Invariant measures for Cartesian powers of Chacon infinite transformation

Furstenberg transformations on Cartesian products of infinite-dimensional tori

We consider in this note Furstenberg transformations on Cartesian products of infinite-dimensional tori. Under some appropriate assumptions, we show that these transformations are uniquely ergodic with respect to the Haar measure and have countable Lebesgue spectrum in a suitable subspace. These results generalise to the infinite-dimensional setting previous results of H. Furstenberg, A. Iwanik, M. Lemanzyk, D. Rudolph and the second author in the one-dimensional setting. Our proofs rely on the use of commutator methods for unitary operators and Bruhat functions on the infinite-dimensional tor…

Rank-one flows of transformations with infinite ergodic index

A rank-one infinite measure preserving flow $T=(T_t)_{t\in\Bbb R}$ is constructed such that for each $t\ne 0$, the Cartesian powers of the transformation $T_t$ are all ergodic.

Disjointness properties for Cartesian products of weakly mixing systems

For $n\geq 1$ we consider the class JP($n$) of dynamical systems whose every ergodic joining with a Cartesian product of $k$ weakly mixing automorphisms ($k\geq n$) can be represented as the independent extension of a joining of the system with only $n$ coordinate factors. For $n\geq 2$ we show that, whenever the maximal spectral type of a weakly mixing automorphism $T$ is singular with respect to the convolution of any $n$ continuous measures, i.e. $T$ has the so-called convolution singularity property of order $n$, then $T$ belongs to JP($n-1$). To provide examples of such automorphisms, we …

D-sets in Arbitrary Semigroup

We define the notion of $D$-set in an arbitrary semigroup, and with some mild restrictions we establish its dynamical and combinatorial characterizations. Assuming a weak form of cancellation in semigroups we have shown that the Cartesian product of finitely many $D$-sets is a $D$-set. A similar partial result has been proved for Cartesian product of infinitely many $D$-sets. Finally, in a commutative semigroup we deduce that $D$-sets (with respect to a Følner net) are $C$-sets.

Infinite symmetric ergodic index and related examples in infinite measure

We define an infinite measure-preserving transformation to have infinite symmetric ergodic index if all finite Cartesian products of the transformation and its inverse are ergodic, and show that infinite symmetric ergodic index does not imply that all products of powers are conservative, so does not imply power weak mixing. We provide a sufficient condition for $k$-fold and infinite symmetric ergodic index and use it to answer a question on the relationship between product conservativity and product ergodicity. We also show that a class of rank-one transformations that have infinite symmetric …

Ergodicity and Conservativity of products of infinite transformations and their inverses

We construct a class of rank-one infinite measure-preserving transformations such that for each transformation $T$ in the class, the cartesian product $T\times T$ of the transformation with itself is ergodic, but the product $T\times T^{-1}$ of the transformation with its inverse is not ergodic. We also prove that the product of any rank-one transformation with its inverse is conservative, while there are infinite measure-preserving conservative ergodic Markov shifts whose product with their inverse is not conservative.

On Infinite Transformations with Maximal Control of Ergodic Two-fold Product Powers

We study the rich behavior of ergodicity and conservativity of Cartesian products of infinite measure preserving transformations. A class of transformations is constructed such that for any subset $R\subset \mathbb Q\cap (0,1)$ there exists $T$ in this class such that $T^p\times T^q$ is ergodic if and only if $\frac{p}{q} \in R$. This contrasts with the finite measure preserving case where $T^p\times T^q$ is ergodic for all nonzero $p$ and $q$ if and only if $T\times T$ is ergodic. We also show that our class is rich in the behavior of conservative products. For each positive integer $k$, a fa…

Projections of cartesian products of the self-similar sets without the irrationality assumption

Let $β>1$. Define a class of similitudes \[S=\left\{f_{i}(x)=\dfrac{x}{β^{n_i}}+a_i:n_i\in \mathbb{N}^{+}, a_i\in \mathbb{R}\right\}.\] Let $\mathcal{A}$ be the collection of all the self-similar sets generated by the similitudes from $S$. In this paper, we prove that for any $θ\in(0,π)$ and $K_1, K_2\in \mathcal{A}$, $Proj_θ(K_1\times K_2)$ is similar to a self-similar set or an attractor of some infinite iterated function system, where $Proj_θ$ denotes the orthogonal projection onto $L_θ$, and $L_θ$ denotes the line through the origin in direction $θ$. As a corollary, $\dim_{P}(Proj_θ(K_1\ti…

无限乘积空间上概率测度的动力熵

Dynamical entropy of probability measures on infinite product spaces

本文提出无限乘积空间上概率测度的无限乘积熵，证明其在平移不变情形下等价于移位熵，并分析其基本性质及与子集拓扑熵的变分关系。

无限乘积空间任意子集的拓扑熵

Topological Entropy for Arbitrary Subsets of Infinite Product Spaces

本文定义无限乘积空间任意子集的广义拓扑熵及广义 Bowen 熵，证明其与连续映射经典熵的一致性，并给出分形内在熵等性质。

Glasner property for linear group actions and their products

A theorem of Glasner from 1979 shows that if $Y \subset \mathbb{T} = \mathbb{R}/\mathbb{Z}$ is infinite then for each $ε> 0$ there exists an integer $n$ such that $nY$ is $ε$-dense. This has been extended in various works by showing that certain irreducible linear semigroup actions on $\mathbb{T}^d$ also satisfy such a \textit{Glasner property} where each infinite set (in fact, arbitrarily large finite set) will have an $ε$-dense image under some element from the acting semigroup. We improve these works by proving a quantitative Glasner theorem for irreducible linear group actions with Zariski…