Abstract: 

We consider operators acting in the space 
$C(X)$ ($X$ is a compact topological space) of the form
$$ Au(x)=\Big(\sum_{k=1}^Ne^{\varphi_k}T_{\alpha_k}\Big)u(x)=
\sum_{k=1}^Ne^{\varphi_k(x)}u(\alpha_k(x)),\;\;u\in C(X),$$
where $\varphi_k\in C(X)$ and $\alpha_k:X\to X$ 
are given continuous mappings ($1\leq k \leq N$).
A new formula on the logarithm of the spectral
radius $r(A)$ is obtained. 
The logarithm of $r(A)$ is defined as a nonlinear functional
$\lambda$ depending on the vector of functions $\varphi=(\varphi_k)_{k=1}^N$. 
We prove that
$$
\ln(r(A)) = \lambda(\varphi)
= \max_{\nu\in Mes}
\bigg\{\sum_{k=1}^N\int_X\varphi_kd\nu_k-\lambda^*(\nu)\bigg\},
$$
where $Mes$ is the set of all probability vectors of measures $\nu=(\nu_k)_{k=1}^N$
on $X\times \{1,\dots ,N\}$
and $\lambda^*$ is some convex lower-semicontinuous functional on
$(C^N(X))^\star$. In other words $\lambda^*$
is the Legendre conjugate to $\lambda$.

1. Institute of Mathematics
   University of Białystok
   Akademicka 2
   15-267 Białystok
   Poland