Multivariate growth and cogrowth

Abstract

We deal with a multivariate growth series $\Gamma_L(\mathbf{z})$, $\mathbf{z} \in \mathbb{C}^d$, associated with a regular language $L$ over an alphabet of cardinality $d \geq 2$. Our focus is on languages coming from subgroups of the free group $F_m$ of finite rank $m$ and from the subshifts of finite type. We suggest a tool for computing the rate of growth $\varphi_L(\mathbf{r})$ of $L$ in the direction $\mathbf{r} \in \mathbb{R}^d$. Using the concave growth condition introduced by the second author in [Comment. Math. Helv. 2002, 77 (3), 563-608] and the results of Convex Analysis we represent $\psi_L(\mathbf{r}) = \log\left(\varphi_L(\mathbf{r})\right)$ as a support function of a convex set that is the closure of the $\textrm{Relog}$ image of the domain of absolute convergence of $\Gamma_L(\mathbf{z})$. This allows us to compute $\psi_L(\mathbf{r})$ in some cases, including a Fibonacci language or a language of freely reduced words representing elements of a free group $F_2$. Also we show that the methods of the Large Deviation Theory can be used as an alternative approach.