Asymptotic measures for skew products of Bernoulli shifts with generalized north pole–south pole diffeomorphisms

Abstract

We study asymptotic measures for a certain class of dynamical systems. In particular, for T : Σ 2 × M → Σ 2 × M T:{\Sigma _2} \times M \to {\Sigma _2} \times M , a skew product of the Bernoulli shift with a generalized north pole-south pole diffeomorphism, we describe the limits of the following two sequences of measures: (1) iterates under T of the product of Bernoulli measure with Lebesgue measure, T ∗ n ( μ × m ) T_\ast ^n(\mu \times m) , and (2) the averages of iterates of point mass measures, 1 n Σ k = 0 n − 1 δ T k ( w , x ) \frac {1}{n}\Sigma _{k = 0}^{n - 1}{\delta _{{T^k}(w,x)}} . We give conditions for the limit of each sequence to exist. We also determine the subsequential limits in case the sequence does not converge. We exploit several properties of null recurrent Markov Chains and apply them to the symmetric random walk on the integers. We also make use of Strassen’s Theorem as an aid in determining subsequential limits.