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Register a new userDynamical properties of $k$-free lattice points
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We revisit the visible points of a lattice in Euclidean $n$-space together with their generalisations, the $k$th-power-free points of a lattice, and study the corresponding dynamical system that arises via the closure of the lattice translation orbit. Our analysis extends previous results obtained by Sarnak and by Cellarosi and Sinai for the special case of square-free integers and sheds new light on previous joint work with Peter Pleasants.
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Recently, the dynamical and spectral properties of square-free integers, visible lattice points and various generalisations have received increased attention. One reason is the connection of one-dimensional examples such as $mathscr B$-free numbers with Sarnaks conjecture on the `randomness of the Mobius function, another the explicit computability of correlation functions as well as eigenfunctions for these systems together with intrinsic ergodicity properties. Here, we summarise some of the results, with focus on spectral and dynamical aspects, and expand a little on the implications for mathematical diffraction theory.
We consider a sequence of matrices that are associated to Markov dynamical systems and use determinant-free linear algebra techniques (as well as some algebra and complex analysis) to rigorously estimate the eigenvalues of every matrix simultaneously without doing any calculations on the matrices themselves. As a corollary, we obtain mixing rates for every system at once, as well as symmetry properties of densities associated to the system; we also find the spectral properties of a sequence of related factor systems.
This paper is aimed at a detailed study of the multifractal analysis of the so-called divergence points in the system of $beta$-expansions. More precisely, let $([0,1),T_{beta})$ be the $beta$-dynamical system for a general $beta>1$ and $psi:[0,1]mapstomathbb{R}$ be a continuous function. Denote by $textsf{A}(psi,x)$ all the accumulation points of $Big{frac{1}{n}sum_{j=0}^{n-1}psi(T^jx): nge 1Big}$. The Hausdorff dimensions of the sets $$Big{x:textsf{A}(psi,x)supset[a,b]Big}, Big{x:textsf{A}(psi,x)=[a,b]Big}, Big{x:textsf{A}(psi,x)subset[a,b]Big}$$ i.e., the points for which the Birkhoff averages of $psi$ do not exist but behave in a certain prescribed way, are determined completely for any continuous function $psi$.
For an infinitely renormalizable negative Schwarzian unimodal map $f$ with non-flat critical point, we analyze statistical properties of periodic points as the periods tend to infinity. Introducing a weight function $varphi$ which is a continuous or a geometric potential $varphi=-betalog|f|$ ($betainmathbb R$), we establish the level-2 Large Deviation Principle for weighted periodic points. From this, we deduce that all weighted periodic points equidistribute with respect to equilibrium states for the potential $varphi$. In particular, it follows that all periodic points are equidistributed with respect to measures of maximal entropy, and all periodic points weighted with their Lyapunov exponents are equidistributed with respect to the post-critical measure supported on the attracting Cantor set.
Let X be a subset of R^n whose interior is connected and dense in X, ordered by a polyhedral cone in R^n with nonempty interior. Let T be a monotone homeomorphism of X whose periodic points are dense. Then T is periodic.
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