Diophantine Properties of Measures Invariant with Respect to the Gauss Map
نویسندگان
چکیده
Motivated by the work of D. Y. Kleinbock, E. Lindenstrauss, G. A. Margulis, and B. Weiss [7, 8], we explore the Diophantine properties of probability measures invariant under the Gauss map. Specifically, we prove that every such measure which has finite Lyapunov exponent is extremal, i.e. gives zero measure to the set of very well approximable numbers. We show on the other hand that there exist examples where the Lyapunov exponent is infinite and the invariant measure is not extremal. We provide a partial converse to B. Weiss’s result of Khinchine type [12] by constructing a large class of measures, which are both conformal and Ahlfors regular, and for which the divergence of Weiss’s series entails the ψ-approximability of almost all numbers.
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