as «binary self-similar functi- ons» by Bl. Sendov. In this paper, we analyze the connections of the object of study with well known classes of fractal functions, with the geometry of numerical series, with distributions of random variables with independent random digits of the two-symbol $Q_2$-representation, with theory of fractals. Structural, variational, integral, diﬀerential and fractal properties are studied for the functions of this class.

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