More on the Dynamics of the Symbolic Square Root Map

Quasiperiods of biinfinite words

We study infinite words u over an alphabet $\mathcal{A}$ satisfying the property $\mathcal{P} :~\mathcal{P}(n)+ \mathcal{P}(n+1) = 1+ \#\mathcal{A} {\rm any} n \in \mathbb{N}$ , where $\mathcal{P}(n)$ denotes the number of palindromic factors of length n occurring in the language of u . We study also infinite words satisfying a stronger property $\mathcal{PE}$ : every palindrome of u has exactly one palindromic extension in u . For binary words, the properties $\mathcal{P}$ and $\mathcal{PE}$ coincide and these properties characterize Sturmian words, i.e., words with the complexity C(n) = n + 1 for any $n \in \mathbb{N}$ . In this paper, we focus on ternary infinite words with the language closed under reversal. For such words u , we prove that if C(n) = 2n + 1 for any $n \in \mathbb{N}$ , then u satisfies the property $\mathcal{P}$ and moreover u is rich in palindromes. Also a sufficient condition for the property $\mathcal{PE}$ is given. We construct a word demonstrating that $\mathcal{P}$ on a ternary alphabet does not imply $\mathcal{PE}$ .

The central problem in dynamical systems is the asymptotic behavior or topological structure of the orbits. Nevertheless only orbits of points with certain recurrence and form a set of full measure are truly of importance. Of course, such a set is desired to be as small (in the sense of set inclusion) as possible. In this paper we discuss such two sets: the set of weakly almost periodic points and the set of quasi-weakly almost periodic points. While the two sets are different from each other by definitions, we prove that their closures both coincide with the measure center (or the minimal center of attraction) of the dynamical systems. Generally, a point may have three levels of orbit-structure: the support of an invariant measure generated by the point, its minimal center of attraction and its ?-limit set. We study the three levels of orbit-structure for weakly almost periodic points and quasi-weakly almost periodic points. We prove that quasi-weakly almost periodic points possess especially rich topological orbit-structures. We also present a necessary and sufficient condition for a point to belong to its own minimal center of attraction.

We show that the separation property fails for the classes Sigma_n of the Rabin-Mostowski index hierarchy of alternating automata on infinite trees. This extends our previous result (obtained with Szczepan Hummel) on the failure of the separation property for the class Sigma_2 (i.e., for co-Buchi sets). It remains open whether the separation property does hold for the classes Pi_n of the index hierarchy. To prove our result, we first consider the Rabin-Mostowski index hierarchy of deterministic automata on infinite words, for which we give a complete answer (generalizing previous results of Selivanov): the separation property holds for Pi_n and fails for Sigma_n-classes. The construction invented for words turns out to be useful for trees via a suitable game.

Prefix normal words are binary words that have no factor with more 1s than the prefix of the same length. Finite prefix normal words were introduced in [Fici and Liptak, DLT 2011]. In this paper, we study infinite prefix normal words and explore their relationship to some known classes of infinite binary words. In particular, we establish a connection between prefix normal words and Sturmian words, between prefix normal words and abelian complexity, and between prefix normality and lexicographic order.

Given an infinite word with values in a finite alphabet that admit frequencies for all its factors, the smallest frequency function associates with a given positive integer the smallest frequency for factors of this length, and the recurrence function measures the gaps between successive occurrences of factors. We develop in this paper the relations between the growth orders of the smallest frequency and of the recurrence functions. As proved by M. Boshernitzan, linearly recurrent words are characterized in terms of the smallest frequency function. In this paper, we see how to extend this result beyond the case of linearly recurrent words. We first establish a general relation between the smallest frequency and the recurrence functions. Given a lower bound for the smallest frequency function, this relation provides an upper bound for the recurrence function which involves the primitive of this lower bound. We then see how to improve this relation in the case of Sturmian words where the product of the smallest frequency and the recurrence function is bounded for most of the lengths of factors. We also indicate how to construct Sturmian words having a prescribed behaviour for the smallest frequency function, and for the product of the smallest frequency and the recurrence function.

In this paper we give a broad unified framework via group actions for constructing complexity functions of infinite words x = x 0 x 1 x 2 ⋯ ∈ A N x=x_0x_1x_2\cdots \in \mathbb {A}^{\mathbb {N}} with values in a finite set A . \mathbb {A}. Factor complexity, Abelian complexity and cyclic complexity are all particular cases of this general construction. We consider infinite sequences of permutation groups ω = ( G n ) n ≥ 1 \omega = (G_n)_{n\geq 1} with each G n ⊆ S n . G_n \subseteq S_n. Associated with every such sequence is a complexity function p ω , x : N → N p_{\omega ,x}:\mathbb {N} \rightarrow \mathbb {N} which counts, for each length n , n, the number of equivalence classes of factors of x x of length n n under the action of G n G_n on A n \mathbb {A}^n given by g ∗ ( u 1 u 2 ⋯ u n ) = u g − 1 ( 1 ) u g − 1 ( 2 ) ⋯ u g − 1 ( n ) . g* (u_1u_2\cdots u_n)=u_{g^{-1}(1)}u_{g^{-1}(2)}\cdots u_{g^{-1}(n)}. Each choice of ω = ( G n ) n ≥ 1 \omega =(G_n)_{n\geq 1} defines a unique complexity function which reflects a different combinatorial property of a given infinite word. For instance, an infinite word x x has bounded Abelian complexity if and only if x x is k k -balanced for some positive integer k , k, while bounded cyclic complexity is equivalent to x x being ultimately periodic. A celebrated result of G.A. Hedlund and M. Morse states that every aperiodic infinite word x ∈ A N x\in \mathbb {A}^{\mathbb {N}} contains at least n + 1 n+1 distinct factors of each length n . n. Moreover x ∈ A N x\in \mathbb {A}^{\mathbb {N}} has exactly n + 1 n+1 distinct factors of each length n n if and only if x x is a Sturmian word, i.e., binary, aperiodic and balanced. We prove that this characterisation of aperiodicity and Sturmian words extends to this general framework.

Let r and s be multiplicatively independent positive integers. We establish that the r-ary expansion and the s-ary expansion of an irrational real number, viewed as infinite words on {0,1,...,r-1} and {0,1,...,s-1}, respectively, cannot have simultaneously a low block complexity. In particular, they cannot be both Sturmian words.

Standard Sturmian infinite words have a curious property, discovered by G. Rauzy. If in such a word we delete all occurrences of each letter, except every pth one, then we get the same infinite word. This property and several generalizations are studied here. In the last part a short and self-contained theory of Sturmian words, using only combinatorial arguments, is presented.

On a combinatorial property of Sturmian words

Some stability properties of motions in pseudo-dynamical systems and semi-systems are studied. Introduction. The purpose of the present paper is to study some properties of motions in dynamical and pseudo-dynamical systems (see definitions below) on metric spaces. We consider uniform stability and semistability (see Section 2) of motions in nonempty subsets of the phase space and discuss in particular properties of mappings of the types: x {the limit set of x} and x {the prolongational limit set of x}, as well as some regularity properties of mappings whose values are some generalized prolongational limit sets (see the formula (1.8)). In Sections 3 and 4 we consider the problem of propagation of stability and uniform stability of motions, from sets onto closures of those sets. In Section 7 we consider certain Lyapunov type functions. The paper is closed by some remarks concerning stability properties of motions preserved when we pass from one dynamical system to another, assuming that those systems satisfy certain conditions of the type of “asymptotic equivalence” in the Wazewski sense. 1. Preliminaries. In order to exclude any misunderstanding we recall fundamental definitions and fix notation used in the sequel. 1991 Mathematics Subject Classification: 34C35, 54H20, 58F10, 58F25, 34D05, 34D20.

We study an automata-theoretic approach to planning for temporally extended goals. Specifically, we devise techniques based on nonemptiness of Buchi automata on infinite words, to synthesize sequential and conditional plans in a generalized setting in which we have that: goals are general temporal properties of desired execution; dynamic systems are represented by finite transition systems; incomplete information on the initial situation is allowed; and states are only partially observable. We prove that the techniques proposed are optimal wrt the worst case complexity of the problem. Thanks to the scalability of the nonemptiness algorithms, the techniques presented here promise to be applicable to fairly large systems, notwithstanding the intrinsic complexity of the problem.

An algebraic approach to the theory of regular languages of finite and infinite words (∞-languages) is presented. It extends the algebraic theory of regular languages of finite words, which is based on finite semigroups. Their role is taken over by a structure called right binoid. A variety theorem is proved: there is a one-to-one correspondence between varieties of ∞-languages and pseudovarieties of right binoids. The class of locally threshold testable languages and several natural subclasses (such as the class of locally testable languages) as well as classes of the Borel hierarchy over the Cantor space (restricted to regular languages) are investigated as examples for varieties of ∞-languages. The corresponding pseudovarieties of right binoids are characterized and in some cases defining equations are derived. The connections with the algebraic description and classification of regular languages of infinite words in terms of finite semigroups are pointed out.

The main elements of the phase space of a dynamical system are singular points and limits sets.1 Singular points and limit sets can both attract all adjacent phase trajectories and repel them. The former are often spoken of as attractors and the latter are said to be repellers.

Natural tangent dynamics with recurrent biorthonormalizations: A geometric computational approach to dynamical systems exhibiting slow manifolds and periodic/chaotic limit sets