On structure of invariant measure for a class of discrete dynamical system (Q2782069)

The author studies the structure of the invariant measure for one class of ergodic discrete dynamical systems. The analytical expression NEWLINE\[NEWLINE \mu_\varphi(A) = \lim\limits_{\lambda\uparrow1\,\,\,(\text{Im}@,\lambda = 0)} \int\limits_0^{2\pi} \frac{2(1-\lambda)^2 d\sigma_\varphi(s;A)}{1-2\lambda\cos s+\lambda^2}\tag{1} NEWLINE\]NEWLINE is obtained, where \(\,A\in\mathcal A(M)\,\) and \(\,\sigma_\varphi\: [0,2\pi]\times \mathcal A(M)\to \mathbb R_+\,\) are the Stieltjes measure on \(\,[0,2\pi]\,\) generated by \textit{a~priori} given dynamical system \(\,\varphi\: M\to M\,\) and the measure \(\,\mu\: \mathcal A(M)\to \mathbb R_+\,\) for the invariant measure \(\,\mu_\varphi\: \mathcal A(M)\to \mathbb R_+\).