Local dimensions of measures of finite type II -
Measures without full support and with non-regular probabilities.
July 12, 2019
Consider a sequence of linear contractions and probabilities with . We are interested in the self-similar measure , of finite type. In this paper we study the multi-fractal analysis of such measures, extending the theory to measures arising from non-regular probabilities and whose support is not necessarily an interval.
Under some mild technical assumptions, we prove that there exists a subset of supp of full and Hausdorff measure, called the truly essential class, for which the set of (upper or lower) local dimensions is a closed interval. Within the truly essential class we show that there exists a point with local dimension exactly equal to the dimension of the support. We give an example where the set of local dimensions is a two element set, with all the elements of the truly essential class giving the same local dimension. We give general criteria for these measures to be absolutely continuous with respect to the associated Hausdorff measure of their support and we show that the dimension of the support can be computed using only information about the essential class.
To conclude, we present a detailed study of three examples. First, we show that the set of local dimensions of the biased Bernoulli convolution with contraction ratio the inverse of a simple Pisot number always admits an isolated point. We give a precise description of the essential class of a generalized Cantor set of finite type, and show that the convolution of the associated Cantor measure has local dimension at tending to 1 as tends to infinity. Lastly, we show that within a maximal loop class that is not truly essential, the set of upper local dimensions need not be an interval. This is in contrast to the case for finite type measures with regular probabilities and full interval support.
In this paper we continue the investigations, begun in , on the multifractal analysis of equicontractive self-similar measures of finite type. For self-similar measures arising from an IFS that satisfies the open set condition the multifractal analysis is well understood. In particular, the set of attainable local dimensions is a closed interval whose endpoints can be computed with the Legendre transform.
For measures that do not satisfy the open set condition, the multifractal analysis is more complicated and the set of local dimensions need not be an interval. This phenomena was discovered first for the 3-fold convolution of the classical Cantor measure in  and was further explored in [2, 9, 17, 22], for example. In , Ngai and Wang introduced the notion of finite type, a property stronger than the weak separation condition (WSC), but satisfied by many interesting self-similar measures that fail the open set condition. Examples include Bernoulli convolutions with contraction factor the inverse of a Pisot number and self-similar Cantor-like measures with ratio the inverse of an integer.
Building on earlier work, such as [10, 13, 16, 21], Feng undertook a study of equicontractive, self-similar measures of finite type in [4, 5, 6], with his main focus being Bernoulli convolutions. Motivated by this research, in  (and ) a general theory was developed for the local dimensions of self-similar measures of finite type assuming the associated self-similar set was an interval and the underlying probabilities generating the measure were regular, meaning . There it was shown that the set of local dimensions at points in the ‘essential class’ (a set of full Lebesgue measure in the support of and often the interior of its support) was a closed interval and that the set of local dimensions at periodic points was dense in this interval. Formulas were given for the local dimensions. These formulae are particularly simple at periodic points.
In contrast to much of the earlier work, in this paper we do not require any assumptions on the probabilities and we relax the requirement that the support of (the self-similar set) is an interval. In Section 2, we give formulas for calculating local dimensions. These are relatively simple for periodic points, although necessarily more complicated than under the previous assumptions. We begin Section 3 by introducing the ‘truly essential class’. We see that this set is the relative interior of the essential class and we prove that it has full and Hausdorff -measure, where is the Hausdorff dimension of the self-similar set. Under a mild technical assumption, that is satisfied in many interesting examples, we prove that the set of local dimensions at the points in the truly essential class is a closed interval and that the set of local dimensions at the periodic points is dense in that interval.
We prove that there is always a point at which the local dimension of coincides with the Hausdorff dimension of and give an example of a measure where this occurs at all the truly essential points (but not at all points of the support). A sufficient condition is given for a finite type measure to be absolutely continuous with respect to the associated Hausdorff measure and an example is seen that satisfies this condition when , even though the self-similar set is not an interval. We also give a formula for calculating the Hausdorff dimension of the support from just the knowledge of the essential class.
Related results were given by Feng in . Here, Feng constructed a (typically, countably infinite) family of closed intervals, with disjoint interiors, where is of full measure and on each of these closed intervals the set of attainable local dimensions of the restricted measure was a closed interval. From his construction one can see that is the essential class and is contained in our truly essential class. The endpoints of the intervals may or may not be truly essential. We note that the local dimension of the restricted measure at an end point of is not necessarily the same as the local dimension of at this point, even when it is a truly essential point.
Feng and Lau in  studied more general IFS that only satisfy the WSC and showed that in this case there is also an open set such that the set of attainable local dimensions of the restricted measure, is a closed interval. In the examples given in that paper, the set is much smaller than the truly essential class.
In , Feng had shown that the set of local dimensions of the uniform Bernoulli convolutions with contraction factor the inverse of a simple Pisot number (meaning the minimal polynomial is ) is always a closed interval. As one application of our work, in Section 4 we prove, in contrast, that biased Bernoulli convolutions with these contraction factors always admit an isolated point in their set of local dimensions.
In Section 5 we present a detailed study of the local dimensions of finite type Cantor-like measures, extending the work done in [2, 11, 22]. In those papers, it was shown, for example, that if for , then the local dimension at is isolated. Here we give further conditions that ensure there is an isolated point. But we also give examples where the measure has no isolated points and we give a family of examples which have exactly two distinct local dimensions. We also show that the local dimensions of the rescaled -fold convolutions of a Cantor-like measure converge to at points in . Previously, in , it had been shown that these local dimensions were bounded.
2. Notation and Preliminary Results
We begin by introducing the definition of finite type, as well as basic notation and terminology that will be used throughout the paper.
2.1. Finite Type
Consider the iterated function system (IFS) consisting of the contractions , , defined by
where , and is an integer. The unique, non-empty, compact set satisfying
is known as the associated self-similar set. By rescaling the if needed, we can assume the convex hull of is . We will not assume that or even that it has non-empty interior.
It was shown in [19, Thm. 1.2] that if and denotes the Hausdorff -measure restricted to , then Upon normalizing we can assume . Further, we note that . We remark that in the special case that , then and is the normalized Lebesgue measure.
Suppose probabilities , satisfy . Throughout this paper, our interest will be in the self-similar measure associated to the family of contractions given above, which satisfies the identity
These non-atomic, probability measures have support .
We put . Given an -tuple , we write for the composition and let
The iterated function system (IFS),
is said to be of finite type if there is a finite set such that for each positive integer and any two sets of indices , , either
where is the diameter of .
If the IFS is of finite type and is an associated self-similar measure satisfying (2), we also say that is of finite type.
Here we have given the general definition of finite type for an equicontractive IFS in . This simplifies to in the case where the convex hull of is . It is worth noting here that the definition of finite type is independent of the choice of probabilities.
Finite type is a property that is stronger than the weak separation condition, but weaker than the open set condition . Examples include (uniform or biased) Bernoulli convolutions with contraction factor the reciprocal of a Pisot number, and Cantor-like measures associated with Cantor sets with contraction factors reciprocals of integers. See Sections 4 and 5 where these are studied in detail.
2.2. Characteristic vectors and the Essential class
For each integer , let be the collection of elements of the set , listed in increasing order. Put
Elements of are called net intervals of level . By definition, a net interval contains net subintervals of every lower level. For each , , there is a unique element which contains called the parent (of child . We will define the left-most child of parent to be the child . We will similarly define the right-most child. It is worth noting that it is possible for a child to be both the left and the right-most child. It is further worth observing that because we are not assuming the self-similar set is the interval , it is possible for a parent to have no left-most child or no right-most child.
Given , we denote the normalized length of by
By the neighbour set of we mean the ordered -tuple
Given (listed in order from left to right) all the net intervals of level which have the same parent and normalized length as , let be the integer with . The characteristic vector of is the triple
Often we suppress giving the reduced characteristic vector .
If the measure is of finite type, there will be only finitely many distinct characteristic vectors. We denote the set of such vectors by
By an admissible path, of length we will mean an ordered -tuple, where for all and the characteristic vector, is the parent of . Each can be uniquely identified by an admissible path of length , say , where , , and for all . This is called the symbolic representation of we will frequently identify with its symbolic representation.
Similarly, the symbolic representation for will mean the sequence
of characteristic vectors where for all and is the parent of . The notation will mean the admissible path consisting of the first characteristic vectors of . We will often write for the net interval in containing ; its symbolic representation is .
If is an endpoint of for some (and then for all larger integers) we call a boundary point. We remark that if is a boundary point, then there can be two different symbolic representations for , one approaching from the left i.e., by taking right-most descendents at all levels beyond level and the other approaching from the right, by taking left-most descendents. If is not a boundary point, then the symbolic representation is unique.
It is worth emphasizing that is defined as the truncation of as opposed to defining it as a sequence with . To see this distinction, recall that it is possible for to have no right-most children. Let be the right-most endpoint of . Then is also the left-most endpoint of the adjacent net interval, . As has no right-most child, we do not have a net interval of depth with . As has no isolated points and , for all we must have net intervals . In such a case, the boundary point has a unique symbolic representation.
A non-empty subset is called a loop class if whenever , then there are characteristic vectors , , such that , and is an admissible path with all . A loop class is called an essential class if, in addition, whenever and is a child of , then . Of course, an essential class is a maximal loop class.
In [6, Lemma 6.4], Feng proved the important fact that there is always precisely one essential class, which we will denote by . If with for all large , we will say that is an essential point (or is in the essential class) and similarly speak of a net interval being essential. A path is in the essential class if all . We similarly speak of a point, net interval or path as being in a given loop class. The finite type property ensures that every element in the support of is contained in a maximal loop class.
We remark that the essential class is dense in the support of . This is because the uniqueness of the essential class ensures that every net interval contains a net subinterval in the essential class. In Proposition 3.6 we will show that the essential class has full measure and full Hausdorff -measure in , where is the Hausdorff dimension of .
2.3. Transition matrices
A very important concept in the multifractal analysis of measures of finite type are the so-called transition matrices. These are defined as follows: Let be a net interval of level with parent . Assume and . The primitive transition matrix, is a matrix whose entry is given by
if and there exists with and , and otherwise. We note that in  the transition matrices are normalized so that the minimal non-zero entry is . That is, we used instead of , where .
We observe that each column of a primitive transition matrix has at least one non-zero entry. The same is true for each row if supp, but not necessarily otherwise, see Example 3.10.
Given an admissible path , we write
and refer to such a product as a transition matrix. We will say the transition matrix is essential if all are essential characteristic vectors.
By the norm of a matrix we mean
A matrix is called positive if all its entries are strictly positive. An admissible path is called positive if is a positive matrix. Here is an elementary lemma which shows the usefulness of positivity.
Assume are transition matrices and is positive.
There are constants , depending on the matrices and respectively, so that and .
If each row of has a non-zero entry, then there is a constant , depending on matrix , such that .
There is a constant such that if is a square matrix, then
Suppose is a square matrix. There is a constant such that
2.4. Basic facts about of local dimensions of measures of finite type
Given a probability measure , by the upper local dimension of at supp we mean the number
Replacing the by gives the lower local dimension, denoted . If the limit exists, we call the number the local dimension of at and denote this by .
It is easy to see that
and similarly for the upper and lower local dimensions.
Notation: Throughout the paper, when we write we mean there are positive constants such that
To calculate local dimensions, it will be helpful to know for net intervals .
Let with . Then
Furthermore, if and
This follows in a similar fashion to Lemma 3.2, Corollary 3.4 and the discussion prior to Corollary 3.10 of , noting that . ∎
This corollary need not be true, however, if the assumptions of supp and regular probabilities, i.e., are not all satisfied. Instead, we proceed as follows.
Terminology: Assume with and suppose is a net interval of level . Let be the empty set if is empty and otherwise, let Similarly, define to be the net interval immediately to the right of (or the empty set), with the understanding that if is the left or right-most net interval in , then (respectively, is the empty set. We refer to as adjacent net intervals (even if some are the empty set).
If belongs to the interior of , we put
If is a boundary point of , we put
where if and if . We will refer to as the other net interval containing , even if it is empty and so formally not a net interval.
Let be a self-similar measure of finite type and let . Then
provided the limit exists. The lower and upper local dimensions of at can be expressed similarly in terms of and .
Assume, first, that
By the finite type assumption, there are constants such that for all . Pick and such that and .
If is a boundary point, then for sufficiently large , is an endpoint of and
where the notation is as in (7). If is not a boundary point, then
In either case,
This in turn implies that
The limit of the left hand side and the right hand side both go to , hence the limit of the middle expression exists and is equal to .
It follows similarly that if exists, then also
The arguments for the lower and upper local dimensions are similar. ∎
2.5. Periodic points
Recall that in , is called a periodic point if has symbolic representation
where is an admissible cycle (a non-trivial path with the same first and last letter) and is the path with the last letter of deleted. We refer to as a period of . Boundary points are necessarily periodic and there are only countably many periodic points. Note that a periodic point is essential if and only if it has a period that is a path in the essential class.
If there is a choice of for which is a positive matrix, we call a positive, periodic point.
Of course, a period for a periodic point is not unique. For example, if is a period, then so is and so is . However, these different choices for the period give the same symbolic representation for . But if is a boundary point, then may have two different symbolic representations, one for which and the other having , and these two representations arise from (fundamentally) different periods.
The notation means the spectral radius of the matrix ,
We note that two periods associated with the same symbolic representation for will have the same spectral radius. This need not be the case for periods associated with different symbolic representations.
Here is the analogue of [11, Proposition 4.14] when there is no assumption of regularity.
If is a periodic point with period , then the local dimension exists and is given by
where if is a boundary point of a net interval with two different symbolic representations given by periods and , then is chosen to satisfy .
First, suppose is a boundary periodic point with two different symbolic representations given by periods and that . There is no loss of generality in assuming the two periods have the same lengths and pre-period path of length . Given large , let , so has symbolic representations
for suitable . From Proposition 2.4,
Lemma 2.2 implies that there are positive constants , independent of , such that
If , then for large enough , and hence
Since Theorem 2.6 gives
If, instead, , then for each ,
the result again follows.
If is a boundary periodic point with only one symbolic representation, then is empty for large and the arguments are similar, but easier.
Now, assume is not a boundary point. Then there is no loss of generality in assuming where the net interval is in the interior of the net interval and is the first letter of . This ensures that is a common ancestor of the two adjacent intervals to
that is at at most levels back. More generally, is a common ancestor of the two adjacent intervals to the net intervals
again at most levels back. Thus, if
then is comparable to with constants of comparability independent of . As above, there are so that
and the argument is completed in a similar fashion to before. ∎
Local Dimensions at Truly Essential Points
In this section we will obtain our main theoretical results on the structure of local dimensions, analogues of those found in Section 5 of . Because local dimensions may depend on adjacent net intervals, and rather than only on , we introduce a subset of the essential class that we call the truly essential class. We will see that this subset has full and Hausdorff -measure for . Our main results state that under a weak technical assumption the local dimensions at periodic points are dense in the set of (upper and lower) local dimensions at truly essential points and that the set of local dimensions at truly essential points is a closed interval. Furthermore, we prove that there is always a truly essential point at which the local dimension agrees with the Hausdorff dimension of the self-similar set and we give criteria for when the measure is absolutely continuous with respect to the Hausdorff measure.
3.1. Truly essential points
Suppose is the self-similar set associated with an IFS of finite type.
We will say that is a boundary essential point if is a boundary point of for some and both and the other ’th level net interval containing , are essential (where if is empty we understand it to be essential).
We will say that is an interior essential point if is not a boundary point and there exists an essential net interval with in its interior.
We call a truly essential point if it is either an interior essential point or a boundary essential point.
Obviously, truly essential points are essential and if is in the interior of some essential net interval, then it is truly essential. In particular, any essential point that is not truly essential must be a boundary point. Hence there can be only countably many of these and they are periodic.
Any point in the relative interior of the essential class (with respect to the space ), is either contained in the interior of some essential interval, or is a boundary essential point. Hence the relative interior of the essential class is equal to the set of truly essential points. If the essential class is a (relatively) open set, then the essential class coincides with the truly essential class. This is the situation, for example, with the Bernoulli convolutions and Cantor-like measures discussed in Sections 4 and 5. Another IFS where the set of essential points is equal to the set of truly essential points is given in Example 3.10.
However, as the example below demonstrates, these two sets need not be equal.
Consider the maps with for , , and . The reduced transition diagram has 4 reduced characteristic vectors. The reduced characteristic vectors are:
Reduced characteristic vector 1:
Reduced characteristic vector 2:
Reduced characteristic vector 3:
Reduced characteristic vector 4:
The transition maps are:
By this we mean, for example, that the reduced characteristic vector 1 has 8 children. Listed in order from left to right, they are the reduced characteristic vectors etc. By we mean the first occurance of the child of type , the second, etc. If there is only one child of that type we do not need to distinquish them. It is possible that the transitions matrices are different for different children of the same type. These also help to distinquish paths unamibiguously. See Example 3.10. It is easy to see from the transition maps that the essential class is . See Figure 1 for the transition diagram.
Consider the boundary periodic point with symbolic representation , being the left most child of . This also has symbolic representation being the right-most child of . One of these symbolic representations is in the essential class, whereas the other is not. As such, this point is an essential point, but it is not a truly essential point.
The significance of an interior essential point is that and have a common essential ancestor for some . Conversely, if and have a common essential ancestor for some , then belongs to the relative interior of the essential class and thus is truly essential.
A periodic point that is an interior essential point admits a period with the property that if and is the first letter of then the net interval (with symbolic representation) is in the interior of the net interval . We will call such a period truly essential. Equivalently, is truly essential if and only if is a path that does not consist solely of right-most descendents or solely of left-most descendents.
It was shown [11, Proposition 4.5] that under the assumption that the self-similar set was an interval, the essential class had full Lebesgue measure. In fact, this is true for the truly essential class, with Lebesgue measure replaced by either the self-similar measure or the (normalized) Hausdorff -measure, where , as the next Proposition shows. To prove this, we first need some preliminary lemmas.
There exists an integer such that for each net interval there exists a with .
Consider a net interval . As there is some in the interior of , there is an index and such that . Since is not isolated in , there must be a level net interval with left end . Choose the index minimal with this property.
Assume has symbolic representation and has representation . Let be any other net interval with symbolic representation ending with the same characteristic vector . It will also have a descendent, , with representation ending with the path . Since is in the neighbour set of , the same is true for and thus its left endpoint is an image of under for some . As the pairs and have the same finite type structure (up to normalization), it follows that . Hence has the same minimal index , in other words, depends only upon the final characteristic vector associated with .
As there are only finitely many characteristic vectors, we can take to be the maximum of these indices taken over all the characteristic vectors. ∎
There exists a positive constant such that for all and all we have
Fix an ’th level net interval . By construction, there exists some such that . Then and hence .
Choose as in the previous lemma. Then there exists some