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The Kotani-Last conjecture states that every ergodic operator in one spacedimension with non-empty absolutely continuous spectrum must have almostperiodic coefficients. This statement makes sense in a variety of settings; forexample, discrete Schr\"odinger operators, Jacobi matrices, CMV matrices, andcontinuum Schr\"odinger operators.  In the main body of this paper we show how to construct counterexamples tothe Kotani-Last conjecture for continuum Schr\"odinger operators by adaptingthe approach developed by Volberg and Yuditskii to construct counterexamples tothe Kotani-Last conjecture for Jacobi matrices. This approach relates thereflectionless operators associated with the prescribed spectrum to a family ofcharacter-automorphic Hardy spaces and then relates the shift action on thespace of operators to the resulting action on the associated characters. Thekey to our approach is an explicit correspondence between the space ofcontinuum reflectionless Schr\"odinger operators associated with a given setand the space of reflectionless Jacobi matrices associated with a derived set.Once this correspondence is established we can rely to a large extent on theprevious work of Volberg and Yuditskii to produce the resulting action on thespace of characters. We analyze this action and identify situations where wecan observe absolute continuity without almost periodicity.  In the appendix we show how to implement this strategy and obtain analogousresults for extended CMV matrices.

Counterexamples to the Kotani-Last Conjecture for Continuum Schr\"odinger Operators via Character-Automorphic Hardy Spaces