Local dimensions

# Local dimensions of measures of finite type

Kathryn E. Hare, Kevin G. Hare, and Kevin R. Matthews
###### 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 -fold sums of Cantor sets with ratio of dissection for integer .

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.

The research of all three authors is supported in part by NSERC.

## 1. Introduction

It is well known that if is a self-similar measure arising from an IFS satisfying the open set condition, then the set of local dimensions of the measure is a closed interval whose endpoints are easily computed. Further, the Hausdorff dimension of the set of points whose local dimension is a given value can be determined using the Legendre transform of the -spectrum of the measure. This is known as the multifractal formalism and we refer the reader to [5] for more details.

For measures that do not satisfy the open set condition, the multifractal analysis is more complicated and, in general, much more poorly understood. In [16], Hu and Lau discovered that the -fold convolution of the classical middle-third Cantor measure fails the multifractal formalism as there is an isolated point in the set of local dimensions. Subsequently, in [2, 26, 28] further examples of this phenomena were explored and it was shown, for example, that there is always an isolated point in the set of local dimensions of the -fold convolution of the Cantor measure associated with a Cantor set with ratio of dissection when the integer . More recently, it was proven in [1] that continuous measures satisfying a weak technical condition have the property that a suitably large convolution power admits an isolated point in its set of local dimensions.

In [21], Ngai and Wang introduced the notion of finite type (see Section 2 for the definition). This property is stronger than the weak separation condition introduced in [18], but is satisfied by many self-similar measures which fail to possess the open set condition. Examples include Bernoulli convolutions, , with contraction factor equal to the reciprocal of a Pisot number [21] and the Cantor-like measures mentioned above.

Building on earlier work (c.f., [12, 15, 19, 25]), Feng undertook a study of equicontractive, self-similar measures of finite type in [7, 8, 9]. His main results were for Bernoulli convolutions. In particular, he proved that despite the failure of the open set condition, the multifractal formalism still holds for the Bernoulli convolutions whose contraction factor was the reciprocal of a simple Pisot number (meaning, a Pisot number whose minimal polynomial is of the form ). A particularly interesting example is when the contraction factor is the golden ratio with minimal polynomial (also called the golden mean).

In this paper we study the local dimension theory of equicontractive, self-similar measures of finite type, whose support is a compact interval and for which the underlying probabilities are regular. We first give a simple formula for the value of the local dimension of at any “periodic” point of its support. The finite type condition leads naturally to a combinatorial notion we call a “loop class”. For a “positive” loop class we prove that the set of attainable local dimensions of the measure is a closed interval and that the set of local dimensions at periodic points in the loop class is a dense subset of this interval.

These self-similar measures have a distinguished positive loop class that is called the “essential class”. Thus the set of attainable local dimensions is the union of a closed interval together with the local dimensions at points in loop classes external to the essential class. We will say that a point is an essential point if it is in the essential class. The set of essential points has full Lebesgue measure on the support of the measure and in many interesting examples the set of essential points is the interior of the support of the measure. This is the case with many Bernoulli convolutions, , including when is the golden ratio (c.f. Section 8.1.1), and with the -fold convolution of the Cantor measure on a Cantor set with ratio of dissection when (see Section 7).

When the essential set is the interior of the support of the measure then has no isolated point in its set of attainable local dimensions if and only if coincides with the local dimension of at an essential point. In that case, the set of attainable local dimensions of is a closed interval. The Bernoulli convolution , with a simple Pisot number has this property.

However, we construct other examples of Bernoulli convolutions (with contraction factor a Pisot inverse) which do have an isolated point in their set of attainable local dimensions (see Subsection 8.1.2). As far as we are aware, these are the first examples of Bernoulli convolutions known to admit an isolated point. We also construct a Cantor-like measure of finite type, whose set of local dimensions consists of (precisely) two distinct points (see Example 6.1). In all of these examples, the essential set is the interior of the support of the measure.

The convolution square of the Bernoulli convolution, , with the golden ratio, is another example of a self-similar measure to which our theory applies. It, too, has exactly one isolated point in its set of attainable local dimensions, although in this case the set of non-essential points is countably infinite (see Subsection 8.2).

The computer was used to help obtain some of these results. In principle, the techniques could be applied to other convolutions of Bernoulli convolutions and other measures of finite type, however even with the simple examples given here, the problem can become computationally difficult.

The paper is organized as follows: In Section 2, we detail the structure of self-similar measures of finite type, introduce terminology and describe a number of examples that we will return to throughout the paper. The notion of transition matrices and properties of local dimensions of measures of finite type are discussed in Section 3. In Section 4 we introduce the notion of loop class, essential class and periodic points. A formula is given for the local dimension at a periodic point and we prove that the essential class is always of positive type. In Section 5 we prove that the set of local dimensions at periodic points in a positive loop class is dense in the set of local dimensions at all points in the loop class. We also show that the set of local dimensions at the points of a positive loop class is a closed interval. In particular, this implies that the set of local dimensions at the essential points is a closed interval.

In Section 6 we give a detailed description of our computer algorithm by means of a worked example. We also explain our main techniques for finding bounds on sets of local dimensions and illustrate these by constructing a Cantor-like measure of finite type whose local dimension is the union of two distinct points. In Section 7 we show that with our approach we can partially recover results from [2, 16, 26] about the local dimensions of Cantor-like measures of finite type. We also show that some facts about the endpoints of the interval portion of the local dimension, that are known to be true for Cantor-like measures in the “small” overlap case, do not hold in general. Bernoulli convolutions, , where is the reciprocal of a Pisot number of degree at most four, are studied in Section 8 and we see that two of these measures admit an isolated point. We also study the convolution square of the Bernoulli convolution with the golden ratio in this final section.

For the examples in this paper, we present only minimal information. A more detailed analysis of all of these examples can found as supplemental information appended to the arXiv version of the paper [14].

## 2. Terminology and examples

### 2.1. Finite Type

Consider the iterated function system (IFS) consisting of the contractions , , defined by

 (1) Sj(x)=ϱx+dj

where , and is an integer. By the associated self-similar set, we mean the unique, non-empty, compact set satisfying

 K=m⋃j=0Sj(K).

Suppose , are probabilities, i.e., for all and . Our interest is in the self-similar measure associated to the family of contractions as above, which satisfies the identity

 (2) μ=m∑j=0pjμ∘S−1j.

These measures are sometimes known as equicontractive, or -equicontractive if we want to emphasize the contraction factor . They are non-atomic, probability measures whose support is the self-similar set.

We put . Given an -tuple , we write for the composition and let

 pσ=pj1⋅⋅⋅pjn.
###### Definition 2.1.

The iterated function system, , 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

 ϱ−n|Sσ(0)−Sσ′(0)|>c or ϱ−n(Sσ(0)−Sσ′(0))∈F,

where is the diameter of .

If is of finite type and is an associated self-similar measure, we also say that is of finite type.

It is worth noting here that the definition of finite type is independent of the choice of probabilities.

In [22], Nguyen showed that any IFS satisfying the open set condition is of finite type, but there are also examples of iterated function systems of finite type that fail the open set condition. Indeed, recall that an algebraic integer greater than is called a Pisot number if all its Galois conjugates are less than in absolute value. Examples include integers greater than and the golden ratio, . In [21, Thm. 2.9], Ngai and Wang showed that if is a Pisot number and all , then the measure satisfying (2) is of finite type. This result allows us to produce many examples of measures of finite type that do not satisfy the open set condition.

The case when the IFS is generated by two contractions is of particular interest.

###### Notation 2.2.

We will use the notation to denote the self-similar measure

 μϱ=12μϱ∘S−10+12μϱ∘S−11,

where for .

###### Example 2.3.

When the measures, are known as Cantor measures (or uniform Cantor measures). Their support is the Cantor set with ratio of dissection and they satisfy the open set condition. When these measures are called Bernoulli convolutions. They fail to satisfy the open set condition, but are of finite type whenever is a Pisot number.

Given two probability measures, , the convolution of and is defined as

 μ∗ν(E)=μ×ν{(x,y):x+y∈E}.

The name “Bernoulli convolution” comes from the fact that

 μϱ=∗∞n=1(δ0+δ(1−ϱ)ϱn2),

where the infinite convolution is understood to converge in a weak sense.

Bernoulli convolutions, , with contraction factor the inverse of a Pisot number, have been long studied. They have unusual properties and are of interest in fractal geometry, number theory and harmonic analysis. For example, although almost every Bernoulli convolution is absolutely continuous with respect to Lebesgue measure, and even has an density function, those with a Pisot inverse as the contraction factor are not only purely singular, but their Fourier transform, , does not even tend to zero as . We refer the reader to [24] and [27] for some of the interesting history of these measures.

###### Example 2.4.

Suppose and are -equicontractive measures, say and where and . Then is the -equicontractive, self-similar measure satisfying

 μ∗ν=∑trt(μ∗ν)∘U−1t

where and . It follows directly from Ngai and Wang’s result [21] that any -fold convolution power of the Bernoulli convolution or Cantor measure, is of finite type when is Pisot.

###### Example 2.5.

Another consequence of [21] is that the IFS

 {Sj(x)=1Rx+jRm(R−1):j=0,…,m},

where is an integer, is of finite type. The convex hull of the self-similar set is and the self-similar set is the full interval when . When , the open set condition is not satisfied. The -fold convolutions of Cantor measures with contraction factor are examples of self-similar measures associated with such an IFS.

These Cantor-like measures were studied in [2, 26] using different methods. In Section 7 we will see how our approach relates to some of their results.

### 2.2. Standard Technical Assumptions

We will refer to the following conditions on a self-similar measure as our standard technical assumptions.

1. The measure is a -equicontractive, self-similar measure, as in (2), that is of finite type.

2. The probabilities, satisfy (we call these regular probabilities).

3. The support of (equivalently, the underlying self-similar set) is a closed interval. By rescaling the appropriately, we can assume without loss of generality that this interval is .

We remark that supp if and only if (the rescaled) satisfy , and for all . In this case, in the definition of finite type.

Although some of what we say is true more generally for self-similar measures of finite type, we make use of the standard technical assumptions at key points throughout the paper.

The Bernoulli convolutions and the -fold convolutions of uniform Cantor measures with are examples of measures satisfying the standard technical assumptions. (See Examples 2.3 and 2.4). The measures of Example 2.5, where , are also examples of such measures when regular probabilities are chosen.

### 2.3. Net intervals and Characteristic vectors

As we have seen, measures that are of finite type need not satisfy the open set condition. Our primary interest is in this case. The finite type property is, however, stronger than the weak separation condition (see [22] for a proof), and the multifractal analysis of self-similar measures of finite type is somewhat more tractable because of their better structure. This structure is explained in detail in [7, 8, 9], but we will give a quick overview here.

For each integer , let be the collection of elements of the set , listed in increasing order. Put

 Fn={[hj,hj+1]:1≤j≤sn}.

Elements of are called net intervals of level . For each , , there is a unique element which contains . We call the parent of and a child of . We denote the normalized length of by

 ℓn(Δ)=ϱ−n(b−a).

Note that by definition there is no with , nor can we have . Furthermore, there must be some with for suitable choices of .

Next, we consider all with . As is a closed interval of length this is the same as the set of all with . We suppose

 {ϱ−n(a−Sσ(0)):σ∈An, \thinspaceΔ⊆Sσ[0,1]}={a1,…,ak}

and assume . We define the neighbour set of as

 Vn(Δ)=(a1,…,ak).

Let be the parent of , and (listed in order from left to right) be all the net intervals of level which are also children of and have the same normalized length and neighbour set as . Define to be the integer with . The characteristic vector of is the triple

 Cn(Δ)=(ℓn(Δ),Vn(Δ),rn(Δ)).

We also speak of the pair of characteristic vectors, as parent and child if and for a parent/child pair . The characteristic vector is important because it carries the neighbourhood information about .

Put

 Ω={Cn(Δ):n∈N, Δ∈Fn}.

If the measure is of finite type, then will contain only finitely many distinct characteristic vectors.

Suppose the net interval, has two children, and that differ only in the value of , that is, they have the same length and the same neighbourhood set. The characteristic vectors for the children of and , will be identical as they depend only on and and not on . For notational reasons, we find it convenient to take advantage of this when drawing the directed graph relating parents to children by suppressing equating these two children on the graph, and allowing multiple edges from to . We will call the characteristic vectors where we have suppressed the numbers the reduced characteristic vectors, and we will call the resulting graph the reduced transition graph.

###### Example 2.6.

In [8, Section 4] and [9, Section 6], Feng studied the Bernoulli convolution , with the golden ratio, and found that there were seven characteristic vectors. Their normalized lengths and neighbourhood sets are given by:

• Characteristic vector 1:

• Characteristic vector 2:

• Characteristic vectors 3a and 3b: and

• Characteristic vector 4:

• Characteristic vector 5:

• Characteristic vector 6:

Notice there are only six reduced characteristic vectors; we label the two characteristic vectors with identical length and neighbourhood set as 3a and 3b. In [8] these were labelled as 3 and 7. The directed graphs in Figure 1 show the parent/children relationships. The term “essential class”, referred to in the Figure, is defined in Section 4.

By an admissible path,  of length we will mean an ordered -tuple, where for all and the characteristic vector, is the parent of .

By the symbolic expression of we mean an admissible path of length , denoted

 [Δ]=(C0(Δ0),…,Cn(Δn)),

where and for each , . Here is . Feng [7] proved that the symbolic expression uniquely determines .

For , the symbolic representation for , denoted will mean the sequence of characteristic vectors where for all and is the parent of . We note that unless is an endpoint of a net interval (in which case there are two representations of ), is unique. The notation will mean the admissible path consisting of the first characteristic vectors of .

We frequently write for the net interval of level containing . Thus is the sequence where the first terms gives the symbolic representation of for each .

## 3. Transition matrices and local Dimensions

### 3.1. Local dimensions of measures of finite type

###### Definition 3.1.

Given a probability measure , by the upper local dimension of  at supp, we mean the number

 ¯¯¯¯¯¯¯¯¯dimμ(x)=limsupr→0+logμ([x−r,x+r])logr.

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 .

Multifractal analysis refers to the study of the local dimensions of measures.

For a -equicontractive measure , it is easy to check that

 (3) dimμ(x)=limn→∞logμ([x−ϱn,x+ϱn])nlogϱ for x∈suppμ,

and similarly for the upper and lower local dimensions.

Our first several lemmas will enable us to show that we can replace the interval by .

###### Lemma 3.2.

Suppose satisfies the standard technical assumptions. Let with . Then

 μ(Δ)=k∑i=1μ[ai,ai+ℓn(Δ)]∑σ∈Anϱ−n(a−Sσ(0))=aipσ.
###### Proof.

This argument can basically be found in Feng [7], but we give the details here for completeness. Iterating (2) times gives

 μ(Δ)=∑σ∈Anpσμ(S−1σ(Δ)).

Since is a non-atomic measure supported on , we have

 μ(Δ) =∑σ∈AnSσ(0,1)∩Δ≠∅pσμ(S−1σ(Δ))

Now, implies that , hence by definition of the neighbourhood set for some . Thus , so . This implies that

 μ(Δ) =k∑i=1∑σ∈Anϱ−n(a−Sσ(0))=aipσμ(S−1σ(Δ))

We observe that , and hence

 Sσ([ai,ai+ℓn(Δ)]) =[aiϱn+a−aiϱn,aiϱn+a−aiϱn+ℓn(Δ)ϱn] =[a,a+ℓn(Δ)ϱn]=[a,b]=Δ.

Hence

 μ(Δ) =k∑i=1∑σ∈Anϱ−n(a−Sσ(0))=aipσμ[ai,ai+ℓn(Δ)] =k∑i=1μ[ai,ai+ℓn(Δ)]∑σ∈Anϱ−n(a−Sσ(0))=aipσ

as claimed. ∎

###### Notation 3.3.

For card, put

 Pin(Δ)=p−n0∑σ∈An:ϱ−n(a−Sσ(0))=aipσ

and

 Pn(Δ)=K∑i=1Pin(Δ).

Here we have chosen to normalize by multiplying by . This is done so that the minimal non-zero entry in the transition matrices (defined in the next subsection) is at least .

###### Corollary 3.4.

There is a constant such that for any and any

 cpn0Pn(Δ)≤μ(Δ)≤pn0Pn(Δ).
###### Proof.

The upper bound is clear from the lemma. For the lower bound we note that each as the support of is the full interval . The finite type condition ensures there are only finitely many choices for . ∎

###### Lemma 3.5.

Suppose satisfies the standard technical assumptions. There are constants such that if , are two adjacent net intervals of level , then

 c11nPn(Δ2)≤Pn(Δ1)≤c2nPn(Δ2).
###### Proof.

The proof is similar to that of [8, Lemma 2.11] and proceeds by induction on . The base case holds as there are only finitely many choices for when . Now assume the result for level and we will verify it holds for level .

If have the same parent , the result follows easily from the observation that

 Pn−1(ˆΔ)≤Pn(Δj)≤mp−10maxpjPn−1(ˆΔ).

Otherwise, and are children of adjacent net intervals of level , respectively, and we can suppose is to the left of . As in [8], put

 D1 ={σ∈An−1:ˆΔ1⊆Sσ[0,1] and they share the same right endpoint\thinspace} D2 ={σ∈An−1:ˆΔ2 ⊆Sσ[0,1] and they share the same left endpoint\thinspace} Ej ={σ∈An−1╲Dj:ˆΔj⊆Sσ[0,1]}, j=1,2.

The definitions ensure that ,

 pn−10Pn−1(ˆΔj)=∑σ∈Djpσ+∑σ∈Ejpσ

and

 pn0Pn(Δ1) ≤∑σ∈D1pσpm+∑σ∈E1pσm∑j=0pj ≤pm∑σ∈D1pσ+pm∑σ∈E1pσ+∑σ∈E2pσ+∑σ∈D2pσ ≤pmpn−10Pn−1(ˆΔ1)+pn−10Pn−1(ˆΔ2)

Applying the induction assumption gives

 p0Pn(Δ1) ≤pmPn−1(ˆΔ1)+Pn−1(ˆΔ2) ≤pmc2(n−1)Pn−1(ˆΔ2)+Pn−1(ˆΔ2) ≤(pmc2(n−1)+1)Pn−1(ˆΔ2)

Taking gives

 p0Pn(Δ1) ≤((n−1)+1)Pn−1(ˆΔ2) ≤nPn−1(ˆΔ2) ≤c2nPn(ˆΔ2)

The other inequality is similar. ∎

Note that in the proof the assumption that were regular probabilities was important.

The following is immediate from the two previous results.

###### Corollary 3.6.

There are constants such that if are adjacent net intervals of level , then

 C11nμ(Δ2)≤μ(Δ1)≤C2nμ(Δ2).

Together these results yield the following useful approach to computing local dimensions.

###### Corollary 3.7.

Suppose satisfies the standard technical assumptions. Let supp and denote a net interval of level containing . Then

 ¯¯¯¯¯¯¯¯¯dimμ(x) =limsupn→∞logμ(Δn(x))nlogϱ (4) =logp0logϱ+limsupn→∞logPn(Δn(x))nlogϱ.

A similar statement holds for the (lower) local dimensions.

###### Proof.

Since any net interval of level has length at most , the interval contains . The finite type property ensures it is contained in a union of a bounded number of ’th level net intervals, say , where is adjacent to and for a suitable index , . Thus for constants , (independent of the choice of and ),

 cpn0Pn(Δn(x)) ≤μ(Δn(x))≤μ([x−ϱn,x+ϱn])≤N∑j=1μ(Δn(xj)) ≤N∑j=1pn0Pn(Δn(xj))≤Npn0CNnNPn(Δn(x)).

Thus the limiting behaviour of the three expressions

 logμ([x−ϱn,x+ϱn])nlogϱ, logμ(Δn(x))nlogϱ

and

 logp0logϱ+logPn(Δn(x))nlogϱ

coincide. ∎

###### Remark 3.8.

If , then and hence . Consequently,  . More generally, since is never empty and is the minimal probability, it follows that for all and . Consequently,

 dimμ(x)≤dimμ(0) for all x∈suppμ.

### 3.2. Transition matrices

The results of the previous subsection show that for studying the local dimensions of these measures it will be helpful to make accurate estimates of . Towards this, slightly modifying [7] we define primitive transition matrices, for a net interval of level and parent as follows:

###### Notation 3.9.

Suppose and . For and we set

 Tjk:=(T(Cn−1(ˆΔ),Cn(Δ)))jk=p−10pℓ

if and there exists with and . This is equivalent to saying

 Tjk=p−10pℓ if c−ϱn−1cj+ϱn−1dℓ=a−ϱnak.

We set otherwise.

As is finite for a measure of finite type, there is an upper bound on the size of these matrices. The entries are non-negative and all non-zero entries are at least one. Each column has at least one non-zero entry because if and only if there is some which “contributes” to it, in the sense defined above.

It is also important to note that the standard technical assumption that supp guarantees that given such that , there exists such that . This means that each row of the matrix also has a non-zero entry.

For card, put

 Qn(Δ)=(P1n(Δ),…,PKn(Δ)).

The same reasoning as in [7, Theorem 3.3] shows that

 Qn(Δ)=Qn−1(ˆΔ)(T(Cn−1(ˆΔ),Cn(Δ))).

Thus if (that is, and ), then since we have

 Pn(Δ)=∥Qn(Δ)∥=∥T(γ0,γ1)⋅⋅⋅T(γn−1,γn)∥,

where by the norm of matrix we mean

 ∥M∥:=∑jk∣∣Mjk∣∣.

Given an admissible path , we write

 T(η)=T(γ1,…,γn)=T(γ1,γ2)⋅⋅⋅T(γn−1,γn)

and refer to such a product as a transition matrix.

With this notation, the results of the previous subsection can be stated as

###### Corollary 3.10.

Suppose satisfies the standard technical assumptions. If supp, then

 (5) ¯¯¯¯¯¯¯¯¯dimμ(x)=logp0logϱ+limsupn→∞log∥T([x|n])∥nlogϱ

and similarly for the (lower) local dimension.

###### Example 3.11.

Again, consider the Bernoulli convolution, , with the golden ratio. Feng [8] showed that has symbolic representation and that . Applying Corollary 3.10 gives another proof that .

Next, we give a useful simple lemma.

###### Lemma 3.12.

Suppose satisfies the standard technical assumptions. Let and be transition matrices. Then and .

###### Proof.

We have

 ∥AB∥=∑ij(∑kAikBkj)=∑jk(∑iAik)Bkj.

Since all the entries of each of the matrices is nonnegative and each column of has an entry , it follows that

The argument for the other inequality is similar, noting that each row of has an entry as a consequence of the standard technical assumptions. ∎

An important consequence of this result is that the local dimension of at depends only on the tail of the symbolic representation of .

###### Corollary 3.13.

Suppose . For any ,

 ¯¯¯¯¯¯¯¯¯dimμ(x)=logp0logϱ+limsupn→∞log∥T(γN,γN+1,…,γN+n)∥nlogϱ

and similarly for the (lower) local dimension.

If , then the (upper or lower) local dimensions of at and agree.

###### Proof.

This holds since

 ∥T(γN,γN+1,…,γN+n)∥ ≤∥T(γ0,…,γN)T(γN,γN+1,…,γN+n)∥ =∥T(γ0,…,γN,γN+1,…,γN+n)∥ ≤∥T(γ0,…γN)∥∥T(γN,γN+1,…,γN+n)∥.

###### Notation 3.14.

By we mean the spectral radius of the square matrix the largest eigenvalue of in absolute value. Recall that

 sp(M)=limn∥Mn∥1/n.

We will call a matrix positive if all its entries are strictly positive and write . We record here some elementary facts about positive matrices that will be useful later.

###### Lemma 3.15.

Suppose satisfies the standard technical assumptions. Assume are transition matrices and is positive.

1. Then .

2. There is a constant such that if is a square matrix, then

 ∥AB∥≤C1sp(AB).
3. Suppose is a square matrix. There is a constant such that

 sp(Bn)≤∥Bn∥≤C2sp(Bn) for all n.
###### Proof.

To see (1), let . As all a simple calculation gives

 ∥ABC∥=∑ijklAijBjkCkl≥∑ijklAijCkl=∥A∥∥C∥.

For (2), assume is a matrix. Let . As the entries of are non-negative and the entries of are at least 1, it is easy to see that

 ∥AB∥ =∑j,k(AB)jk=∑j,k,lAjlBlk≤bq∑j,k=1p∑l=1Ajl≤bqq∑j=1p∑l=1AjlBlj =bqq∑j=1(AB)jj≤bq2sp(AB),

with the final inequality holding because the sum of the diagonal entries of is the sum of the eigenvalues of , counted by multiplicity.

For (3), let be the Jordan decomposition of and let . By the Perron-Frobenius theory, is a simple root of the characteristic polynomial of and all other eigenvalues of are strictly less than in modulus. Since all entries of are at least , it can be easily seen that . As , it is enough to prove where depends on , but not .

Assume is of size . Since the Jordan block for is , all entries of , other than the entry which is , are either or of the form where and is an eigenvalue of with . Thus

 ∥Jn∥≤