On the Interplay of Direct Topological Factorizations and Cellular Automata Dynamics on Beta-Shifts

We consider the range of possible dynamics of cellular automata (CA) on two-sided beta-shifts [Formula: see text] and its relation to direct topological factorizations. We show that any reversible CA [Formula: see text] has an almost equicontinuous direction whenever [Formula: see text] is not sofic. This has the corollary that non-sofic beta-shifts are topologically direct prime, i.e. they are not conjugate to direct topological factorizations [Formula: see text] of two nontrivial subshifts [Formula: see text] and [Formula: see text]. We also give a simple criterion to determine whether [Formula: see text] is conjugate to [Formula: see text] for a given integer [Formula: see text] and a given real [Formula: see text] when [Formula: see text] is a subshift of finite type. When [Formula: see text] is strictly sofic, we show that such a conjugacy is not possible at least when [Formula: see text] is a quadratic Pisot number of degree [Formula: see text]. We conclude by using direct factorizations to give a new proof for the classification of reversible multiplication automata on beta-shifts with integral base and ask whether nontrivial multiplication automata exist when the base is not an integer.

Conjugates of Pisot numbers

In this paper, we investigate the Galois conjugates of a Pisot number [Formula: see text], [Formula: see text]. In particular, we conjecture that for [Formula: see text] we have [Formula: see text] for all conjugates [Formula: see text] of [Formula: see text]. Further, for [Formula: see text], we conjecture that for all Pisot numbers [Formula: see text] we have [Formula: see text]. A similar conjecture if made for [Formula: see text]. We conjecture that all of these bounds are tight. We provide partial supporting evidence for this conjecture. This evidence is both of a theoretical and computational nature. Lastly, we connect this conjecture to a result on the dimension of Bernoulli convolutions parameterized by [Formula: see text], whose conjugate is the reciprocal of a Pisot number.

On the co-factors of degree 6 Salem number beta expansions

For [Formula: see text], a sequence [Formula: see text] with [Formula: see text] is the beta expansion of [Formula: see text] with respect to [Formula: see text] if [Formula: see text]. Defining [Formula: see text] to be the greedy beta expansion of [Formula: see text] with respect to [Formula: see text], it is known that [Formula: see text] is eventually periodic as long as [Formula: see text] is a Pisot number. It is conjectured that the same is true for Salem numbers, but is only currently known to be true for Salem numbers of degree 4. Heuristic arguments suggest that almost all degree 6 Salem numbers admit periodic expansions but that a positive proportion of degree 8 Salem numbers do not. In this paper, we investigate the degree 6 case. We present computational methods for searching for families of degree 6 numbers with eventually periodic greedy expansions by studying the co-factors of their expansions. We also prove that the greedy expansions of degree 6 Salem numbers can have arbitrarily large periods. In addition, computational evidence is compiled on the set of degree 6 Salem numbers with [Formula: see text]. We give examples of numbers with [Formula: see text] whose expansions have period and preperiod lengths exceeding [Formula: see text], yet are still eventually periodic.

Periodic Intermediate β-Expansions of Pisot Numbers

The subshift of finite type property (also known as the Markov property) is ubiquitous in dynamical systems and the simplest and most widely studied class of dynamical systems are β -shifts, namely transformations of the form T β , α : x ↦ β x + α mod 1 acting on [ − α / ( β − 1 ) , ( 1 − α ) / ( β − 1 ) ] , where ( β , α ) ∈ Δ is fixed and where Δ ≔ { ( β , α ) ∈ R 2 : β ∈ ( 1 , 2 ) and 0 ≤ α ≤ 2 − β } . Recently, it was shown, by Li et al. (Proc. Amer. Math. Soc. 147(5): 2045–2055, 2019), that the set of ( β , α ) such that T β , α has the subshift of finite type property is dense in the parameter space Δ . Here, they proposed the following question. Given a fixed β ∈ ( 1 , 2 ) which is the n-th root of a Perron number, does there exists a dense set of α in the fiber { β } × ( 0 , 2 − β ) , so that T β , α has the subshift of finite type property? We answer this question in the positive for a class of Pisot numbers. Further, we investigate if this question holds true when replacing the subshift of finite type property by the sofic property (that is a factor of a subshift of finite type). In doing so we generalise, a classical result of Schmidt (Bull. London Math. Soc., 12(4): 269–278, 1980) from the case when α = 0 to the case when α ∈ ( 0 , 2 − β ) . That is, we examine the structure of the set of eventually periodic points of T β , α when β is a Pisot number and when β is the n-th root of a Pisot number.

Generic point equivalence and Pisot numbers

Let $\unicode[STIX]{x1D6FD}>1$ be an integer or, generally, a Pisot number. Put $T(x)=\{\unicode[STIX]{x1D6FD}x\}$ on $[0,1]$ and let $S:[0,1]\rightarrow [0,1]$ be a piecewise linear transformation whose slopes have the form $\pm \unicode[STIX]{x1D6FD}^{m}$ with positive integers $m$. We give a sufficient condition for $T$ and $S$ to have the same generic points. We also give an uncountable family of maps which share the same set of generic points.

Local Dimensions of Measures of Finite Type II: Measures Without Full Support and With Non-regular Probabilities

Consider a finite sequence of linear contractions Sj(x) = px + dj and probabilities pj > 0 with ∑Pj = 1. We are interested in the self-similar measure , of finite type. In this paper we study the multi-fractal analysis of such measures, extending the theory to measures arising from non-regular probabilities and whose support is not necessarily an interval.Under some mild technical assumptions, we prove that there exists a subset of supp μ of full μ and Hausdorff measure, called the truly essential class, for which the set of (upper or lower) local dimensions is a closed interval. Within the truly essential class we show that there exists a point with local dimension exactly equal to the dimension of the support. We give an example where the set of local dimensions is a two element set, with all the elements of the truly essential class giving the same local dimension. We give general criteria for these measures to be absolutely continuous with respect to the associated Hausdorff measure of their support, and we show that the dimension of the support can be computed using only information about the essential class.To conclude, we present a detailed study of three examples. First, we show that the set of local dimensions of the biased Bernoulli convolution with contraction ratio the inverse of a simple Pisot number always admits an isolated point. We give a precise description of the essential class of a generalized Cantor set of finite type, and show that the k-th convolution of the associated Cantor measure has local dimension at x ∊ (0,1) tending to 1 as ft: tends to infinity. Lastly, we show that within a maximal loop class that is not truly essential, the set of upper local dimensions need not be an interval. This is in contrast to the case for finite type measures with regular probabilities and full interval support.

The Pisot conjecture for -substitutions

We prove the Pisot conjecture for$\unicode[STIX]{x1D6FD}$-substitutions: if$\unicode[STIX]{x1D6FD}$is a Pisot number, then the tiling dynamical system$(\unicode[STIX]{x1D6FA}_{\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D6FD}}},\mathbb{R})$associated with the$\unicode[STIX]{x1D6FD}$-substitution has pure discrete spectrum. As corollaries: (1) arithmetical coding of the hyperbolic solenoidal automorphism associated with the companion matrix of the minimal polynomial of any Pisot number is almost everywhere one-to-one; and (2) all Pisot numbers are weakly finitary.

On the Conjecture of Lehmer, Limit Mahler Measure of Trinomials and Asymptotic Expansions

Let n ≥ 2 be an integer and denote by θn the real root in (0, 1) of the trinomial Gn(X) = −1 + X + Xn. The sequence of Perron numbers $(\theta _n^{ - 1} )_{n \ge 2} $ tends to 1. We prove that the Conjecture of Lehmer is true for $\{ \theta _n^{ - 1} |n \ge 2\} $ by the direct method of Poincaré asymptotic expansions (divergent formal series of functions) of the roots θn, zj,n, of Gn(X) lying in |z| < 1, as a function of n, j only. This method, not yet applied to Lehmer’s problem up to the knowledge of the author, is successfully introduced here. It first gives the asymptotic expansion of the Mahler measures ${\rm{M}}(G_n ) = {\rm{M}}(\theta _n ) = {\rm{M}}(\theta _n^{ - 1} )$ of the trinomials Gn as a function of n only, without invoking Smyth’s Theorem, and their unique limit point above the smallest Pisot number. Comparison is made with Smyth’s, Boyd’s and Flammang’s previous results. By this method we obtain a direct proof that the conjecture of Schinzel-Zassenhaus is true for $\{ \theta _n^{ - 1} |n \ge 2\} $, with a minoration of the house , and a minoration of the Mahler measure M(Gn) better than Dobrowolski’s one. The angular regularity of the roots of Gn, near the unit circle, and limit equidistribution of the conjugates, for n tending to infinity (in the sense of Bilu, Petsche, Pritsker), towards the Haar measure on the unit circle, are described in the context of the Erdős-Turán-Amoroso-Mignotte theory, with uniformly bounded discrepancy functions.