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Moreover, αn+1 = αn and αn+1 − αn > 1. 242. This Growth Rate conjecture strengthens Exponential conjecture. Thus, the threshold languages are supposed to be big enough in spite of the fact that it was tough to prove that they are even infinite. 376. 6 Extending the Techniques to Related Classes of Languages In this section we aim to show two possible applications of the methods described in Sect. 3, 4 beyond the class of power-free languages. Regular approximations (see Sect. 3) can be used to estimate the growth rate of a factorial language L whenever the finite approximations of the antidictionary of L can be calculated.
Establish asymptotic laws of behaviour of the growth rate for large alphabets. 5. Extend the line of Dejean’s conjecture; in particular, prove the following Exponential conjecture: threshold languages over three and more letters have exponential complexity 1 . 2 Small Languages: Polynomial Plateau In this section, all words and languages are binary. It is no surprise that binary power-free languages attract great attention of researchers. The interest was warmed up by the first two results that appears almost simultaneously.
Now we roughly describe the algorithm of [31, 33], which drastically improves the situation. We split G into strong components and process each component separately, because the index of a digraph equals the maximum index of its strong component. , we calculate the sequence x, Ax, . . , An x, where x is some initial vector and A is the adjacency matrix of the processed component. The convergence of the iteration method can be guaranteed by some additional argument, and the convergence rate is exponential for the adjacency matrices of all strongly connected digraphs.