TY - JOUR
T1 - Equivalence of positive Hausdorff measure and the open set condition for self-conformal sets
AU - Peres, Y.
AU - Rams, M.
AU - Simon, K.
AU - Solomyak, B.
PY - 2001
Y1 - 2001
N2 - A compact set is self-conformal if it is a finite union of its images by conformal contractions. It is well known that if the conformal contractions satisfy the ``open set condition'' (OSC), then has positive -dimensional Hausdorff measure, where is the solution of Bowen's pressure equation. We prove that the OSC, the strong OSC, and positivity of the -dimensional Hausdorff measure are equivalent for conformal contractions; this answers a question of R. D. Mauldin. In the self-similar case, when the contractions are linear, this equivalence was proved by Schief (1994), who used a result of Bandt and Graf (1992), but the proofs in these papers do not extend to the nonlinear setting. - See more at: http://www.ams.org/journals/proc/2001-129-09/S0002-9939-01-05969-X/#sthash.Pd60169C.dpuf
AB - A compact set is self-conformal if it is a finite union of its images by conformal contractions. It is well known that if the conformal contractions satisfy the ``open set condition'' (OSC), then has positive -dimensional Hausdorff measure, where is the solution of Bowen's pressure equation. We prove that the OSC, the strong OSC, and positivity of the -dimensional Hausdorff measure are equivalent for conformal contractions; this answers a question of R. D. Mauldin. In the self-similar case, when the contractions are linear, this equivalence was proved by Schief (1994), who used a result of Bandt and Graf (1992), but the proofs in these papers do not extend to the nonlinear setting. - See more at: http://www.ams.org/journals/proc/2001-129-09/S0002-9939-01-05969-X/#sthash.Pd60169C.dpuf
UR - http://www.ams.org/journals/proc/2001-129-09/S0002-9939-01-05969-X/#sthash.Pd60169C.dpuf
M3 - Article
VL - 129
SP - 2689
EP - 2699
JO - Proc. Amer. Math. Soc.
JF - Proc. Amer. Math. Soc.
ER -