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We investigate the spectral properties of the discrete one-dimensionalSchr\"odinger operators whose potentials are generated by continuous samplingalong the orbits of a minimal translation of a Cantor group. We show that forgiven Cantor group and minimal translation, there is a dense set of continuoussampling functions such that the spectrum of the associated operators has zeroHausdorff dimension and all spectral measures are purely singular continuous.The associated Lyapunov exponent is a continuous strictly positive function ofthe energy. It is possible to include a coupling constant in the model andthese results then hold for every non-zero value of the coupling constant.

Limit-Periodic Schr\"odinger Operators in the Regime of Positive Lyapunov Exponents