Local dimensions of measures of finite type

Abstract

Abstract. We study the multifractal analysis of a class of equicontractive, self-similar measures of finite type, whose support is an interval. Finite type is a property weaker than the open set condition, but stronger than the weak open set condition. Examples include Bernoulli convolutions with contraction factor the inverse of a Pisot number and self-similar measures associated with m-fold sums of Cantor sets with ratio of dissection 1/R for integer R ≤ m.We introduce a combinatorial notion called a loop class and prove that the set of attainable local dimensions of the measure at points in a positive loop class is a closed interval. We prove that the local dimensions at the periodic points in the loop class are dense and give a simple formula for those local dimensions. These self-similar measures have a distinguished positive loop class called the essential class. The set of points in the essential class has full Lebesgue measure in the support of the measure and is often all but the two endpoints of the support. Thus many, but not all, measures of finite type have at most one isolated point in their set of local dimensions.We give examples of Bernoulli convolutions whose sets of attainable local dimensions consist of an interval together with an isolated point. As well, we give an example of a measure of finite type that has exactly two distinct local dimensions.

Local dimensions of measures of finite type on the torus

“…We will call a quotient loop class positive if there is some path η in the loop class for which T (η) has strictly positive entries. With the preliminary results we have established, the same arguments as in [9,Section 5] prove the following important result. The details are left to the reader.…”

“…with a similar formula for upper and lower local dimensions. In the special case that the probabilities defining the self-similar measure are regular and π(K) = T, the same arguments as given in [9, show that if [x] = (γ 0 , γ 1 , ...), then we have the simpler formula,…”

“…Using the computer, we determined that any corresponding quotient measure has 10 reduced quotient characteristic vectors. From the reduced transition diagram, Figure 1, one can see that there are two different essential classes, the first from the reduced characteristic vector labelled 7 and the second consisting of the three reduced characteristic vectors 5,9,10. Remark 2.11.…”

“…In [3,4,5], Feng conducted a detailed study of equicontractive, self-similar measures of finite type on R, focussing primarily on Bernoulli convolutions with contraction factors the inverse of a simple Pisot number. In [9,11], the authors, together with Ng, showed that if µ is a measure of finite type on R that arises from regular probabilities and has full interval support, then the set of local dimensions of µ at the points in any positive loop class is a closed interval and the local dimensions at periodic points in the loop class are dense within this interval. See Section 2.1 for a definition of regular probabilities.…”

“…Measures which satisfy the open set condition are of finite type on R and measures of finite type on R satisfy the weak open set condition [14]. The structure of finite type measures and aspects of their multi-fractal analysis is explained in detail in [3,4,5,9,11].…”