The smallest Mealy automaton of intermediate growth

In this paper we study the automaton I2I2, the smallest Mealy automaton of intermediate growth, first considered by the last two authors [I.I. Reznykov, V.I. Sushchansky, The two-state Mealy automata over the two-symbol alphabet of the intermediate growth, Mat. Zametki 72 (2002) 102–117]. We describe the automatic transformation monoid defined by I2I2, give a formula for the generating series for the (ball volume) growth function of I2I2, and give sharp asymptotics for the growth function of I2I2, namelyγI2(n)∼25/233/4π−2n1/4exp(πn/6), with the ratios of left- to right-hand side tending to 1 as n→∞n→∞.