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Register a new userNumber-theoretic positive entropy shifts with small centraliser and large normaliser
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Higher-dimensional binary shifts of number-theoretic origin with positive topological entropy are considered. We are particularly interested in analysing their symmetries and extended symmetries. They form groups, known as the topological centraliser and normaliser of the shift dynamical system, which are natural topological invariants. Here, our focus is on shift spaces with trivial centralisers, but large normalisers. In particular, we discuss several systems where the normaliser is an infinite extension of the centraliser, including the visible lattice points and the $k$-free integers in some real quadratic number fields.
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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 symbolic dynamics on sets of full measure (w.r.t. an ergodic measure of positive entropy) for $C^{1+epsilon}$ flows on compact smooth three-dimensional manifolds. One consequence is that the geodesic flow on the unit tangent bundle of a compact $C^infty$ surface has at least const $times(e^{hT}/T)$ simple closed orbits of period less than $T$, whenever the topological entropy $h$ is positive -- and without further assumptions on the curvature.
We study approximation schemes for shift spaces over a finite alphabet using (pseudo)metrics connected to Ornsteins $bar{d}$ metric. This leads to a class of shift spaces we call $bar{d}$-approachable. A shift space $bar{d}$-approachable when its canonical sequence of Markov approximations converges to it also in the $bar{d}$ sense. We give a topological characterisation of chain mixing $bar{d}$-approachable shift spaces. As an application we provide a new criterion for entropy density of ergodic measures. Entropy-density of a shift space means that every invariant measure $mu$ of such a shift space is the weak$^*$ limit of a sequence $mu_n$ of ergodic measures with the corresponding sequence of entropies $h(mu_n)$ converging to $h(mu)$. We prove ergodic measures are entropy-dense for every shift space that can be approximated in the $bar{d}$ pseudometric by a sequence of transitive sofic shifts. This criterion can be applied to many examples that were out of the reach of previously known techniques including hereditary $mathscr{B}$-free shifts and some minimal or proximal systems. The class of symbolic dynamical systems covered by our results includes also shift spaces where entropy density was established previously using the (almost) specification property.
We study the topological entropy of hom tree-shifts and show that, although the topological entropy is not conjugacy invariant for tree-shifts in general, it remains invariant for hom tree higher block shifts. In doi:10.1016/j.tcs.2018.05.034 and doi:10.3934/dcds.2020186, Petersen and Salama demonstrated the existence of topological entropy for tree-shifts and $h(mathcal{T}_X) geq h(X)$, where $mathcal{T}_X$ is the hom tree-shift derived from $X$. We characterize a necessary and sufficient condition when the equality holds for the case where $X$ is a shift of finite type. In addition, two novel phenomena have been revealed for tree-shifts. There is a gap in the set of topological entropy of hom tree-shifts of finite type, which makes such a set not dense. Last but not least, the topological entropy of a reducible hom tree-shift of finite type is equal to or larger than that of its maximal irreducible component.
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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