The smallest Mealy automaton of intermediate growth

In this paper we study the smallest Mealy automaton of intermediate growth, first considered by the last two authors. We describe the automatic transformation monoid it defines, give a formula for the generating series for its (ball volume) growth function, and give sharp asymptotics for its growth function, namely [ F(n) \sim 2^{5/2} 3^{3/4} \pi^{-2} n^{1/4} \exp{\pi\sqrt{n/6}} ] with the ratios of left- to right-hand side tending to 1 as $n \to \infty$

On the 3-state Mealy Automata over an m-symbol Alphabet of Growth Order [ n ^{{\log n}/{2 \log m}} ]

We consider the sequence ${J_m,m \ge 2}$ of the 3-state Mealy automata over an m-symbol alphabet such that the growth function of $J_m$ has the intermediate growth order $[n ^{{\log n}/{2 \log m}} ]$. For each automaton $J_m$ we describe the automaton transformation monoid $S_{J_m}$, defined by it, provide generating series for the growth functions, and consider primary properties of $S_{J_m}$ and $J_m$.Comment: 38 pages, 5 Postscript figure

Semigroups of matrices of intermediate growth

Finitely generated linear semigroups over a field K that have intermediate growth are considered. New classes of such semigroups are found and a conjecture on the equivalence of the subexponential growth of a finitely generated linear semigroup S and the nonexistence of free noncommutative subsemigroups in S, or equivalently the existence of a nontrivial identity satisfied in S, is stated. This 'growth alternative' conjecture is proved for linear semigroups of degree 2, 3 or 4. Certain results supporting the general conjecture are obtained. As the main tool, a new combinatorial property of groups is introduced and studied

Growth of Algebras and Codes

This dissertation is devoted to the study of the growth of algebras and formal languages. It consists of three parts. The first part is devoted to the growth of finitely presented quadratic algebras. The study of these algebras was motivated by the question about the growth types of Koszul algebras which are a special subclass of finitely presented quadratic algebras. We show that there exist finitely presented quadratic algebras of intermediate growth and give two concrete examples of such algebras with their presentations. The second part focuses on the study of the growth of metabelian Lie algebras and their universal enveloping algebras. Our motivation was to construct finitely presented algebras of different intermediate growth types. As an outcome of this investigation we prove that for any d 2 N there exists a finitely presented algebra whose growth function is equivalent to e^n^d=(d+1). The last part focuses on infinite codes over finite alphabets, their properties and growth. A special attention is paid to S-codes, weak S-codes and Markov codes which play an important role in coding theory and ergodic theory. We investigate what types of codes may have maximal growth. Also, we prove that S-codes covering Bernoulli schemes are maximal