A strongly irreducible affine iterated function system with two invariant measures of maximal dimension
A classical theorem of Hutchinson asserts that if an iterated function system acts on by similitudes and satisfies the open set condition then it admits a unique self-similar measure with Hausdorff dimension equal to the dimension of the attractor. In the class of measures on the attractor which arise as the projections of shift-invariant measures on the coding space, this self-similar measure is the unique measure of maximal dimension. In the context of affine iterated function systems it is known that there may be multiple shift-invariant measures of maximal dimension if the linear parts of the affinities share a common invariant subspace, or more generally if they preserve a finite union of proper subspaces of . In this note we give an example where multiple invariant measures of maximal dimension exist even though the linear parts of the affinities do not preserve a finite union of proper subspaces.
We recall that an iterated function system is by definition a tuple of contracting transformations of some metric space , which in this article will be equipped with the Euclidean distance. To avoid trivialities it will be assumed throughout this article that . If is an iterated function system acting on then it is well-known that there exists a unique nonempty compact set with the property , called the attractor or limit set of the iterated function system. If we define with the infinite product topology, there exists moreover a well-defined coding map characterised by the property
for all and , and this coding map is a continuous surjection from to the attractor.
We recall that is said to satisfy the open set condition if there exists a nonempty open set such that the sets are pairwise disjoint subsets of ; if the same statements are true for a nonempty compact set in place of , we say that satisfies the strong separation condition. We observe that if the strong separation condition is satisfied, the coding map is a homeomorphism from to the attractor.
If satisfies the open set condition and the transformations are similarities, it is a classical result of J.E. Hutchinson  that there exists a probability measure on the attractor of with Hausdorff dimension equal to that of the attractor; moreover, this measure has the form where is a Bernoulli measure on the coding space . In particular is invariant with respect to the shift transformation . In the more general context in which the transformations are invertible affine transformations of it is thus natural to ask when there exists an invariant measure on the coding space which projects to a measure with dimension equal to that of the attractor, and if such a measure exists, how many such measures there might be; this problem was first investigated extensively in . It was shown recently by D.-J. Feng in  that if is an ergodic shift-invariant measure on and is an affine iterated function system then is necessarily exact-dimensional: this means that the limit
exists for -a.e. and is -almost-everywhere constant, where denotes the open Euclidean ball with centre and radius . This almost sure value will be called the dimension of the measure and is equal to its upper and lower Hausdorff and packing dimensions, see [5, §2].
In order to describe progress on the problem of finding measures of maximal dimension for affine iterated function systems it is useful to recall some definitions. We recall that the singular values of a real invertible matrix are defined to be the positive square roots of the eigenvalues of the positive definite matrix . We write the singular values of as with the convention . We have and for all , where denotes the operator norm induced by the Euclidean norm. If is a positive integer and a non-negative real number, following  we define
for all real matrices . The inequality is valid for all , and and was originally noted in . If is given then for each we define the -pressure of to be the quantity
which is well-defined by subadditivity. The function is continuous with respect to for fixed . When is fixed and has the property that for some norm on , the function has a unique zero which we call the affinity dimension of . If is an iterated function system of the form where , we define the affinity dimension of to be .
The affinity dimension is always an upper bound for the box dimension of the attractor of , see . If is an ergodic -invariant measure on then we define its Lyapunov dimension to be the unique zero of the function defined by
The Hausdorff dimension of is always bounded above by the Lyapunov dimension of , which is bounded above by the affinity dimension of , see  and [11, §4]. We say that a shift-invariant measure on is a -equilibrium state for if it satisfies
and a Käenmäki measure if it is a -equilibrium state with . For every and there exists at least one -equilibrium state for , a point which we discuss in more detail in §2 below. A shift-invariant measure is a Käenmäki measure if and only if it has Lyapunov dimension equal to .
In certain highly degenerate cases it is possible for the Hausdorff dimension of the attractor of an iterated function system to exceed the dimension of every invariant measure supported on it, and even to exceed the supremum of the dimensions of such measures: see . However, in generic cases the attractor of an affine iterated function system has Hausdorff dimension equal to the affinity dimension [1, 6, 7], and for generic affine iterated functions it is also the case that every Käenmäki measure on projects to a measure on the attractor which has dimension equal to the affinity dimension [7, 11] and is fully supported on the attractor . We refer the reader to the articles cited for the various precise meanings of “generic” with respect to which these statements are true. It is therefore of interest to ask how many measures of the form may achieve this maximal dimension value. Since any convex combination of measures with maximal dimension will also have maximal dimension, we ask specifically how many pairwise mutually singular measures of the form may have dimension equal to that of the attractor, where is shift-invariant. In generic cases this is equivalent to asking how many ergodic Käenmäki measures a given iterated function system may have. This latter question was first raised by A. Käenmaki  and is the subject of the present article.
Let us say that is reducible if there exists a nonzero proper subspace of such that for every , and otherwise is irreducible. We also say that is strongly irreducible if there does not exist a finite collection of nonzero proper subspaces such that for every . We extend the notions of irreducibility and strong irreducibility to subsets of in the obvious fashion. It is not difficult to see that a subset of is (strongly) irreducible if and only if the subsemigroup of which it generates is (strongly) irreducible. We will say that an affine iterated function system is (strongly) irreducible if it has the form where is (strongly) irreducible.
It is easy to show that every has a unique -equilibrium state when . There exist reducible tuples which have as many as mutually singular -equilibrium states (see ) and it is believed that this is the maximum possible number of mutually singular -equilibrium states for any tuple . This number is known to be a sharp upper bound for the number of mutually singular -equilibrium states in dimensions up to four  and for simultaneously upper triangularisable tuples , but in the general case the best upper bound which has been obtained so far for the number of mutually singular -equilibrium states is , see . When the maximum possible number of mutually singular -equilibrium states can be shown to equal using the techniques of [8, 13] although this result does not seem to have been explicitly stated in the literature.
If is irreducible, it was shown by D.-J. Feng and A. Käenmäki in  that has a unique -equilibrium state for all , and their argument easily extends to cover the case . In particular if is an irreducible affine iterated function system acting on then it has a unique Käenmäki measure. It was shown by the first named author and A. Käenmäki in  that in three dimensions strong irreducibility is sufficient for the uniqueness of -equilibrium states (and hence of Käenmäki measures) but irreducibility is not. A criterion for uniqueness of -equilibrium states in terms of irreducibility and strong irreducibility of successive exterior powers was also given in that article, and is discussed further in §2 below. In dimensions higher than two irreducibility does not suffice for the uniqueness of the Käenmäki measure: using the arguments of [13, §9] together with the results of  one may show that the example
is irreducible with and has exactly two ergodic Käenmäki measures.
These examples leave open the question of whether or not strong irreducibility is sufficient for the uniqueness of -equilibrium states and Käenmäki measures in dimensions higher than three. The purpose of this article is to show that in four dimensions strong irreducibility does not suffice for the uniqueness of -equilibrium states. We give the following example:
Let be even with , let such that and let . Let where for we have
and for we have
Then is strongly irreducible and for every there exist exactly two distinct ergodic -equilibrium states for . These equilibrium states are both fully supported on .
Here the symbol represents the Kronecker product of the two matrices and , which is a standard mechanism for representing the tensor product of two linear maps in terms of their matrices; for a more detailed explanation see §2 below. The fact that the equilibrium states are fully supported will be easily obtained during the proof, but also follows from the far more general results of . It is somewhat easier to prove Theorem 1 under the weaker hypothesis that , but we include the more general statement for completeness.
The algebraic idea underlying Theorem 1 is that there exists a natural irreducible representation from the group to the group defined by . One may show that is an irreducible subgroup of and since it contains the irreducible connected subgroup it is not difficult to deduce that it is strongly irreducible. Intuitively, if the matrices generate a subsemigroup of which is sufficiently large in an appropriate sense, this subsemigroup should be expected to inherit the strong irreducibility of and therefore will be strongly irreducible. On the other hand, the exterior power representation takes to a subset of which is not irreducible, but instead preserves two three-dimensional complementary subspaces of . Together with the various symmetries built into the construction of the matrices above this will allow us to construct two distinct equilibrium states with equal, maximal pressure. The relevance of exterior powers to the singular value function will be discussed in §2 below.
Theorem 1 implies the existence of strongly irreducible affine iterated function systems in four dimensions where there exists more than one fully-supported measure on the attractor with maximal dimension:
Let be as defined in Theorem 1 with , , and arbitrary . Then there exists such that the iterated function system defined by satisfies the strong separation condition, has , and admits two mutually singular invariant measures , with Hausdorff dimension equal to , each of which is fully supported on the attractor.
2.1. Linear algebra
For the remainder of the article will denote either the Euclidean norm defined by the standard inner product or a specified inner product, or the operator norm on matrices defined by such a Euclidean norm. If and are represented by the matrices
then their Kronecker product may be understood to be the linear map with matrix given by
This construction satisfies the identities and for all and . The identity follows from the first of these two identities. If are the eigenvalues of and the eigenvalues of then the eigenvalues of are precisely the products with and . Combining these observations it follows that the singular values of are the products such that and and in particular for all and . For proofs of these identities we direct the reader to [9, §4.2]. The Kronecker product may be understood algebraically as the matrix representation of the tensor product of the linear maps and , but this interpretation will not be needed explicitly in the present work.
For each the exterior power of , denoted , is a -dimensional real vector space spanned by the set of all vectors of the form where , where the symbol “” is subject to the identities
for all , and . If is any basis for then the vectors such that form a basis for . The standard inner product on induces an inner product on by
If then induces a linear map on by . By considering appropriate bases it is easy to see that if are the eigenvalues of then the eigenvalues of are the numbers such that . The identity follows directly from the definition of the inner product on , and combining these observations we see that the singular values of are precisely the products such that . In particular we have . The significance of exterior powers to the present article arises from the following identity: if and , then
by the identity previously remarked.
2.2. Thermodynamic formalism
If is understood, let which we equip with the infinite product topology. This topological space is compact and metrisable. We define the shift transformation by which is a continuous surjection. We let denote the set of all -invariant Borel probability measures on equipped with the weak-* topology, which is the smallest topology such that is continuous for every . With respect to this topology is a nonempty, compact, metrisable topological space.
We will say that a word is any finite sequence . We define the length of the word to be and denote the length of any word by . When is understood we denote the set of all words by . If then we define to be the word . If then we define the corresponding cylinder set to be the set . The set of all cylinder sets is a basis for the topology of . If are arbitrary words then we define their concatenation in the obvious fashion: it is the word such that for and for . If is understood then we define for every .
We will find it convenient in proofs to appeal to more general notions of pressure and equilibrium state than those defined in the introduction. If is understood let us say that a potential is any function . We will say that a potential is submultiplicative if it has the property for all . All potentials considered in this article will be submultiplicative. If is a submultiplicative potential then the sequence of functions defined by satisfies the submultiplicativity relation for all and . Each is continuous since it depends only on finitely many co-ordinates. For every we define
which is well-defined by subadditivity. By the subadditive ergodic theorem, if is ergodic then we have for -a.e. .
If is a submultiplicative potential then we define its pressure to be the quantity
which is well-defined by subadditivity. By the subadditive variational principle of D.-J. Feng, Y.-L. Cao and W. Huang we have
see [3, Theorem 1.1]). Since the map is continuous for each and the map is upper semi-continuous, the map is upper semi-continuous. In particular the supremum above is always attained. We call a measure which attains this supremum an equilibrium state for .
If and then we may define a submultiplicative potential by . Clearly in this case and the notion of equilibrium state for coincides with the notion of -equilibrium state for introduced in the introduction. Our mechanism for studying equilibrium states in this article will be the following result from :
Let be strongly irreducible and let be irreducible, and let and . Define a submultiplicative potential by
for all . Then there exists a unique equilibrium state for . It is ergodic, and there exists a constant such that
for every .
The result is a special case of [2, Corollary 2.2] with if and with if .∎
If has the property that and are both irreducible and at least one of them is strongly irreducible then it follows from (2) and Theorem 3 that has a unique -equilibrium state, a result first observed in . (For the purposes of this observation the tuples and should be understood to always be strongly irreducible as a matter of definition.) The example (1) shows that strong irreducibility cannot here be weakened to irreducibility.
By considering the definitions of and it is not difficult to see that if two submultiplicative potentials , satisfy for all and some constant then they have the same pressure and the same equilibrium states as one another. In particular when considering potentials of the form the pressure and equilibrium states are independent of the choice of norm or norms used to define the potential. It likewise follows that if , and then the tuple has the same pressure and -equilibrium states as .
Let and . Then is a -equilibrium state of if and only if it is a -equilbrium state of the tuple defined by
for all .
Define by and observe that is a homomorphism and satisfies . The result follows. ∎
3. Proof of Theorem 1
For the sake of simplicity we will give the proof in the case , the modifications required for higher being straightforward. We will begin the proof by showing that in order to prove the full statement of the theorem it is sufficient to consider only those cases in which .
Suppose that Theorem 1 has been proved for all pairs as described in the statement of Theorem 1, for all . Consider a particular pair as given in the statement of that theorem and let . By Lemma 2.1 the -equilibrium states of are precisely the -equilibrium states of the pair defined by
and . It is straightforward to check that
so that in particular satisfies the hypotheses of Theorem 1. It therefore has precisely 2 ergodic -equilibrium states, since and the theorem is assumed to have been proved in that parameter range. These equilibrium states are precisely the -equilibrium states of , which are precisely the -equilibrium states of . Thus the full statement of Theorem 1 will follow from our establishing the theorem only in the case .
We next wish to show that every satisfying the hypotheses of Theorem 1 is irreducible. We will use an elementary argument: a shorter but more technical argument using representation theory will be indicated in a subsequent footnote. Fix such a pair . The eigenvalues of and are and where is not an integer multiple of , so in particular neither nor has any real eigenvalues. It follows that if is a proper nonzero linear subspace of which is invariant under then the dimension of cannot be odd: if it were odd then the characteristic polynomial of the restriction of to would be a real polynomial of odd degree and would therefore have at least one real root, implying the existence of a real eigenvalue of with associated eigenvector in . This is clearly a contradiction, so if a proper nonzero linear subspace of exists which is invariant under both and then its dimension must be .
We will show that there cannot exist a two-dimensional subspace of which is invariant under both and . Let us define
It is straightforward to verify the following observations: preserves both and ; preserves and ; for all , for all , for all and for all .
Let be any nonzero subspace of such that . We claim that contains a nonzero element of either or . If is nonzero and , then we may write with , and . We then have
In particular every accumulation point of the sequence