scholarly journals Weighted Birkhoff ergodic theorem with oscillating weights

2017 ◽  
Vol 39 (5) ◽  
pp. 1275-1289 ◽  
Author(s):  
AI-HUA FAN
Keyword(s):  
Dynamical Systems ◽  
Ergodic Theorem ◽  
Almost Everywhere ◽  
Birkhoff Ergodic Theorem ◽  
Good Sequences

We consider sequences of Davenport type or Gelfond type and prove that sequences of Davenport exponent larger than$\frac{1}{2}$are good sequences of weights for the ergodic theorem, and that the ergodic sums weighted by a sequence of strong Gelfond property are well controlled almost everywhere. We prove that for any$q$-multiplicative sequence, the Gelfond property implies the strong Gelfond property and that sequences realized by dynamical systems can be fully oscillating and have the Gelfond property.

Ergodic fractal measures and dimension conservation

2008 ◽  
Vol 28 (2) ◽  
pp. 405-422 ◽  
Author(s):  
HILLEL FURSTENBERG
Keyword(s):  
Dynamical Systems ◽  
Hausdorff Dimension ◽  
Euclidean Space ◽  
Ergodic Theorem ◽  
Compact Set ◽  
Linear Map ◽  
Compactly Supported ◽  
Almost Everywhere ◽  
Fractal Measures

AbstractA linear map from one Euclidean space to another may map a compact set bijectively to a set of smaller Hausdorff dimension. For ‘homogeneous’ fractals (to be defined), there is a phenomenon of ‘dimension conservation’. In proving this we shall introduce dynamical systems whose states represent compactly supported measures in which progression in time corresponds to progressively increasing magnification. Application of the ergodic theorem will show that, generically, dimension conservation is valid. This ‘almost everywhere’ result implies a non-probabilistic statement for homogeneous fractals.

Induced transformations of recurrent a.m.s. dynamical systems

2015 ◽  
Vol 15 (02) ◽  
pp. 1550010
Author(s):  
Sheng Huang ◽  
Mikael Skoglund
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Random Process ◽  
Ergodic Theorem ◽  
Finite Measure ◽  
Stationary Random Process ◽  
Finite State ◽  
Ratio Ergodic Theorem

This note proves that an induced transformation with respect to a finite measure set of a recurrent asymptotically mean stationary dynamical system with a sigma-finite measure is asymptotically mean stationary. Consequently, the Shannon–McMillan–Breiman theorem, as well as the Shannon–McMillan theorem, holds for all reduced processes of any finite-state recurrent asymptotically mean stationary random process. As a by-product, a ratio ergodic theorem for asymptotically mean stationary dynamical systems is presented.

ON THE ABSOLUTE REGULARITY OF LINEAR RANDOM DYNAMICAL SYSTEMS

2003 ◽  
Vol 03 (04) ◽  
pp. 453-461 ◽  
Author(s):  
LUU HOANG DUC
Keyword(s):  
Dynamical Systems ◽  
Ergodic Theorem ◽  
Random Dynamical Systems ◽  
Absolute Regularity ◽  
Integrability Conditions ◽  
Multiplicative Ergodic Theorem ◽  
Lyapunov Regularity ◽  
The Absolute

We introduce a concept of absolute regularity of linear random dynamical systems (RDS) that is stronger than Lyapunov regularity. We prove that a linear RDS that satisfies the integrability conditions of the multiplicative ergodic theorem of Oseledets is not merely Lyapunov regular but absolutely regular.

scholarly journals Uniform Convergence of Cesaro Averages for Uniquely Ergodic C*-Dynamical Systems

Entropy ◽  
2018 ◽  
Vol 20 (12) ◽  
pp. 987 ◽  
Author(s):  
Francesco Fidaleo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Uniform Convergence ◽  
Unit Circle ◽  
Ergodic Theorem ◽  
Ergodic Theory ◽  
C Algebra ◽  
Cesaro Averages ◽  
Uniformly Convergent ◽  
Uniquely Ergodic

Consider a uniquely ergodic C * -dynamical system based on a unital *-endomorphism Φ of a C * -algebra. We prove the uniform convergence of Cesaro averages 1 n ∑ k = 0 n − 1 λ − n Φ ( a ) for all values λ in the unit circle, which are not eigenvalues corresponding to “measurable non-continuous” eigenfunctions. This result generalizes an analogous one, known in commutative ergodic theory, which turns out to be a combination of the Wiener–Wintner theorem and the uniformly convergent ergodic theorem of Krylov and Bogolioubov.

Akcoglu's Ergodic Theorem for Uniform Sequences

1980 ◽  
Vol 32 (4) ◽  
pp. 880-884
Author(s):  
James H. Olsen
Keyword(s):  
Ergodic Theorem ◽  
Measure Space ◽  
Finite Measure ◽  
Positive Contraction ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Image Position

Let (X, F,) be a sigma-finite measure space. In what follows we assume p fixed, 1 < p < ∞ . Let T be a contraction of Lp(X, F, μ) (‖T‖,p ≦ 1). If ƒ ≧ 0 implies Tƒ ≧ 0 we will say that T is positive. In this paper we prove that if is a uniform sequence (see Section 2 for definition) and T is a positive contraction of Lp, thenexists and is finite almost everywhere for every ƒ ∊ Lp(X, F, μ).

The Individual Ergodic Theorem for Contractions with Fixed Points

1980 ◽  
Vol 23 (1) ◽  
pp. 115-116 ◽  
Author(s):  
James H. Olsen
Keyword(s):  
Fixed Points ◽  
Ergodic Theorem ◽  
Measure Space ◽  
Finite Measure ◽  
Individual Ergodic Theorem ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
The Individual ◽  
Image Position

Let (X, I, μ) be a σ-finite measure space and let T take Lp to Lp, p fixed, 1<p<∞,‖t‖p≤1. We shall say that the individual ergodic theorem holds for T if for any uniform sequence K1, k2,… (for the definition, see [2]) and for any f∊LP(X), the limitexists and is finite almost everywhere.

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