On circle-valued cocycles of an ergodic measure-preserving transformation

1988 ◽  
Vol 61 (1) ◽  
pp. 29-38 ◽  
Author(s):  
Larry Baggett
Keyword(s):  
Ergodic Measure ◽  
Measure Preserving Transformation

scholarly journals Entropy of systems

2016 ◽  
Vol 38 (3) ◽  
pp. 1118-1126
Author(s):  
RADU B. MUNTEANU
Keyword(s):  
Positive Integer ◽  
Probability Space ◽  
Bernoulli Shift ◽  
Ergodic Measure ◽  
Measure Preserving Transformation ◽  
Zero Entropy ◽  
Standard Probability

In this paper we show that any ergodic measure preserving transformation of a standard probability space which is $\text{AT}(n)$ for some positive integer $n$ has zero entropy. We show that for every positive integer $n$ any Bernoulli shift is not $\text{AT}(n)$. We also give an example of a transformation which has zero entropy but does not have property $\text{AT}(n)$ for any integer $n\geq 1$.

Mixing on Sequences

1983 ◽  
Vol 35 (2) ◽  
pp. 339-352 ◽  
Author(s):  
Nathaniel A. Friedman
Keyword(s):  
Ergodic Measure ◽  
General Definition ◽  
Weak Mixing ◽  
Independent Sequence ◽  
Density Zero ◽  
Measure Preserving Transformation ◽  
Upper Density ◽  
Mixing Sequences ◽  
Definition Of ◽  
Single Sequence

Our aim is to study the mixing sequences of a weak mixing transformation. An ergodic measure preserving transformation is weak mixing if and only if for each pair of sets there exists a sequence of density one on which the transformation mixes the sets [9]. An unpublished result of S. Kakutani implies there actually exists a single sequence of density one on which the transformation is mixing for all sets (see Section 3). This result motivated the general définition of a transformation being mixing on a sequence, as well as mixing of higher order on a sequence. Given a weak mixing transformation, there exist sequences along which it is mixing of all degrees. In particular, this is the case for an eventually independent sequence [7].In Section 3 it will be shown that if T is weak mixing but not mixing, then a sequence on which T is two-mixing must have upper density zero.

scholarly journals A Transformation with Simple Spectrum which is not Rank One

1977 ◽  
Vol 29 (3) ◽  
pp. 655-663 ◽  
Author(s):  
Andrés Del Junco
Keyword(s):  
Ergodic Measure ◽  
Negative Answer ◽  
Simple Spectrum ◽  
Spectral Multiplicity ◽  
Measure Preserving Transformation ◽  
Rank One

Following [10] an ergodic measure-preserving transformation is called rank one if it admits a sequence of approximating stacks. Rank one transformations have been studied in [1] and [2] where it was shown that any rank one transformation has simple spectrum. More generally it has been shown by Chacon [4] that a transformation of rank n has spectral multiplicity at most n. M. A. Akcoglu and J. R. Baxter have asked whether the converse is true. In particular: does simple spectrum imply rank one? In this paper we give a negative answer to this question.

scholarly journals (Uniform) convergence of twisted ergodic averages

2015 ◽  
Vol 36 (7) ◽  
pp. 2172-2202 ◽  
Author(s):  
TANJA EISNER ◽  
BEN KRAUSE
Keyword(s):  
Uniform Convergence ◽  
Probability Space ◽  
Elementary Proof ◽  
Ergodic Measure ◽  
Ergodic Averages ◽  
Almost Everywhere ◽  
Measure Preserving Transformation ◽  
Image Position ◽  
Badly Approximable

Let$T$be an ergodic measure-preserving transformation on a non-atomic probability space$(X,\unicode[STIX]{x1D6F4},\unicode[STIX]{x1D707})$. We prove uniform extensions of the Wiener–Wintner theorem in two settings: for averages involving weights coming from Hardy field functions $p$,$$\begin{eqnarray}\displaystyle \bigg\{\frac{1}{N}\mathop{\sum }_{n\leq N}e(p(n))T^{n}f(x)\bigg\}; & & \displaystyle \nonumber\end{eqnarray}$$and for ‘twisted’ polynomial ergodic averages,$$\begin{eqnarray}\displaystyle \bigg\{\frac{1}{N}\mathop{\sum }_{n\leq N}e(n\unicode[STIX]{x1D703})T^{P(n)}f(x)\bigg\} & & \displaystyle \nonumber\end{eqnarray}$$for certain classes of badly approximable$\unicode[STIX]{x1D703}\in [0,1]$. We also give an elementary proof that the above twisted polynomial averages converge pointwise$\unicode[STIX]{x1D707}$-almost everywhere for$f\in L^{p}(X),p>1,$and arbitrary$\unicode[STIX]{x1D703}\in [0,1]$.

On Ergodic Extensions of Stationary Measures with Minimal Support

1983 ◽  
Vol 26 (1) ◽  
pp. 20-25
Author(s):  
William B. Krebs ◽  
James B. Robertson
Keyword(s):  
Positive Integer ◽  
Discrete Spectrum ◽  
Ergodic Measure ◽  
Finite Partition ◽  
Minimal Support ◽  
Stationary Measures ◽  
Root Of Unity ◽  
Measure Preserving Transformation

AbstractLet T be an ergodic measure preserving transformation with the following property: there exists a positive integer n and a finite partition α such that the number of atom of is one more than that of , and the probability of at least one of the atoms is irrational. Then there exists a unique (up to conjugacy) transformation S such that there is a partition β with S restricted to isomorphic to T restricted to and the number of atoms in is one more than the number of atoms in for all m ≥ n. Moreover this transformation has discrete spectrum with at most two generators. If there are two generators, one of them must be a root of unity.

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.

In General a Measure Preserving Transformation is Mixing

Selecta ◽  
1983 ◽  
pp. 68-74
Author(s):  
Paul R. Halmos
Keyword(s):  
Measure Preserving Transformation

Invariant subspaces of a measure preserving transformation

1964 ◽  
Vol 2 (3) ◽  
pp. 198-200 ◽  
Author(s):  
S. R. Foguel
Keyword(s):  
Invariant Subspaces ◽  
Measure Preserving Transformation
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