Cost and dimension of words of zero topological entropy

Let $A^*$ denote the free monoid generated by a finite nonempty set $A.$ In this paper we introduce a new measure of complexity of languages $L\subseteq A^*$ defined in terms of the semigroup structure on $A^*.$ For each $L\subseteq A^*,$ we define its {\it cost} $c(L)$ as the infimum of all real numbers $\alpha$ for which there exist a language $S\subseteq A^*$ with $p_S(n)=O(n^\alpha)$ and a positive integer $k$ with $L\subseteq S^k.$ We also define the {\it cost dimension} $d_c(L)$ as the infimum of the set of all positive integers $k$ such that $L\subseteq S^k$ for some language $S$ with $p_S(n)=O(n^{c(L)}).$ We are primarily interested in languages $L$ given by the set of factors of an infinite word $x=x_0x_1x_2\cdots \in A^\omega$ of zero topological entropy, in which case $c(L)<+\infty.$ We establish the following characterisation of words of linear factor complexity: Let $x\in A^\omega$ and $L=$Fac$(x)$ be the set of factors of $x.$ Then $p_x(n)=\Theta(n)$ if and only $c(L)=0$ and $d_c(L)=2.$ In other words, $p_x(n)=O(n)$ if and only if Fac$(x)\subseteq S^2$ for some language $S\subseteq A^+$ of bounded complexity (meaning $\limsup p_S(n)<+\infty).$ In general the cost of a language $L$ reflects deeply the underlying combinatorial structure induced by the semigroup structure on $A^*.$ For example, in contrast to the above characterisation of languages generated by words of sub-linear complexity, there exist non factorial languages $L$ of complexity $p_L(n)=O(\log n)$ (and hence of cost equal to $0)$ and of cost dimension $+\infty.$ In this paper we investigate the cost and cost dimension of languages defined by infinite words of zero topological entropy

Cost and dimension of words of zero topological entropy

Let A∗ denote the free monoid generated by a finite nonempty set A. In this paper we introduce a new measure of complexity of languages L⊆A∗ defined in terms of the semigroup structure on A∗. For each L⊆A∗, we define its {\it cost} c(L) as the infimum of all real numbers α for which there exist a language S⊆A∗ with pS(n)=O(nα) and a positive integer k with L⊆Sk. We also define the {\it cost dimension} dc(L) as the infimum of the set of all positive integers k such that L⊆Sk for some language S with pS(n)=O(nc(L)). We are primarily interested in languages L given by the set of factors of an infinite word x=x0x1x2⋯∈Aω of zero topological entropy, in which case c(L)<+∞. We establish the following characterisation of words of linear factor complexity: Let x∈Aω and L=Fac(x) be the set of factors of x. Then px(n)=Θ(n) if and only c(L)=0 and dc(L)=2. In other words, px(n)=O(n) if and only if Fac(x)⊆S2 for some language S⊆A+ of bounded complexity (meaning limsuppS(n)<+∞). In general the cost of a language L reflects deeply the underlying combinatorial structure induced by the semigroup structure on A∗. For example, in contrast to the above characterisation of languages generated by words of sub-linear complexity, there exist non factorial languages L of complexity pL(n)=O(logn) (and hence of cost equal to 0) and of cost dimension +∞. In this paper we investigate the cost and cost dimension of languages defined by infinite words of zero topological entropy

Cost and dimension of words of zero topological entropy

International audienceLet A * denote the free monoid generated by a finite nonempty set A. In this paper we introduce a new measure of complexity of languages L ⊆ A * defined in terms of the semigroup structure on A *. For each L ⊆ A * , we define its cost c(L) as the infimum of all real numbers α for which there exist a language S ⊆ A * with p S (n) = O(n α) and a positive integer k with L ⊆ S k. We also define the cost dimension d c (L) as the infimum of the set of all positive integers k such that L ⊆ S k for some language S with p S (n) = O(n c(L)). We are primarily interested in languages L given by the set of factors of an infinite word x = x 0 x 1 x 2 • • • ∈ A N of zero topological entropy, in which case c(L) 2 there exist infinite words x of positive cost and of complexity p x (n) = O(n α)