![]() ![]() We then state Lochs-type theorems which work even in the case of zero entropy, in particular for several important log-balanced sequences of partitions of a number-theoretic nature. These are sequences of interval partitions such that the logarithms of the measures of their intervals at each depth are roughly the same. In order to deal with sequences of partitions beyond positive entropy, this paper introduces the notion of log-balanced sequences of partitions, together with their weight functions. Such conversion results can also be stated for sequences of interval partitions under suitable assumptions, with results holding almost everywhere, or in measure, involving the entropy. ![]() In its original version, Lochs' theorem related decimal expansions with continued fraction expansions. Lochs' theorem and its generalizations are conversion theorems that relate the number of digits determined in one expansion of a real number as a function of the number of digits given in some other expansion. Mardi, 14 heures 30, Online Vilmos Komornik (Université de Strasbourg et Shenzhen University) Topology of univoque sets in real base expansions This is joint work with Célia Cisternino, Zuzana Masáková and Edita Pelantová. For our purposes, we use a generalized concept of spectrum associated with a complex base and complex digits, and we study its topological properties. An important tool in our study is the spectrum of numeration systems associated with alternate bases. Under some suitable condition (i.e., our previous sufficient condition for being a sofic system), we prove that the normalization function is computable by a finite Büchi automaton, and furthermore, we effectively construct such an automaton. The second aim is to provide an analogue of Frougny's result concerning normalization of real bases representations. Comparing the setting of alternate bases to that of one real base, these conditions exhibit a new phenomenon: the bases should be expressible as rational functions of their product. We exhibit two conditions: one necessary and one sufficient. However, the difficulty of computing the automorphism group in general is such that any new non-trivial examples are useful and instructive.The first aim of this work is to give information about the algebraic properties of alternate bases determining sofic systems. It is not clear whether the techniques used in this paper can be generalized to a significantly wider class of systems. Another result is that for a minimal system with complexity that grows at most linearly the quotient of the automorphism group by the group generated by $\sigma$ is finite. The proof uses dynamical methods to reduce the problem to combinatorial arguments. An interesting result in the paper is an algorithm to compute the automorphism group in this situation, along with the use of this algorithm to compute all conjugacies between two shifts generated by constant length substitutions. ![]() In this paper, the authors focus on the particular class of substitution systems of constant length, which are the systems whose infinite words are obtained by iterating a substitution infinitely many times on some letter in the alphabet. However, even in the lowest nontrivial complexity (that of nonconstant, sublinear complexity), we do not have a complete understanding of the automorphism group, and in general there is no method that gives a complete characterization of this group. In the past few years, there has been a lot of work on showing that dynamical systems $(X,\sigma)$ for which the complexity function grows slowly have automorphism groups that are in some sense small. A trivial example of an automorphism is $\sigma$ itself, or indeed any power of $\sigma$. In particular, the rate of growth of this function is important.Īn _automorphism_ of a dynamical system $(X,\sigma)$ is a continuous bijection from $X$ to $X$ that commutes with $\sigma$. We prove that for any transitive subshift $X$ with word complexity function $c_n(X)$, if $\liminf \frac)$. ![]()
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