Quasi-Markovian transformations

Abstract

Let $(f,\alpha)$ be the process given by an endomorphism $f$ and by a finite partition $\alpha=\{A\}_{i=1}^{s}$ of a Lebesgue space. Let $E(f,\alpha)$ be the set of densities of absolutely continuous invariant measures for skew products with the base $(f,\alpha)$. We prove that a process $(f,\alpha)$ is Markovian if and only if $E(f,\alpha)\subset \{g: g=\sum_{i=1}^{s} 1_{A_{i}}\otimes g_{i}\}$. If $f$ is the Lasota–Yorke or Misiurewicz type map with the Markovian partition $\alpha$, then $(f,\alpha)$ is quasi-Markovian, i.e. $E(f,\alpha)\subset \{g:\supp g=\bigcup_{i=1}^{s} A_{i}\times B_{i}\}$. Moreover, we give the characterization of quasi-Markovian processes.