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Mariusz Lemańczyk

Publications and source records attributed to Mariusz Lemańczyk.

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arXiv2018

Möbius disjointness for models of an ergodic system and beyond

Given a topological dynamical system $(X,T)$ and an arithmetic function $\boldsymbol{u}\colon\mathbb{N}\to\mathbb{C}$, we study the strong MOMO property (relatively to $\boldsymbol{u}$) which is a strong version of $\boldsymbol{u}$-disjointness with all observable sequences in $(X,T)$. It is proved that, given an ergodic measure-preserving system $(Z,\mathcal{D},κ,R)$, the strong MOMO property (relatively to $\boldsymbol{u}$) of a uniquely ergodic model $(X,T)$ of $R$ yields all other uniquely ergodic models of $R$ to be $\boldsymbol{u}$-disjoint. It follows that all uniquely ergodic models of: ergodic unipotent diffeomorphisms on nilmanifolds, discrete spectrum automorphisms, systems given by some substitutions of constant length (including the classical Thue-Morse and Rudin-Shapiro substitutions), systems determined by Kakutani sequences are Möbius (and Liouville) disjoint. The validity of Sarnak's conjecture implies the strong MOMO property relatively to $\boldsymbolμ$ in all zero entropy systems, in particular, it makes $\boldsymbolμ$-disjointness uniform. The absence of strong MOMO property in positive entropy systems is discussed and, it is proved that, under the Chowla conjecture, a topological system has the strong MOMO property relatively to the Liouville function if and only if its topological entropy is zero.

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arXiv2018

Rationally almost periodic sequences, polynomial multiple recurrence and symbolic dynamics

A set $R\subset \mathbb{N}$ is called rational if it is well-approximable by finite unions of arithmetic progressions. Examples of rational sets include many classical sets of number-theoretical origin such as the set of squarefree numbers, the set of abundant numbers, or sets of the form $Φ_x:=\{n\in\mathbb{N}: \frac{\boldsymbolφ(n)}{n} 0$, then the following are equivalent: (a) $R$ is divisible, i.e. $\overline{d}(R\cap u \mathbb{N})>0$ for all $u\in\mathbb{N}$. (b) $R$ is an averaging set of polynomial single recurrence. (c) $R$ is an averaging set of polynomial multiple recurrence. As an application, we show that if $R$ is rational and divisible, then for any set $E\subset\mathbb{N}$ with $\overline{d}(E)>0$ and any polynomials $p_i\in\mathbb{Q}[t]$,$i=1,\ldots,\ell$, which satisfy $p_i(\mathbb{Z})\subset\mathbb{Z}$ and $p_i(0)=0$ for all $i\in\{1,\ldots,\ell\}$, there exists $β>0$ such that the set $$\{n\in R:\overline{d}( E\cap (E-p_1(n))\cap\ldots\cap(E-p_\ell(n)))>β\}$$ has positive lower density. Ramsey-theoretical applications naturally lead to problems in symbolic dynamics, which involve rationally almost periodic sequences. We prove that if $\mathcal{A}$ is a finite alphabet, $η\in\mathcal{A}^\mathbb{N}$ is rationally almost periodic, $S$ denotes the left-shift on $\mathcal{A}^\mathbb{Z}$ and $$X:=\{y\in \mathcal{A}^\mathbb{Z} : \text{each finite word appearing in $y$ appears in }η\},$$ then $η$ is a generic point for an $S$-invariant probability measure $ν$ on $X$ such that $(X,ν,S)$ is ergodic and has rational discrete spectrum.

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arXiv2017

Möbius disjointness along ergodic sequences for uniquely ergodic actions

We show that there are an irrational rotation $Tx=x+α$ on the circle $\mathbb{T}$ and a continuous $φ\colon\mathbb{T}\to\mathbb{R}$ such that for each (continuous) uniquely ergodic flow $\mathcal{S}=(S_t)_{t\in\mathbb{R}}$ acting on a compact metric space $Y$, the automorphism $T_{φ,\mathcal{S}}$ acting on $(X\times Y,μ\otimesν)$ by the formula $T_{φ,\mathcal{S}}(x,y)=(Tx,S_{φ(x)}(y))$, where $μ$ stands for Lebesgue measure on $\mathbb{T}$ and $ν$ denotes the unique $\mathcal{S}$-invariant measure, has the property of asymptotically orthogonal powers. This gives a class of relatively weakly mixing extensions of irrational rotations for which Sarnak's conjecture on Möbius disjointness holds for all uniquely ergodic models of $T_{φ,\mathcal{S}}$. Moreover, we obtain a class of "random" ergodic sequences $(c_n)\subset\mathbb{Z}$ such that if $\boldsymbolμ$ denotes the Möbius function, then $$ \lim_{N\to\infty}\frac1N\sum_{n\leq N}g(S_{c_n}y)\boldsymbolμ(n)=0 $$ for all (continuous) uniquely ergodic flows $\mathcal{S}$, all $g\in C(Y)$ and $y\in Y$.

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arXiv2017

Dynamics of $\mathcal B$-free sets: a view through the window

Let $\mathcal B$ be an infinite subset of $\{1,2,\dots\}$. We characterize arithmetic and dynamical properties of the $\mathcal B$-free set $\mathcal F_{\mathcal B}$ through group theoretical, topological and measure theoretic properties of a set $W$ (called the window) associated with $\mathcal B$. This point of view stems from the interpretation of the set $\mathcal F_{\mathcal B}$ as a weak model set. Our main results are: $\mathcal B$ is taut if and only if the window is Haar regular; the dynamical system associated to $\mathcal F_{\mathcal B}$ is a Toeplitz system if and only if the window is topologically regular; the dynamical system associated to $\mathcal F_{\mathcal B}$ is proximal if and only if the window has empty interior; and the dynamical system associated to $\mathcal F_{\mathcal B}$ has the "naïvely expected" maximal equicontinuous factor if and only if the interior of the window is aperiodic.

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arXiv2016

Approximate orthogonality of powers for ergodic affine unipotent diffeomorphisms on nilmanifolds

Let $ G $ be a connected, simply connected nilpotent Lie group and $ Γ< G $ a lattice. We prove that each ergodic diffeomorphism $ ϕ(xΓ)=uA(x)Γ$ on the nilmanifold $ G/Γ$, where $ u\in G $ and $ A:G\to G $ is a unipotent automorphism satisfying $ A(Γ)=Γ$, enjoys the property of asymptotically orthogonal powers (AOP). Two consequences follow: (i) Sarnak's conjecture on Möbius orthogonality holds in every uniquely ergodic model of an ergodic affine unipotent diffeomorphism; (ii) For ergodic affine unipotent diffeomorphisms themselves, the Möbius orthogonality holds on so called typical short interval: $ \frac1 M\sum_{M\leq m<2M}\left|\frac1H\sum_{m\leq n<m+H} f(ϕ^n(xΓ))μ(n)\right|\to 0$ as $ H\to\infty $ and $ H/M\to0 $ for each $ xΓ\in G/Γ$ and each $ f\in C(G/Γ) $. In particular, the results in (i) and (ii) hold for ergodic nil-translations. Moreover, we prove that each nilsequence is orthogonal to the Möbius function $μ$ on a typical short interval. We also study the problem of lifting of the AOP property to induced actions and derive some applications on uniform distribution.

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arXiv2015

$\mathscr{B}$-free sets and dynamics

Let $B\subset \mathbb{N}$ and let $η\in \{0,1\}^\mathbb{Z}$ be the characteristic function of the set $F_B:=\mathbb{Z}\setminus\bigcup_{b}b\mathbb{Z}$ of B-free numbers. Consider $(S,X_η)$, where $X_η$ is the closure of the orbit of $η$ under the left shift S. When $B=\{p^2 : p\in P\}$, $(S,X_η)$ was studied by Sarnak. This case + some generalizations, including the case (*) of B infinite, coprime with $\sum_{b}1/b<\infty$, were discussed by several authors. For general B, contrary to (*), we may have $X_η\subsetneq X_B:=\{x\in \{0,1\}^\mathbb{Z} : |\text{supp }x\bmod b|\leq b-1 \forall_b\}$. Also, $X_η$ may not be hereditary (heredity means that if $x\in X$ and $y\leq x$ coordinatewise then $y\in X$). We show that $η$ is quasi-generic for a natural measure $ν_η$. We solve the problem of proximality by showing first that $X_η$ has a unique minimal (Toeplitz) subsystem. Moreover B-free system is proximal iff B contains an infinite coprime set. B is taut when $δ(F_B)<δ(F_{B\setminus \{b\} })$ for each b. We give a characterization of taut B in terms of the support of $ν_η$. Moreover, for any B there exists a taut B' with $ν_η=ν_{η'}$. For taut sets B,B', we have B=B' iff $X_B=X_{B'}$. For each B there is a taut B' with $\tilde{X}_{η'}\subset \tilde{X}_η$ and all invariant measures for $(S,\tilde{X}_η)$ live on $\tilde{X}_{η'}$. $(S,\tilde{X}_η)$ is shown to be intrinsically ergodic for all B. We give a description of all invariant measures for $(S,\tilde{X}_η)$. The topological entropies of $(S,\tilde{X}_η)$ and $(S,X_B)$ are both equal to $\overline{d}(F_B)$. We show that for a subclass of taut B-free systems proximality is the same as heredity. Finally, we give applications in number theory on gaps between consecutive B-free numbers. We apply our results to the set of abundant numbers.

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arXiv2015

Hereditary subshifts whose simplex of invariant measures is Poulsen

We give a sufficient condition for the simplex of invariant measures for a hereditary system to be Poulsen. In particular, we show that this simplex is Poulsen in case of positive entropy $\mathscr{B}$-free systems. We also give an example of a positive entropy hereditary system whose simplex of invariant measures is not Poulsen.

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arXiv2014

The Moebius function and continuous extensions of rotations

Let $f\colon \mathbb{T}\to \mathbb{R}$ be of class $C^{1+δ}$ for some $δ>0$ and let $c\in\mathbb{Z}$. We show that for a generic $α\in\mathbb{R}$, the extension $T_{c,f}\colon \mathbb{T}^2\to\mathbb{T}^2$ of the irrational rotation $Tx=x+α$, given by $T_{c,f}(x,u)=(x+α, u+cx+f(x))$ ($\bmod\ 1$) satisfies Sarnak's conjecture.

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arXiv2014

On invariant measures for $\mathscr{B}$-free systems

We show that the $\mathscr{B}$-free subshift $(S,X_{\mathscr{B}})$ associated to a $\mathscr{B}$-free system is intrinsically ergodic, i.e.\ it has exactly one measure of maximal entropy. Moreover, we study invariant measures for such systems. It is proved that each ergodic invariant measure is of joining type, determined by a joining of the Mirsky measure of a $\mathscr{B}'$-free subshift contained in $(S,X_{\mathscr{B}})$ and an ergodic invariant measure of the full shift on $\{0,1\}^{\mathbb{Z}}$. Moreover, each ergodic joining type measure yields a measure-theoretic dynamical system with infinite rational part of the spectrum corresponding to the above Mirsky measure. Finally, we show that, in general, hereditary systems may not be intrinsically ergodic.

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arXiv2014

Polynomial actions of unitary operators and idempotent ultrafilters

Let $p$ be an idempotent ultrafilter over $\mathbb{N}$. For a positive integer $N$, let ${\cal P}_{\leq N}$ denote the additive group of polynomials $P\in\mathbb{Z}[x]$ with ${\rm deg}\, P\leq N$ and $P(0)=0$. Given a unitary operator $U$ on a Hilbert space ${\cal H}$, we prove, for each $N\geq1$, the existence of a unique decomposition ${\cal H}=\bigoplus_{r\geq 1}{\cal H}^{(N)}_r$ into closed, $U$-invariant subspaces such that (a) for any polynomial $P\in{\cal P}_{\leq N}$, we have $$ p\, \text{-}\!\lim_{n\in\mathbb{N}} \left(U|_{{\cal H}_r^{(N)}}\right)^{P(n)}=0_{{\cal H}_r^{(N)}}\;\mbox{or}\; Id_{{\cal H}_r^{(N)}},\; \mbox{for each}\; r\geq1 ; $$ (b) for each $r\neq s$ there exists $Q\in{\cal P}_{\leq N}$ such that $$ p\,\text{-}\!\lim_{n\in\mathbb{N}} \left(U|_{{\cal H}_r^{(N)}}\right)^{Q(n)}\neq p\,\text{-}\!\lim_{n\in\mathbb{N}} \left(U|_{{\cal H}_s^{(N)}}\right)^{Q(n)}. $$ In connection with this result we introduce the notion of rigidity group. Namely, a subgroup $G\subset {\cal P}_{\leq N}$ is called an $N$-rigidity group if there exist an idempotent ultrafilter $p$ over $\mathbb{N}$ and a unitary operator $U$ on a Hilbert space $\cal H$ such that $$\label{ab1} G=\{P\in{\cal P}_{\leq N}:\: p\,\text{-}\!\lim_{n\in\mathbb{N}} U ^{P(n)}=Id\}$$ and $p\,\text{-}\!\lim_{n\in\mathbb{N}} U ^{Q(n)}=0\;\;\mbox{for each}\;\;Q\in{\cal P}_{\leq N}\setminus G.$ The main result of the paper states that a subgroup $G\subset {\cal P}_{\leq N}$ satisfying $\max\{{\rm deg}\, P:\:P\in G\}=N$ is an $N$-rigidity group if and only if $G$ has finite index in ${\cal P}_{\leq N}$.

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arXiv2013

0-1 sequences of the Thue-Morse type and Sarnak's conjecture

We show that the images via $z\mapsto z^m$ of the continuous part of the spectral measures of the dynamical systems generated by the 0-1 sequences of the Thue-Morse type are pairwise mutually singular for different odd numbers $m\in\N$. Sarnak's conjecture on orthogonality with the Möbius function is shown to hold for such dynamical systems. The same conjecture is shown to hold for all systems induced by regular Toeplitz sequences. A non-regular Toeplitz sequence for which Sarnak's conjecture fails is constructed.

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arXiv2010

Spectral multiplicities for ergodic flows

Let $E$ be a subset of positive integers such that $E\cap\{1,2\}\ne\emptyset$. A weakly mixing finite measure preserving flow $T=(T_t)_{t\in\Bbb R}$ is constructed such that the set of spectral multiplicities (of the corresponding Koopman unitary representation generated by $T$) is $E$. Moreover, for each non-zero $t\in\Bbb R$, the set of spectral multiplicities of the transformation $T_t$ is also $E$. These results are partly extended to actions of some other locally compact second countable Abelian groups.

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arXiv2005

On the ergodicity of cylindrical transformations given by the logarithm

Given $\a \in [0,1]$ and $φ: \T \to \R$ measurable, the {\it cylindircal cascade} $S_{\a,φ}$ is the map from $\T \times \R$ to itself given by $S_{\a,φ} (x,y) = (x+\a,y+φ(x))$ that naturally appears in the study of some ordinary differential equations on $\R^3$. In this paper, we prove that for a set of full Lebesgue measure of $\a \in [0,1]$ the cylindrical cascades $S_{\a,φ}$ are ergodic for every smooth function $φ$ with a logarithmic singularity, provided that the average of $φ$ vanishes. Closely related to $S_{\a,φ}$ are the special flows constructed above $R_\a$ and under $φ+c$ where $c \in \R$ is such that $φ+c>0$. In the case of a function $φ$ with an asymmetric logarithmic singularity our result gives the first examples of ergodic cascades $S_{\a,φ}$ with the corresponding special flows being mixing. Indeed, when the latter flows are mixing the usual techniques used to prove the {\it essential value criterion} for $S_{\a,φ}$, that is equivalent to ergodicity, fail and we device a new method to prove this criterion that we hope could be useful in tackling other problems of ergodicity for cocycles preserving an infinite measure.

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