Ruelle operator with weakly contractive iterated function systems (Q2842236)

Let NEWLINE\[NEWLINE\Sigma =\{1,\dots,N\}^{\mathbb N} =\{ \omega =i_0 i_1 \dots i_{n-1} \dots|i_{n-1} \in \{1,\dots,N \}, n=1,2,\dots \}NEWLINE\]NEWLINE be the one-sided symbolic space and \(\sigma\) the map defined by NEWLINE\[NEWLINE\sigma :\omega =i_0 i_1 \dots i_{n-1} \dots \rightarrow \sigma (\omega )= i_1 \dots i_{n-1}\dots.NEWLINE\]NEWLINE \((\Sigma, \sigma )\) is called a symbolic system. Let \(\phi\) be a Hölder continuous function on \(\Sigma\). The Ruelle operator is defined by NEWLINE\[NEWLINE{\mathcal T}f(x)=\sum_{y\in \sigma^{-1} (x)} e^{\phi (y)} f(y),\quad f\in C(\Sigma ).NEWLINE\]NEWLINE It is a positive operator. NEWLINENEWLINENEWLINE A Dini (resp. Lipschitz) system is a triple \((X,\{w_j \}_{j=1} ^m ,\{p_j \}_{j=1} ^m)\) where the \(w_j\)'s are functions \(w_j : X \rightarrow X\) that are both continuously differentiable and weakly contractive, and the \(p_j\)'s are potentials \(p_j : X \rightarrow \mathbb R^+\), that are positive Dini (resp. Lipschitz) continuous. Throughout the paper it is assumed that the maps \(w_j :X\rightarrow X,1 \leq j \leq m\), are both weakly contractive and continuously differentiable. The main results of the paper are the following:NEWLINENEWLINETheorem 1.2. Let \((X,\{w_j \}_{j=1} ^m ,\{p_j \}_{j=1} ^m )\) be a Dini system with \(w_j ' (x) \neq 0\), \(\forall j\). Suppose that the following condition is fulfilled: NEWLINE\[NEWLINE\sum_{j=1} ^m p_j (x) \dot |w_j ' (x)|< \rho \,\, \text{for all}\,\, x \in X. \tag{1}NEWLINE\]NEWLINE Then the Ruelle operator theorem holds for this Dini system.NEWLINENEWLINETheorem 1.4. Let \((X,\{w_j \}_{j=1} ^m ,\{p_j \}_{j=1} ^m )\) be a Lipschitz system. Suppose that condition (1) is satisfied. Then \(\rho_e (T) < \rho (T)\). Moreover, for any Lipschitz continuous function \(f\) defined on some subset \(K\) of \(X\), the sequence \(\rho^{-n} T^n f\) converges with a specific geometric rate.