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Register a new userA strictly ergodic, positive entropy subshift uniformly uncorrelated to the Moebius function
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A recent result of Downarowicz and Serafin (DS) shows that there exist positive entropy subshifts satisfying the assertion of Sarnaks conjecture. More precisely, it is proved that if $y=(y_n)_{nge 1}$ is a bounded sequence with zero average along every infinite arithmetic progression (the Mobius function is an example of such a sq $y$) then for every $Nge 2$ there exists a subshift $Sigma$ over $N$ symbols, with entropy arbitrarily close to $log N$, uncorrelated to $y$. In the present note, we improve the result of (DS). First of all, we observe that the uncorrelation obtained in (DS) is emph{uniform}, i.e., for any continuous function $f:Sigmato {mathbb R}$ and every $epsilon>0$ there exists $n_0$ such that for any $nge n_0$ and any $xinSigma$ we have $$ left|frac1nsum_{i=1}^{n}f(T^ix),y_iright|<epsilon. $$ More importantly, by a fine-tuned modification of the construction from (DS) we create a emph{strictly ergodic} subshift, with all the desired properties of the example in (DS) (uniformly uncorrelated to $y$ and with entropy arbitrarily close to $log N$). The question about these two additional properties (uniformity of uncorrelation and strict ergodicity) has been posed by Mariusz Lemanczyk in the context of the so-called strong MOMO (Mobius Orthogonality on Moving Orbits) property. Our result shows, among other things, that strong MOMO is essentially stronger than uniform uncorrelation, even for strictly ergodic systems.
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The classical theorem of Jewett and Krieger gives a strictly ergodic model for any ergodic measure preserving system. An extension of this result for non-ergodic systems was given many years ago by George Hansel. He constructed, for any measure preserving system, a strictly uniform model, i.e. a compact space which admits an upper semicontinuous decomposition into strictly ergodic models of the ergodic components of the measure. In this note we give a new proof of a stronger result by adding the condition of purity, which controls the set of ergodic measures that appear in the strictly uniform model.
Let $f:Xto X$ be a dominating meromorphic map of a compact Kahler surface of large topological degree. Let $S$ be a positive closed current on $X$ of bidegree $(1,1)$. We consider an ergodic measure $ u$ of large entropy supported by $mathrm{supp}(S)$. Defining dimensions for $ u$ and $S$, we give inequalities `a la Ma~ne involving the Lyapunov exponents of $ u$ and those dimensions. We give dynamical applications of those inequalities.
We construct a minimal dynamical system of mean dimension equal to $1$, which can be embedded in the shift action on the Hilbert cube $[0,1]^mathbb{Z}$. Our result clarifies a seemingly plausible impression and finally enables us to have a full understanding of (a pair of) the exact ranges of all possible values of mean dimension, within which there will always be a minimal dynamical system that can be (resp. cannot be) embedded in the shift action on the Hilbert cube. The key ingredient of our idea is to produce a dense subset of the alphabet $[0,1]$ more gently.
The goal of this chapter is to illustrate a generalization of the Fibonacci word to the case of 2-dimensional configurations on $mathbb{Z}^2$. More precisely, we consider a particular subshift of $mathcal{A}^{mathbb{Z}^2}$ on the alphabet $mathcal{A}={0,dots,18}$ for which we give three characterizations: as the subshift $mathcal{X}_phi$ generated by a 2-dimensional morphism $phi$ defined on $mathcal{A}$; as the Wang shift $Omega_mathcal{U}$ defined by a set $mathcal{U}$ of 19 Wang tiles; as the symbolic dynamical system $mathcal{X}_{mathcal{P}_mathcal{U},R_mathcal{U}}$ representing the orbits under some $mathbb{Z}^2$-action $R_mathcal{U}$ defined by rotations on $mathbb{T}^2$ and coded by some topological partition $mathcal{P}_mathcal{U}$ of $mathbb{T}^2$ into 19 polygonal atoms. We prove their equality $Omega_mathcal{U} =mathcal{X}_phi=mathcal{X}_{mathcal{P}_mathcal{U},R_mathcal{U}}$ by showing they are self-similar with respect to the substitution $phi$. This chapter provides a transversal reading of results divided into four different articles obtained through the study of the Jeandel-Rao Wang shift. It gathers in one place the methods introduced to desubstitute Wang shifts and to desubstitute codings of $mathbb{Z}^2$-actions by focussing on a simple 2-dimensional self-similar subshift. SageMath code to find marker tiles and compute the Rauzy induction of $mathbb{Z}^2$-rotations is provided allowing to reproduce the computations.
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