Semi-uniform sub-additive ergodic theorems for discontinuous skew-product transformations (Q2845468)

They establish some semi-uniform ergodic theorems for skew product transformations with the discontinuity sets of transformations.NEWLINENEWLINESuppose that \(\Omega\) and \(X\) are compact metric spaces, and \(\phi\colon\Omega\to\Omega\) is a continuous transformation. Let \(\Phi\colon\Omega\times X\to\Omega\times X\) be a skew-product transformation with the base \(\phi\). Denote the discontinuity set by NEWLINE\[NEWLINED_\Phi=\{(\omega,x)\in\Omega\times X\colon\Phi\hbox{ is discontinuous at }(\omega,x)\},NEWLINE\]NEWLINE and the set \(\mathcal M(\Omega,\phi)\) be a \(\phi\)-invariant Borel probability on \(\Omega\). Assume that NEWLINE\[NEWLINE \nu(\pi(D_\Phi))=0\quad\forall\nu\in\mathcal M(\Omega,\phi),\leqno{(\mathrm{H1})}NEWLINE\]NEWLINE where \(\pi\) is the projection from \(\Omega\times X\) to \(\Omega\).NEWLINENEWLINEThe main theorem is: NEWLINENEWLINE\medskip { Theorem 1.2.} Let \(\Phi\) be an skew-product transformation with the base \(\phi\) on \(\Omega\) fulfilling (H1). Suppose that \(a\in\mathbb R\) and \(h\in C(\Omega\times X)\) satisfies NEWLINE\[NEWLINE\int_{\Omega\times X}h\,d\mu\leq a\quad\forall\mu\in\mathcal M(\Omega\times X,\Phi).NEWLINE\]NEWLINE Then, for any \(\varepsilon>0\), there exists some \(N_\varepsilon\in\mathbb N\) such that for all \(n\geq N_\varepsilon\), the following holds NEWLINE\[NEWLINE{1\over n}\sum_{i=0}^{n-1}h(\Phi^i(\omega,x))\leq a+\varepsilon\quad \forall(\omega,x)\in\Omega\times X.NEWLINE\]NEWLINE NEWLINENEWLINE\medskip An example of skew-product transformation fulfilling the above conditions is given, namely ``linear Schrödinger equations with fixed phase transitions''. They also prove the following. NEWLINENEWLINENEWLINE\medskip { Theorem 1.4.} Suppose that \(a\in\mathbb R\) and \(\{\varphi_n\}_{n\in\mathbb N}\subset C(\Omega\times X)\) is a sub-additive sequence with respect to \(\Phi\) such that for every ergodic measure \(\lambda\in\mathcal M(\Omega\times X,\Phi)\), there exists some \(m\in\mathbb N\) such that NEWLINE\[NEWLINE{1\over m}\int_{\Omega\times X}\varphi_m\,d\lambda\leq a.NEWLINE\]NEWLINE Then under the same conditions as in Theorem 1.2. and for any \(\varepsilon>0\), there exists some \(N_\varepsilon\in\mathbb N\) such that for all \(n\geq N_\varepsilon\), the following inequality holds NEWLINE\[NEWLINE{1\over n}\varphi_n(\omega,x)\leq a+\varepsilon\quad\forall(\omega,x)\in\Omega\times X.NEWLINE\]