Certain properties of smooth cocycles over flows with an invariant ergodic measure

1987 ◽  
Vol 39 (1) ◽  
pp. 67-73
Author(s):  
V. A. Pliss
Keyword(s):  
Ergodic Measure ◽  
Invariant Ergodic Measure

Bi-invariant sets and measures have integer Hausdorff dimension

1999 ◽  
Vol 19 (2) ◽  
pp. 523-534 ◽  
Author(s):  
DAVID MEIRI ◽  
YUVAL PERES
Keyword(s):  
Hausdorff Dimension ◽  
Invariant Measure ◽  
Ergodic Measure ◽  
Invariant Set ◽  
Invariant Sets ◽  
Dimensional Torus ◽  
Uniformly Distributed Sequence ◽  
Invariant Ergodic Measure

Let $A,B$ be two diagonal endomorphisms of the $d$-dimensional torus with corresponding eigenvalues relatively prime. We show that for any $A$-invariant ergodic measure $\mu$, there exists a projection onto a torus ${\mathbb T}^r$ of dimension $r\ge\dim\mu$, that maps $\mu$-almost every $B$-orbit to a uniformly distributed sequence in ${\mathbb T}^r$. As a corollary we obtain that the Hausdorff dimension of any bi-invariant measure, as well as any closed bi-invariant set, is an integer.

APPLICATIONS OF A RELATIVE VARIATIONAL PRINCIPLE TO DIMENSIONS OF NONCONFORMAL EXPANDING MAPS

2011 ◽  
Vol 11 (04) ◽  
pp. 643-679 ◽  
Author(s):  
YUKI YAYAMA
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Hausdorff Dimension ◽  
Ergodic Measure ◽  
Sierpinski Carpet ◽  
3 Dimensional ◽  
Sofic Shift ◽  
Full Shift ◽  
Sierpiński Carpet ◽  
Invariant Ergodic Measure

Zhao and Cao (2008) showed the relative variational principle for subadditive potentials in random dynamical systems. Applying their result, we find the Hausdorff dimension of an n (≥3)-dimensional general Sierpiński carpet which has an irreducible sofic shift in symbolic representation and study an invariant ergodic measure of full Hausdorff dimension. These generalize the results of Kenyon and Peres (1996) on the Hausdorff dimension of an n-dimensional general Sierpiński carpet represented by a full shift.

scholarly journals ANALYSIS OF THE STOCHASTIC FITZHUGH–NAGUMO SYSTEM

2008 ◽  
Vol 11 (03) ◽  
pp. 427-446 ◽  
Author(s):  
STEFANO BONACCORSI ◽  
ELISA MASTROGIACOMO
Keyword(s):  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Stochastic Differential Equations ◽  
Continuous Dependence ◽  
Infinitesimal Generator ◽  
Ergodic Measure ◽  
Transition Semigroup ◽  
Dissipative Nonlinearity ◽  
Invariant Ergodic Measure

In this paper we study a system of stochastic differential equations with dissipative nonlinearity which arise in certain neurobiology models. Besides proving existence, uniqueness and continuous dependence on the initial datum, we shall mainly be concerned with the asymptotic behaviour of the solution. We prove the existence of an invariant ergodic measure ν associated with the transition semigroup Pt; further, we identify its infinitesimal generator in the space L2 (H; ν).

Variations on a central limit theorem in infinite ergodic theory

2014 ◽  
Vol 35 (5) ◽  
pp. 1610-1657 ◽  
Author(s):  
DAMIEN THOMINE
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Continuous Time ◽  
Ergodic Measure ◽  
Continuous Time Systems ◽  
Infinite Ergodic Theory ◽  
Time Systems ◽  
Negative Sectional Curvature ◽  
Invariant Ergodic Measure

In a previous paper the author proved a distributional convergence for the Birkhoff sums of functions of null average defined over a dynamical system with an infinite, invariant, ergodic measure, akin to a central limit theorem. Here we extend this result to larger classes of observables, with milder smoothness conditions, and to larger classes of dynamical systems, which may no longer be mixing. A special emphasis is given to continuous time systems: semi-flows, flows, and $\mathbb{Z}^{d}$-extensions of flows. The latter generalization is applied to the geodesic flow on $\mathbb{Z}^{d}$-periodic manifolds of negative sectional curvature.

scholarly journals Infinite-step nilsystems, independence and complexity

2011 ◽  
Vol 33 (1) ◽  
pp. 118-143 ◽  
Author(s):  
PANDENG DONG ◽  
SEBASTIÁN DONOSO ◽  
ALEJANDRO MAASS ◽  
SONG SHAO ◽  
XIANGDONG YE
Keyword(s):  
Haar Measure ◽  
Inverse Limit ◽  
Ergodic Measure ◽  
Minimal System ◽  
Open Cover ◽  
Topological Complexity ◽  
One To One ◽  
Uniquely Ergodic ◽  
Invariant Ergodic Measure

AbstractAn∞-step nilsystem is an inverse limit of minimal nilsystems. In this article, it is shown that a minimal distal system is an∞-step nilsystem if and only if it has no non-trivial pairs with arbitrarily long finite IP-independence sets. Moreover, it is proved that any minimal system without non-trivial pairs with arbitrarily long finite IP-independence sets is an almost one-to-one extension of its maximal∞-step nilfactor, and each invariant ergodic measure is isomorphic (in the measurable sense) to the Haar measure on some∞-step nilsystem. The question if such a system is uniquely ergodic remains open. In addition, the topological complexity of an∞-step nilsystem is computed, showing that it is polynomial for each non-trivial open cover.

scholarly journals Pointwise multiple averages for systems with two commuting transformations

2017 ◽  
Vol 38 (6) ◽  
pp. 2132-2157 ◽  
Author(s):  
SEBASTIÁN DONOSO ◽  
WENBO SUN
Keyword(s):  
Ergodic Measure ◽  
Image Position

We show that for every ergodic measure-preserving system $(X,{\mathcal{X}},\unicode[STIX]{x1D707},S,T)$ with commuting transformations $S$ and $T$, the average $$\begin{eqnarray}\frac{1}{N^{3}}\mathop{\sum }_{i,j,k=0}^{N-1}f_{0}(S^{j}T^{k}x)f_{1}(S^{i+j}T^{k}x)f_{2}(S^{j}T^{i+k}x)\end{eqnarray}$$ converges for $\unicode[STIX]{x1D707}$-almost every $x\in X$ as $N\rightarrow \infty$ for all $f_{0},f_{1},f_{2}\in L^{\infty }(\unicode[STIX]{x1D707})$. We also show that if $(X,{\mathcal{X}},\unicode[STIX]{x1D707},S,T)$ is an ergodic measurable distal system, then the average $$\begin{eqnarray}\frac{1}{N}\mathop{\sum }_{i=0}^{N-1}f_{1}(S^{i}x)f_{2}(T^{i}x)\end{eqnarray}$$ converges for $\unicode[STIX]{x1D707}$-almost every $x\in X$ as $N\rightarrow \infty$ for all $f_{1},f_{2}\in L^{\infty }(\unicode[STIX]{x1D707})$.

scholarly journals LIMIT LAWS FOR DISTORTED CRITICAL RETURN TIME PROCESSES IN INFINITE ERGODIC THEORY

2007 ◽  
Vol 07 (01) ◽  
pp. 103-121 ◽  
Author(s):  
MARC KESSEBÖHMER ◽  
MEHDI SLASSI
Keyword(s):  
Asymptotic Behaviour ◽  
Uniform Distribution ◽  
Ergodic Theory ◽  
Ergodic Measure ◽  
Return Time ◽  
Limit Laws ◽  
Measure Spaces ◽  
Distribution Laws ◽  
Infinite Ergodic Theory ◽  
Measure Preserving Transformations

We consider conservative ergodic measure preserving transformations on infinite measure spaces and investigate the asymptotic behaviour of distorted return time processes with respect to sets satisfying a type of Darling–Kac condition. We identify two critical cases for which we prove uniform distribution laws.

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