Yu-Mei Xue

Language structure of pattern Sturmian words

Pattern Sturmian words introduced by Kamae and Zamboni [Sequence entropy and the maximal pattern complexity of infinite words, Ergodic Theory Dynamical Systems 22 (2002) 1191-1199; Maximal pattern complexity for discrete systems, Ergodic Theory Dynamical Systems 22 (2002) 1201-1214] are an analogy of Sturmian words for the maximal pattern complexity instead of the block complexity. So far, two kinds of recurrent pattern Sturmian words are known, namely, rotation words and Toeplitz words. But neither a structural characterization nor a reasonable classification of the recurrent pattern Sturmian words is known. In this paper, we introduce a new notion, pattern Sturmian sets, which are used to study the language structure of pattern Sturmian words. We prove that there are exactly two primitive structures for pattern Sturmian words. Consequently, we suggest a classification of pattern Sturmian words according to structures of pattern Sturmian sets and prove that there are at most three classes in this classification. Rotation words and Toeplitz words fall into two different classes, but no examples of words from the third class are known.

Super-stationary set, subword problem and the complexity

Let Ω⊂{0,1}NΩ⊂{0,1}N be a nonempty closed set with N={0,1,2,…}N={0,1,2,…}. For N={N0<N1<N2<⋯}⊂NN={N0<N1<N2<⋯}⊂N and ω∈{0,1}Nω∈{0,1}N, define ω[N]∈{0,1}Nω[N]∈{0,1}N by ω[N](n)≔ω(Nn)(n∈N) and Ω[N]≔{ω[N]∈{0,1}N;ω∈Ω}. We call ΩΩ a super-stationary set if Ω[N]=ΩΩ[N]=Ω holds for any infinite subset NN of NN. Denoting Ω′Ω′ the derived set (i.e. the set of accumulating points) of ΩΩ and degΩ=inf{d;Ω(d+1)=0} with Ω(1)=Ω′,Ω(2)=(Ω′)′,…, it is known [T. Kamae, Uniform set and complexity, preprint, (downloadable from http://www14.plala.or.jp/kamae/e-kamae.htm)] that for any nonempty closed subset ΩΩ of {0,1}N{0,1}N such that there exists an infinite subset NN of NN with degΩ[N]<∞degΩ[N]<∞, there exists an infinite subset MM such that Ω[M]Ω[M] is a super-stationary set. Moreover, if degΩ[N]=∞degΩ[N]=∞ for any infinite subset NN of NN, then the maximal pattern complexity [T. Kamae, Uniform set and complexity, preprint, (downloadable from http://www14.plala.or.jp/kamae/e-kamae.htm)] pΩ∗(k) is 2k(k=1,2,…). Thus, the uniform complexity functions are realized by the super-stationary sets [T. Kamae, Uniform set and complexity, preprint, (downloadable from http://www14.plala.or.jp/kamae/e-kamae.htm)]. We call ξ∈{0,1}∗ξ∈{0,1}∗ a super-subword of ω∈{0,1}Nω∈{0,1}N if there exists S={s1<s2<⋯<sk}S={s1<s2<⋯<sk} with k=|ξ|k=|ξ| such that ξ=ω[S]≔ω(s1)ω(s2)⋯ω(sk)ξ=ω[S]≔ω(s1)ω(s2)⋯ω(sk). Let P(ξ)P(ξ) be the set of ω∈{0,1}Nω∈{0,1}N having no super-subword ξξ. Denote Q(Ξ)=∪ξ∈ΞP(ξ)andP(Ξ)=∩ξ∈ΞP(ξ), where Ξ⊂{0,1}∗Ξ⊂{0,1}∗. In this paper, we prove that the class of super-stationary sets other than {0,1}N{0,1}N coincides with the class of Q(Ξ)Q(Ξ) for nonempty finite sets Ξ⊂{0,1}+Ξ⊂{0,1}+. Moreover, it also coincides with the class of P(L(Ξ)) for nonempty finite sets Ξ⊂{0,1}+Ξ⊂{0,1}+, where L(Ξ) is the set of minimal covers of ΞΞ. Using these expressions, we can calculate the complexity of super-stationary sets and prove that the complexity function of a super-stationary set in kk is either 2k2k or a polynomial function of kk for large kk. We also discuss the word problems related to the super-subwords.

Maximal pattern complexity of two-dimensional words

The maximal pattern complexity of one-dimensional words has been studied in several papers [T. Kamae, L. Zamboni, Sequence entropy and the maximal pattern complexity of infinite words, Ergodic Theory Dynam. Systems 22(4) (2002) 1191-1199; T. Kamae, L. Zamboni, Maximal pattern complexity for discrete systems, Ergodic Theory Dynam. Systems 22(4) (2002) 1201-1214; T. Kamae, H. Rao, Pattern Complexity over l letters, E. Comb. J., to appear; T. Kamae, Y.M. Xue, Two dimensional word with 2k maximal pattern complexity, Osaka J. Math. 41(2) (2004) 257-265]. We study the maximal pattern complexity pα(k) of two-dimensional words α. A two-dimensional version of the notion of strong recurrence is introduced. It is shown that if α is strongly recurrent, then either α is doubly periodic or pα(k≥2k (k = 1, 2,...). Accordingly, we define a two-dimensional pattern Sturmian word as a strongly recurrent word α with pα(k = 2k. Examples of pattern Sturmian words are given.

Maximal pattern complexity of higher dimensional words

This paper studies the pattern complexity of n-dimensional words. We show that an n-recurrent but not n-periodic word has pattern complexity at least 2k, which generalizes the result of [T. Kamae, H. Rao, Y.-M. Xue, Maximal pattern complexity of two dimension words, Theoret. Comput. Sci. 359 (1-3) (2006) 15-27] on two-dimensional words. Analytic directions of a word are defined and its topological properties play a crucial role in the proof. Accordingly n-dimensional pattern Sturmian words are defined. Irrational rotation words are proved to be pattern Sturmian. A new class of higher dimensional words, the simple Toeplitz words, are introduced. We show that they are also pattern Sturmian words.

Partitions by congruent sets and optimal positions

Let X be a metrizable space with a continuous group or semi-group action G. Let D be a nonempty subset of X. Our problem is how to

Holder equivalence of homogeneous Moran sets

For two homogeneous Moran sets E = C([0, 1], {nk}, {ck}) and E′ = C([0, 1], {nk}, {ck}) with Hausdorff dimensions s and s′ with s′ < s such that {nk} and {nk} are bounded and the spacings are uniform in some sense, we prove that there exists a homeomorphism f : E → E′ such that f is ( s′ s − ) -Hölder continuous but not ( s′ s + ) -Hölder continuous for any > 0.

Dynamical behavior of a food chain model with prey toxicity

Abstract This paper deals with a three-dimensional plant–herbivore–predator model that incorporates explicitly the plant toxicity in plant–herbivore interactions. The existence and stability conditions of all the feasible equilibria are established. Our results indicate that plant toxicity may play a key role in the dynamical behavior of the system. By adding another plant species with a different toxicity level to this system, we derive threshold conditions on the invasion of the second plant species. The analysis indicates that several parameters may be critical to determine successful invasion. Numerical simulations are also provided to reinforce the theoretical conclusions.