Abstract
We prove that if {varvec{mu }} is a finitely supported measure on {varvec{SL}}_{textbf{2}}({mathbb {R}}) with positive Lyapunov exponent but not uniformly hyperbolic, then the Lyapunov exponent function is not {varvec{alpha }}-Hölder around {varvec{mu }} for any {varvec{alpha }} exceeding the Shannon entropy of {varvec{mu }} over the Lyapunov exponent of {varvec{mu }}.
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