Invariant measures and asymptotics for some skew products (Q701358)

Let \(T_\phi: X\times G\to X\times G\), \(T_\phi(x, g)= (\tau(x), g+\phi(x))\) be a skew product transformation over a measure-preserving dynamical system \((X,{\mathcal B},\tau,m)\) and a measurable map \(\phi: X\to G\), where \(G\) is a locally compact, Abelian and Polish group. A measure \(\mu\) on \(X\times G\) is called finite if it is finite on all sets \(X\times K\) with \(K\) compact. Consider the pair \((\tau,G)\) which is either \(\tau(x)= x+\alpha\text{ mod }1\) and \(G= \mathbb{R}\) or \(\tau\) the adic transformation and \(G\) arbitrary. For specifically defined skewing functions (in each of the above cases) the authors show the following results: 1. For each continuous homomorphism \(\alpha: G\to\mathbb{R}\), there exists a unique \(\exp[\alpha\circ \phi]\)-conformal probability measure \(m_\alpha\) on \(X\). 2. For each continuous homomorphism \(\alpha: G\to \mathbb{R}\) the Maharam measure \(d\mu_\alpha(x, g)= e^{-\alpha(g)} dm_\alpha(x) dg\) is ergodic. 3. Any ergodic \(T_\phi\)-invariant locally finite measure is a multiple of some Maharam measure. The last two sections deal with asymptotic properties of aperiodic Hölder continuous functions defined on \(X\) when considering the second of the above cases. It is shown that the skew product is boundedly rationally ergodic with return sequence \(\asymp n/(\log n)^{d/2}\) and that a logarithmic ergodic theorem holds.