  We investigate a multivariate growth series $\Gamma_L({\bf z}), {\bf z} \in
\mathbb{C}^d$ associated with a regular language $L$ over an alphabet of
cardinality $d.$ Our focus is on languages coming from subgroups of the free
group and from subshifts of finite type. We develop a mechanism for computing
the rate of growth $\varphi_L({\bf r})$ of $L$ in the direction ${\bf r} \in
\mathbb{R}^d$. Using the concave growth condition (CG) introduced by the second
author in \cite{quint2002divergence} and the results of Convex Analysis we
represent $\psi_L({\bf r}) = \log\left(\varphi_L({\bf r})\right)$ as a support
function of a convex set that is a closure of the $\textrm{Relog}$ image of the
domain of absolute convergence of $\Gamma_L({\bf z})$. This allows us to
compute $\psi_L({\bf r})$ in some important cases, like 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. Finally, we suggest some open problems directed on the
possibility of extensions of the results of the first author from
\cite{grigorchuk1980symmetrical} on multivariate cogrowth.
