Spectral radius of operators associated with dynamical systems in the spaces $C(X)$
Tom 67 / 2005
Streszczenie
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 $\lambda$.
