Multifractal spectrum of self-similar measures with overlap

Abstract

It is well known that the multifractal spectrum of a self-similar measure satisfying the open set condition is a closed interval. Recently, there has been interest in the overlapping case and it is known that in this case there can be isolated points. We prove that for an interesting class of self-similar measures with overlap the spectrum consists of a closed interval union together with at most two isolated points. In the case of convolutions of uniform Cantor measures we determine the end points of the interval and the isolated points. We also give an example of a related self-similar measure where the spectrum is a union of two disjoint intervals. In contrast, we prove that if one considers quotient measures of this class on the quotient group [0, 1], rather than the real line, the multifractal spectrum is a closed interval.

Local Dimensions of Overlapping Self-Similar Measures

“…This completes the proof. We remark that ( √ m 2 + 4−m)/2 < 1/m, so this improves upon the fact that the local dimension of the m-fold convolution of the uniform Cantor measure on the Cantor set with ratio 1/m has an isolated point at 0, as shown in [2,20]. Note that when ̺ = 1/(m + 1), the IFS satisfies the open set condition and hence there is no isolated point.…”

Local dimensions of measures of finite type on the torus

“…Indeed, in [12], Hu and Lau discovered that when µ is the three-fold convolution of the classic middle-third Cantor measure on R, then there is an isolated point in the set of local dimensions, namely at x = 0, 3. This fact was later established for other 'overlapping' Cantor-like measures in [1,6,7,15].…”

“…In [1], it was shown that the sets of local dimensions for quotients of k-fold convolutions of Cantor measures with contraction factor 1/d are intervals whenever k ≥ d. Although these quotient measures do not have an essential class of positive type, we are able to modify our general approach to give a new proof of this fact. Moreover, we extend this result to what we call complete quotient Cantor-like measures and also prove that the set of local dimensions is the closure of the set of local dimensions of periodic points.…”

“…Given a measure µ on R, we let µ π be the quotient measure defined by µ π (E) = µ(π −1 (E)) for any Borel set E ⊆ T. Of course, if µ has support K, then µ π has support π(K). If µ has support contained in [0,1], the only difference between the local dimensions of µ and µ π is that (identifying T with [0, 1)) dim loc µ π (0) = min(dim loc µ(0), dim loc µ(1)), so this situation is trivial.…”

The entropy of Cantor-like measures

“…We call these self-similar Cantor-like measures, (m, d)-measures, and refer to them as uniform if all p i are equal. The dimensional properties of these measures are also of much interest; see, for example, [5,13,17,20]. We use combinatorial techniques to find an (explicit) analytic function T with the property that the Garsia entropy of the uniform (m, d)-measure is given by T (1)/ log 2 d when 2 ≤ d < m ≤ 2d − 1.…”