The dynamics of the square root map on a Sturmian subshift are well understood. We proved that the square root map preserves the languages of Sturmian words (which are optimal squareful words). We apply our results to the symbolic square root map $\sqrt\) of s is the infinite word \(X_1 X_2 \cdots \) obtained by deleting half of each square. This result can be interpreted as a yet another characterization for standard Sturmian words. ![]() A particular and remarkable consequence is that a word $w$ is a standard word if and only if its reversal is a solution to the word equation and $\gcd(|w|, |w|_1) = 1$. We apply a method developed by the second author for studying word equations and prove that there are exactly two families of solutions: reversed standard words and words obtained from reversed standard words by a simple substitution scheme. We consider solutions of the word equation $X_1^2 \dotsm X_n^2 = (X_1 \dotsm X_n)^2$ such that the squares $X_i^2$ are minimal squares found in optimal squareful infinite words. We also give a characterization of 2-repetitive sequences and solve the values of M(α) for 1≤α≤15/7. In this paper, we study optimal 2-repetitive sequences and optimal 2 -repetitive sequences, and show that Sturmian words belong to both classes. ![]() We call the everywhere α-repetitive sequences witnessing this property optimal. In both cases, the number of distinct minimal α-repetitions (or α -repetitions) occurring in the sequence is finite.A natural question regarding global regularity is to determine the least number, denoted by M(α), of distinct minimalα-repetitions such that an α-repetitive sequence is not necessarily ultimately periodic. ![]() If each repetition is of order strictly larger than α, then the sequence is called everywhereα -repetitive. Such a sequence is defined by the property that there exists an integer N≥2 such that every length-N factor has a repetition of order α as a prefix. We consider this topic by studying everywhereα-repetitive sequences. The author's easily-readable style combined with the profusion of exercises and references, summaries, historical remarks, and heuristic discussions make this book useful either as a text for graduate students or self-study, or as a reference work for the initiated.Local constraints on an infinite sequence that imply global regularity are of general interest in combinatorics on words. Among the advanced topics are a thorough treatment of maximal functions and their usefulness in ergodic theory, analysis, and probability, an introduction to almost-periodic functions and topological dynamics, a proof of the Jewett-Krieger Theorem, an introduction to multiple recurrence and the Szemeredi-Furstenberg Theorem, and the Keane-Smorodinsky proof of Ornstein's Isomorphism Theorem for Bernoulli shifts. At the introductory level, the book provides clear and complete discussions of the standard examples, the mean and pointwise ergodic theorems, recurrence, ergodicity, weak mixing, strong mixing, and the fundamentals of entropy. Each of the four basic aspects of ergodic theory - examples, convergence theorems, recurrence properties, and entropy - receives first a basic and then a more advanced, particularized treatment. By selecting one or more of these topics to focus on, the reader can quickly approach the specialized literature and indeed the frontier of the area of interest. Karl Petersen has written a book which presents the fundamentals of the ergodic theory of point transformations and then several advanced topics which are currently undergoing intense research. The study of dynamical systems forms a vast and rapidly developing field even when one considers only activity whose methods derive mainly from measure theory and functional analysis. On the other hand, for the so-called "consecutive" ordering, the Vershik map is not strongly mixing on all finite rank diagrams. We then prove that the Vershik map on the path space of an exact finite rank diagram cannot be strongly mixing, independent of the ordering. Several examples illustrate the broad range of possible behavior of finite type diagrams and invariant measures on them. One of these is a condition of exact finite rank, which parallels a similar notion in measurable dynamics. A number of sufficient conditions for unique ergodicity are obtained. ![]() We further investigate quantitative properties of these measures, which are mainly determined by the asymptotic behavior of products of incidence matrices. It is shown that every ergodic invariant measure (finite or "regular" infinite) is obtained by an extension from a simple subdiagram. We consider Bratteli diagrams of finite rank (not necessarily simple) and ergodic invariant measures with respect to the cofinal equivalence relation on their path spaces.
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