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Register a new userDecomposition of multicorrelation sequences and joint ergodicity
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We show that, under finitely many ergodicity assumptions, any multicorrelation sequence defined by invertible measure preserving $mathbb{Z}^d$-actions with multivariable integer polynomial iterates is the sum of a nilsequence and a null sequence, extending a recent result of the second author. To this end, we develop a new seminorm bound estimate for multiple averages by improving the results in a previous work of the first, third and fourth authors. We also use this approach to obtain new criteria for joint ergodicity of multiple averages with multivariable polynomial iterates on $mathbb{Z}^{d}$-systems.
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We study multicorrelation sequences arising from systems with commuting transformations. Our main result is a refinement of a decomposition result of Frantzikinakis and it states that any multicorrelation sequences for commuting transformations can be decomposed, for every $epsilon>0$, as the sum of a nilsequence $phi(n)$ and a sequence $omega(n)$ satisfying $lim_{Ntoinfty}frac{1}{N}sum_{n=1}^N |omega(n)|<epsilon$ and $lim_{Ntoinfty}frac{1}{|mathbb{P}cap [N]|}sum_{pin mathbb{P}cap [N]} |omega(p)|<epsilon$.
We study mean convergence of multiple ergodic averages, where the iterates arise from smooth functions of polynomial growth that belong to a Hardy field. Our results include all logarithmico-exponential functions of polynomial growth, such as the functions $t^{3/2}, tlog t$ and $e^{sqrt{log t}}$. We show that if all non-trivial linear combinations of the functions $a_1,...,a_k$ stay logarithmically away from rational polynomials, then the $L^2$-limit of the ergodic averages $frac{1}{N} sum_{n=1}^{N}f_1(T^{lfloor{a_1(n)}rfloor}x)cdots f_k(T^{lfloor{a_k(n)}rfloor}x)$ exists and is equal to the product of the integrals of the functions $f_1,...,f_k$ in ergodic systems, which establishes a conjecture of Frantzikinakis. Under some more general conditions on the functions $a_1,...,a_k$, we also find characteristic factors for convergence of the above averages and deduce a convergence result for weak-mixing systems.
Exploiting the recent work of Tao and Ziegler on the concatenation theorem on factors, we find explicit characteristic factors for multiple averages along polynomials on systems with commuting transformations, and use them to study the criteria of joint ergodicity for sequences of the form $(T^{p_{1,j}(n)}_{1}cdotldotscdot T^{p_{d,j}(n)}_{d})_{ninmathbb{Z}},$ $1leq jleq k$, where $T_{1},dots,T_{d}$ are commuting measure preserving transformations on a probability measure space and $p_{i,j}$ are integer polynomials. To be more precise, we provide a sufficient condition for such sequences to be jointly ergodic. We also give a characterization for sequences of the form $(T^{p(n)}_{i})_{ninmathbb{Z}}, 1leq ileq d$ to be jointly ergodic, answering a question due to Bergelson.
We determine the Krieger type of nonsingular Bernoulli actions $G curvearrowright prod_{g in G} ({0,1},mu_g)$. When $G$ is abelian, we do this for arbitrary marginal measures $mu_g$. We prove in particular that the action is never of type II$_infty$ if $G$ is abelian and not locally finite, answering Krengels question for $G = mathbb{Z}$. When $G$ is locally finite, we prove that type II$_infty$ does arise. For arbitrary countable groups, we assume that the marginal measures stay away from $0$ and $1$. When $G$ has only one end, we prove that the Krieger type is always I, II$_1$ or III$_1$. When $G$ has more than one end, we show that other types always arise. Finally, we solve the conjecture of [VW17] by proving that a group $G$ admits a Bernoulli action of type III$_1$ if and only if $G$ has nontrivial first $L^2$-cohomology.
We define interacting particle systems on configurations of the integer lattice (with values in some finite alphabet) by the superimposition of two dynamics: a substitution process with finite range rates, and a circular permutation mechanism(called cut-and-paste) with possibly unbounded range. The model is motivated by the dynamics of DNA sequences: we consider an ergodic model for substitutions, the RN+YpR model ([BGP08]), with three particular cases, the models JC+cpg,T92+cpg, and RNc+YpR. We investigate whether they remain ergodic with the additional cut-and-paste mechanism, which models insertions and deletions of nucleotides. Using either duality or attractiveness techniques, we provide various sets of sufficient conditions, concerning only the substitution rates, for ergodicity of the superimposed process. They imply ergodicity of the models JC+cpg, T92+cpg as well as the attractive RNc+YpR, all with an additional cut-and-paste mechanism.
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