We see from the previous theorem that there is a one - to - one correspondence. The results show that while there is some similarity it is possible for the square root map to exhibit quite different behavior compared to the Sturmian case. Let us call a subshift Sturmian if it is generated by a Sturmian sequence. Quasi-Sturmian subshifts extending the corresponding results in 19 for certain. The main results are characterizations of periodic points and the limit set. Here, a subshift is called periodic if there exists an and a p. In this paper, we study these dynamical systems in more detail and compare their properties to the Sturmian case. The (factor) complexity function p : Z+ Z+ is defined by p(n)W(. In our earlier work, we introduced another type of subshift of optimal squareful words which together with the square root map form a dynamical system. exhibit a large class of effective Sturmian subshifts which can be realized by. The dynamics of the square root map on a Sturmian subshift are well understood. by projective subaction of a subshift of finite type or a sofic. We proved that the square root map preserves the languages of Sturmian words (which are optimal squareful words). The square root s of s is the infinite word X 1 X 2 ⋯ obtained by deleting half of each square. Section3uses notions from computability theory (also called recursion theory). See Lind and Marcus 5 for an introduction to subshifts. An example Sturmian subshifts: These are subshifts whose languages. Then because this conjugacy simply exchanges 0’s and 1’s, we see that y and z have the same anomaly size. For a presentation of Bipartite codes and conjugacy for subshifts Nasus paper. Every optimal squareful infinite word s contains exactly six minimal squares and can be written as a product of these squares: s = X 1 2 X 2 2 ⋯. S nN of all sequences that avoid Sis called a subshift (or a shift space) and the elements of Sare called forbidden words. By Proposition 3.3, the subshift generated by y is conjugate by symbol reversal to a subshift generated by a skew Sturmian sequence z with frequency p q and type S. In this paper, we attempt to extend the definition of the entropic chaos degree on a d -dimensional Euclidean space to improve the ability to measure the stength of chaos of the dynamical system and show several relations between the extended entropic chaos degree and the Lyapunov exponent.In our earlier paper ], we introduced a symbolic square root map. for T to admit a subshift cover is that T must have finite entropy. Moreover, we study the zeta-function of the spectral triple and relate its abscissa of convergence to the complexity exponent of the subshift or the. for expansive systems Re and for some or all group translations (e.g., Sturmian. For repetitive tilings we show that if their patches have equi-distributed frequencies then the two metrics are Lipschitz equivalent. It may seem like a kind of finite space Kolmogorov-Sinai entropy, which then indicates the relation between the entropic chaos degree and the Lyapunov exponent. For Sturmian subshifts this is equivalent to linear recurrence. The entropic chaos degree can be used to compute the strength of chaos with a practical time series. ![]() strength of chaos of the dynamical system. Suppose there is a metric G-flow (X, f) such that every Sturmian subshift is. Information dynamics introduces the entropic chaos degree to measure the. Then there is no metric G-flow that has all. However, we cannot directly compute the Lyapunov exponent for a dynamical system without its dynamical equation, although some estimation methods do exist. For instance the spectrum of a Sturmian subshift MH40 with an irrational angle is the cone whose base is the degree of the angle of the rotation. described the automorphism group of Sturmian shifts, and generalizations are given. The Lyapunov exponent is used to quantify the chaos of a dynamical system, by characterizing the exponential sensitivity of an initial point on the dynamical system. For a subshift over a finite alphabet, a measure of the complexity.
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