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We show that the measure of the spectrum of Schr\"odinger operator withpotential defined by non-constant function over any minimal aperiodic finitesubshift tends to zero, as the coupling constant tends to infinity. We alsoobtained a quantitative upper bound for the measure of the spectrum. Thisfollows from a result we proved for ergodic Schr\"odinger operators withfinitely valued potentials under two conditions on the recurrence property ofthe shift. We also show that one of these conditions is necessary for suchresult.

On the spectrums of ergodic Schrodinger operators with finitely valued potentials