Following a recent paper of Siboni's (1994) we study a one-parameter family of skew-products of the torus and prove that for irrational values of the parameter each one is an exact endomorphism and for characters we have exponential decay of correlations. We pose the problem of measure-theoretic classification. When the Lebesgue measure is altered to other Bernoulli measures the problem is solved and isomorphisms are specified. Uncountably many are non-isomorphic; in fact +or- epsilon ( epsilon the parameter) is a complete invariant.

This paper demonstrates that for irrational values of a parameter in a one-parameter family of skew-products, each one is an exact endomorphism and for characters there is exponential decay of correlations.

Ergodic properties of a one-parameter family of skew-products

Key Takeaway: This paper demonstrates that for irrational values of a parameter in a one-parameter family of skew-products, each one is an exact endomorphism and for characters there is exponential decay of correlations.