Generic behavior of a measure-preserving transformation

Del Junco–Lemańczyk [Generic spectral properties of measure-preserving maps and applications. Proc. Amer. Math. Soc., 115 (3) (1992)] showed that a generic measure-preserving transformation satisfies certain orthogonality conditions. More precisely, there is a dense $G_{\unicode[STIX]{x1D6FF}}$ subset of measure preserving transformations such that, for every $T\in G$ and $k(1),k(2),\ldots ,k(l)\in \mathbb{Z}^{+}$, $k^{\prime }(1),k^{\prime }(2),\ldots ,k^{\prime }(l^{\prime })\in \mathbb{Z}^{+}$, the convolutions $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{T^{k(1)}}\ast \cdots \ast \unicode[STIX]{x1D70E}_{T^{k(l)}}\quad \text{and}\quad \unicode[STIX]{x1D70E}_{T^{k^{\prime }(1)}}\ast \cdots \ast \unicode[STIX]{x1D70E}_{T^{k^{\prime }(l^{\prime })}},\end{eqnarray}$$ where $\unicode[STIX]{x1D70E}_{T^{k}}$ is the maximal spectral type of $T^{k}$, are mutually singular, provided that $(k(1),k(2),\ldots ,k(l))$ is not a rearrangement of $(k^{\prime }(1),k^{\prime }(2),\ldots ,k^{\prime }(l^{\prime }))$. We will introduce analogous orthogonality conditions for continuous unitary representations of the group of all measurable functions with values in the circle, $L^{0}(\unicode[STIX]{x1D707},\mathbb{T})$, which we denote by the DL-condition. We connect the DL-condition with a result of Solecki [Unitary representations of the groups of measurable and continuous functions with values in the circle. J. Funct. Anal., 267 (2014), pp. 3105–3124] which identifies continuous unitary representations of $L^{0}(\unicode[STIX]{x1D707},\mathbb{T})$ with a collection of measures $\{\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D705}}\}$, where $\unicode[STIX]{x1D705}$ runs over all increasing finite sequence of non-zero integers. In particular, we show that the ‘probabilistic’ DL-condition translates to ‘deterministic’ orthogonality conditions on the measures $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D705}}$. As a corollary, we show that the same orthogonality conditions as in the result by Del Junco–Lemańczyk hold for a generic unitary operator on a separable infinite-dimensional Hilbert space.

Entropy of systems

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 $n\geq 1$.

Ergodicity Space for Measure-Preserving Transformations

We introduce the concept of ergodicity space of a measure-preserving transformation and will present some of its properties as an algebraic weight for measuring the size of the ergodicity of a measure-preserving transformation. We will also prove the invariance of the ergodicity space under conjugacy of dynamical systems.

(Uniform) convergence of twisted ergodic averages

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$\unicode[STIX]{x1D703}\in [0,1]$.

Closed subgroups generated by generic measure automorphisms

We prove that for a generic measure-preserving transformation $T$, the closed group generated by $T$ is a continuous homomorphic image of a closed linear subspace of $L_0(\lambda , {\mathbb R})$, where $\lambda $ is the Lebesgue measure, and that the closed group generated by $T$contains an increasing sequence of finite-dimensional tori whose union is dense.

Almost Everywhere Convergence of Convolution Measures

Let (X, ℬ, m, τ) be a dynamical system with (X, ℬ, m) a probability space and τ an invertible, measure preserving transformation. This paper deals with the almost everywhere convergence in L1(X) of a sequence of operators of weighted averages. Almost everywhere convergence follows once we obtain an appropriate maximal estimate and once we provide a dense class where convergence holds almost everywhere. The weights are given by convolution products of members of a sequence of probability measures {vi} defined on ℤ. We then exhibit cases of such averages where convergence fails.

Infinite partitions and Rokhlin towers

We find a countable partition P on a Lebesgue space, labeled {1,2,3,…}, for any non-periodic measure-preserving transformation T such that P generates T and, for the T,P process, if you see an n on time −1 then you only have to look at times −n,1−n,…−1 to know the positive integer i to put at time 0 . We alter that proof to extend every non-periodic T to a uniform martingale (i.e. continuous g function) on an infinite alphabet. If T has positive entropy and the weak Pinsker property, this extension can be made to be an isomorphism. We pose remaining questions on uniform martingales. In the process of proving the uniform martingale result we make a complete analysis of Rokhlin towers which is of interest in and of itself. We also give an example that looks something like an independent identically distributed process on ℤ2 when you read from right to left but where each column determines the next if you read left to right.

Non-intersecting splitting σ-algebras in a non-Bernoulli transformation

Given a measure-preserving transformation T on a Lebesgue σ-algebra, a complete T-invariant sub-σ-algebra is said to split if there is another complete T-invariant sub-σ-algebra on which T is Bernoulli which is completely independent of the given sub-σ-algebra and such that the two sub-σ-algebras together generate the entire σ-algebra. It is easily shown that two splitting sub-σ-algebras with nothing in common imply T to be K. Here it is shown that T does not have to be Bernoulli by exhibiting two such non-intersecting σ-algebras for the T,T−1 transformation, negatively answering a question posed by Thouvenot in 1975.