Grammatical complexity of unimodal maps with eventually periodic kneading sequences

We study complexity problems of the languages L(KS) generated from eventually periodic kneading sequences (abbreviated as KS) of unimodal maps on an interval. Two approaches have been used in this paper. The first approach is to analyse the set of all distinct excluded blocks of L(KS). It is proved that this set is infinite and has a special semi-linear structure. A method of calculating it is given and explained through examples. The second approach is to find the finite automata accepting L(KS). The minimal DFA accepting L(KS) is obtained. A formula is given to calculate the regular language complexities of L(KS). Finally, we solve the problem of where all generalized composition rules, including *-composition rules, come from. Many new self-similar mappings from the set of all KSs into itself are obtained.

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