Lq DENSITIES FOR MEASURES ASSOCIATED WITH PARABOLIC IFS WITH OVERLAPS
نویسنده
چکیده
We study parabolic iterated function systems (IFS) with overlaps on the real line and measures associated with them. A Borel probability measure μ on the coding space projects into a measure ν on the limit set of the IFS. We consider families of IFS satisfying a transversality condition. In [SSU2] sufficient conditions were found for the measure ν to be absolutely continuous for Lebesgue-a.e. parameter value. Here we investigate when ν has a density in Lq(R) for q > 1. A necessary condition is that the q-dimension of μ (computed with respect to a certain metric associated with the IFS) is greater or equal to one. We prove that this is sharp for 1 < q ≤ 2 in the following sense: if μ is a Gibbs measure with a Hölder continuous potential, then ν has a density in Lq(R) for Lebesgue-a.e. parameter value such that the q-dimension of μ is greater than one. This result is applied to a family of random continued fractions studied by R. Lyons.
منابع مشابه
ABSOLUTELY CONTINUOUS MEASURES WITH Lq DENSITY ARISING FROM PARABOLIC IFS AND RANDOM CONTINUED FRACTIONS
We continue to study parabolic iterated function systems (IFS) with overlaps on the real line and invariant measures associated with them. A shift-invariant Borel probability measure on the coding space projects into a measure on the limit set of the IFS. We consider families of IFS depending on parameter and satisfying a certain \transversality" condition. In SSU2] suucient conditions were fou...
متن کاملHausdorff Dimension of Limit Sets for Parabolic Ifs with Overlaps
We study parabolic iterated function systems with overlaps on the real line. We show that if a d-parameter family of such systems satisfies a transversality condition, then for almost every parameter value the Hausdorff dimension of the limit set is the minimum of 1 and the least zero of the pressure function. Moreover, the local dimension of the exceptional set of parameters is estimated. If t...
متن کاملCarleson Measure Problems for Parabolic Bergman Spaces and Homogeneous Sobolev Spaces
Let bα(R 1+n + ) be the space of solutions to the parabolic equation ∂tu+ (−△)u = 0 (α ∈ (0, 1]) having finite L(R 1+n + ) norm. We characterize nonnegative Radon measures μ on R + having the property ‖u‖Lq(R1+n + ,μ) . ‖u‖ Ẇ1,p(R + ) , 1 ≤ p ≤ q < ∞, whenever u(t, x) ∈ bα(R 1+n + ) ∩ Ẇ 1.p(R + ). Meanwhile, denoting by v(t, x) the solution of the above equation with Cauchy data v0(x), we chara...
متن کاملFiner Diophantine and Regularity Properties of 1-dimensional Parabolic Ifs
Recall that a Borel probability measure μ on IR is called extremal if μ-almost every number in IR is not very well approximable. In this paper, we prove extremality (and implying it the exponentially fast decay property (efd)) of conformal measures induced by 1-dimensional finite parabolic iterated function systems. We also investigate the doubling property of these measures and we estimate fro...
متن کامل