arXiv subjects

Illya I. Reznykov

Publications and source records attributed to Illya I. Reznykov.

4 recordsLinked to original sources

arXiv2007

A Mealy machine with polynomial growth of irrational degree

We consider a very simple Mealy machine (three states over a two-symbol alphabet), and derive some properties of the semigroup it generates. In particular, this is an infinite, finitely generated semigroup; we show that the growth function of its balls behaves asymptotically like n^2.4401..., where this constant is 1 + log(2)/log((1+sqrt(5))/2); that the semigroup satisfies the identity g^6=g^4; and that its lattice of two-sided ideals is a chain.

math.GR↗

arXiv2006

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 $J_m$.

math.AG↗

arXiv2005

On composite and non-monotonic growth functions of Mealy automata

We introduce the notion of composite growth function and provide examples that illustrate the primary properties of these growth functions. There are provided examples of Mealy automata that have composite non-monotonic growth functions of the polynomial growth order. There are described examples of Mealy automata that have composite monotonic growth functions of intermediate and exponential growth. The questions concerning the interrelation between the notions ``composite'' and ``non-monotonic'' of a Mealy automaton growth function are formulated.

math.AG↗

arXiv2004

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} π^{-2} n^{1/4} \exp{π\sqrt{n/6}} ] with the ratios of left- to right-hand side tending to 1 as $n \to \infty$.

math.GR↗