Totally dissipative measures for the shift and conformal \(\sigma \)-finite measures for the stable holonomies (Q968022)

Suppose \((\Sigma_A, \sigma)\) is a mixing two-sided subshift of finite type with alphabet \(E\). For \(e\in E\), let \(C\) be a cylinder \(\{ x\in \Sigma_A\mid x(0)=e\}\), \(\phi:\Sigma_A\to R\) Hölder, \(\beta\in R\), \(m_\beta\) the equilibrium state for the potential \(\phi - \beta \chi_C\), and \(g\) the first return map of \(\sigma\) from \(C\) to \(C\). There is a \(g\)-invariant probability measure \(\mu_\beta\) on \(C\) induced by \(m_\beta\). The paper is largely concerned with behavior of \(\mu_\beta\) as \(\beta\to \infty\). It is shown that \(\mu_\beta\) converges to a probability measure \(\mu\), the support of \(\mu\) is disjoint from its image under \(\sigma^n\) for any \(n\neq 0\), yet for any continuous function \(f\), the time average of \(f\) for almost all \(x\) with respect to the measure \(\sum_{k=0}^\infty \sigma^k_*\mu\) converges to the integral of \(f\) with respect to one of the ergodic components of the limiting measure \(\mu\). Moreover, if \(\{m_x^u\}\) is a system of conditional measures with respect to the partition of \(\Sigma_A\) by the local unstable leaves \(L_x = \{ y\in \Sigma_A\mid y_n = x_n, n= 0, -1, -2, \dots\}\), the paper gives the construction of a systems of measures equivalent to \(\{m_x^u\}\) that are quasi-invariant under stable holonomy.