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Regularity of Minkowski's question mark measure, its inverse and a class of IFS invariant measures
Author(s) -
Mantica Giorgio,
Totik Vilmos
Publication year - 2019
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms.12197
Subject(s) - mathematics , iterated function system , minkowski space , measure (data warehouse) , conjecture , logarithm , invariant (physics) , pure mathematics , iterated function , class (philosophy) , inverse , invariant measure , discrete mathematics , cardinality (data modeling) , combinatorics , mathematical analysis , ergodic theory , attractor , geometry , database , artificial intelligence , computer science , mathematical physics , data mining
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.