We prove the recent conjecture that Minkowski's question mark measure is regular in the sense of logarithmic potential theory. The proof employs: an Iterated Function System composed of Möbius maps, which yields the classical Stern–Brocot sequences, an estimate of the cardinality of large spacings between numbers in these sequences and a criterion due to Stahl and Totik. We also generalize this result to a class of balanced measures of Iterated Function Systems in one dimension.
Regularity of Minkowski's question mark measure, its inverse and a class of IFS invariant measures
Mantica G.
;
2018-01-01
Abstract
We prove the recent conjecture that Minkowski's question mark measure is regular in the sense of logarithmic potential theory. The proof employs: an Iterated Function System composed of Möbius maps, which yields the classical Stern–Brocot sequences, an estimate of the cardinality of large spacings between numbers in these sequences and a criterion due to Stahl and Totik. We also generalize this result to a class of balanced measures of Iterated Function Systems in one dimension.
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