ABSOLUTELY CONTINUOUS MEASURES WITH Lq DENSITY ARISING FROM PARABOLIC IFS AND RANDOM CONTINUED FRACTIONS
نویسنده
چکیده
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 found for the measure to be absolutely continuous for a.e. parameter value. Here we investigate when this measure has a density in L q , for q 2 (1; 2]. We prove that if the q-dimension of the measure , with respect to a certain metric, is greater than one, then has a density in L q for a.e. parameter value. On the other hand, if the q-dimension of is less than one, then the density, if it exists, cannot belong to L q. These results are applied to a family of random continued fractions studied by R. Lyons. He proved that the distribution cannot have a density in L 2 above a certain threshold; we show that this threshold is sharp and establish the existence of L 2 density for a.e. parameter in some interval below the threshold.
منابع مشابه
Lq DENSITIES FOR MEASURES ASSOCIATED WITH PARABOLIC IFS WITH OVERLAPS
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. p...
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