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Register a new userJoint ergodicity of Hardy field sequences
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Konstantinos Tsinas
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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.
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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.
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 study equidistribution properties of translations on nilmanifolds along functions of polynomial growth from a Hardy field. More precisely, if $X=G/Gamma$ is a nilmanifold, $a_1,ldots,a_kin G$ are commuting nilrotations, and $f_1,ldots,f_k$ are functions of polynomial growth from a Hardy field then we show that $bullet$ the distribution of the sequence $a_1^{f_1(n)}cdotldotscdot a_k^{f_k(n)}Gamma$ is governed by its projection onto the maximal factor torus, which extends Leibmans Equidistribution Criterion form polynomials to a much wider range of functions; and $bullet$ the orbit closure of $a_1^{f_1(n)}cdotldotscdot a_k^{f_k(n)}Gamma$ is always a finite union of sub-nilmanifolds, which extends some of the previous work of Leibman and Frantzikinakis on this topic.
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.
An textit{algebraic} action of a discrete group $Gamma $ is a homomorphism from $Gamma $ to the group of continuous automorphisms of a compact abelian group $X$. By duality, such an action of $Gamma $ is determined by a module $M=widehat{X}$ over the integer group ring $mathbb{Z}Gamma $ of $Gamma $. The simplest examples of such modules are of the form $M=mathbb{Z}Gamma /mathbb{Z}Gamma f$ with $fin mathbb{Z}Gamma $; the corresponding algebraic action is the textit{principal algebraic $Gamma $-action} $alpha _f$ defined by $f$. In this note we prove the following extensions of results by Hayes cite{Hayes} on ergodicity of principal algebraic actions: If $Gamma $ is a countably infinite discrete group which is not virtually cyclic, and if $finmathbb{Z}Gamma $ satisfies that right multiplication by $f$ on $ell ^2(Gamma ,mathbb{R})$ is injective, then the principal $Gamma $-action $alpha _f$ is ergodic (Theorem ref{t:ergodic2}). If $Gamma $ contains a finitely generated subgroup with a single end (e.g. a finitely generated amenable subgroup which is not virtually cyclic), or an infinite nonamenable subgroup with vanishing first $ell ^2$-Betti number (e.g., an infinite property $T$ subgroup), the injectivity condition on $f$ can be replaced by the weaker hypothesis that $f$ is not a right zero-divisor in $mathbb{Z}Gamma $ (Theorem ref{t:ergodic1}). Finally, if $Gamma $ is torsion-free, not virtually cyclic, and satisfies Linnells textit{analytic zero-divisor conjecture}, then $alpha _f$ is ergodic for every $fin mathbb{Z}Gamma $ (Remark ref{r:analytic zero divisor}).
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