A strongly irreducible affine iterated function system with two invariant measures of maximal dimension
aa r X i v : . [ m a t h . D S ] S e p A STRONGLY IRREDUCIBLE AFFINE ITERATED FUNCTIONSYSTEM WITH TWO INVARIANT MEASURES OF MAXIMALDIMENSION
IAN D. MORRIS AND CAGRI SERT
Abstract.
A classical theorem of Hutchinson asserts that if an iterated func-tion system acts on R d by similitudes and satisfies the open set condition thenit admits a unique self-similar measure with Hausdorff dimension equal to thedimension of the attractor. In the class of measures on the attractor whicharise as the projections of shift-invariant measures on the coding space, thisself-similar measure is the unique measure of maximal dimension. In the con-text of affine iterated function systems it is known that there may be multipleshift-invariant measures of maximal dimension if the linear parts of the affini-ties share a common invariant subspace, or more generally if they preservea finite union of proper subspaces of R d . In this note we construct exampleswhere multiple invariant measures of maximal dimension exist even though thelinear parts of the affinities do not preserve a finite union of proper subspaces. Introduction
We recall that an iterated function system is by definition a tuple ( T , . . . , T N )of contracting transformations of some metric space X , which in this article will be R d equipped with the Euclidean distance. To avoid trivialities it will be assumedthroughout this article that N ≥
2. If ( T , . . . , T N ) is an iterated function systemacting on R d then it is well-known that there exists a unique nonempty compactset Z ⊂ R d with the property Z = S Ni =1 T i Z , called the attractor or limit set ofthe iterated function system. If we define Σ N := { , . . . , N } N with the infiniteproduct topology, there exists moreover a well-defined coding map Π : Σ N → R d characterised by the propertyΠ [( x k ) ∞ k =1 ] = lim n →∞ T x · · · T x n v for all v ∈ R d and ( x k ) ∞ k =1 ∈ Σ N , and this coding map is a continuous surjectionfrom Σ N to the attractor.We recall that ( T , . . . , T N ) is said to satisfy the open set condition if there existsa nonempty open set U ⊆ R d such that the sets T U, . . . , T N U are pairwise disjointsubsets of U ; if the same condition holds with a nonempty compact set X ⊆ R d instead of an open set U then we say that ( T , . . . , T N ) satisfies the strong separationcondition . It is not difficult to show that if the strong separation condition issatisfied then the coding map is a homeomorphism from Σ N to the attractor.It was shown recently by D.-J. Feng in [12] that if µ is an ergodic shift-invariantmeasure on Σ N and ( T , . . . , T N ) is an affine iterated function system then Π ∗ µ isnecessarily exact-dimensional: this means that the limitlim r → log Π ∗ µ ( B r ( v ))log r exists for Π ∗ µ -a.e. v ∈ R d and is Π ∗ µ -almost-everywhere constant, where B r ( v )denotes the open Euclidean ball with centre v and radius r . This almost sure valuewill be called the dimension of the measure Π ∗ µ and is equal to its upper andlower Hausdorff and packing dimensions, see [9, § T , . . . , T N ) satisfies theopen set condition and the transformations T i are similarity transformations, it is aclassical result of J.E. Hutchinson [16] that there exists a probability measure on theattractor of ( T , . . . , T N ) with dimension equal to that of the attractor; moreover,this measure has the form Π ∗ µ where µ is a Bernoulli measure on the codingspace Σ N . In particular µ is invariant with respect to the shift transformation σ [( x k ) ∞ k =1 ] := ( x k +1 ) ∞ k =1 . In the more general context in which the transformations T i are invertible affine transformations of R d it is thus natural to ask when thereexists an invariant measure µ on the coding space which projects to a measure withdimension equal to that of the attractor, and if such a measure exists, how manysuch measures there might be. This question was posed explicitly, in somewhatdiffering forms, by D. Gatzouras and Y. Peres in [14] and by A. K¨aenm¨aki in [19].In order to describe progress on the problem of finding measures of maximaldimension for affine iterated function systems it is useful to recall some definitions.We recall that the singular values of a real invertible d × d matrix A are definedto be the positive square roots of the eigenvalues of the positive definite matrix A ⊤ A . We write the singular values of A as σ ( A ) , . . . , σ d ( A ) with the convention σ ( A ) ≥ · · · ≥ σ d ( A ). We have k A k = σ ( A ) and | det A | = σ ( A ) · · · σ d ( A ) for all A ∈ GL d ( R ), where k ·k denotes the operator norm induced by the Euclidean norm.If d is a positive integer and s a non-negative real number, following [10] we define ϕ s ( A ) := (cid:26) σ ( A ) · · · σ ⌊ s ⌋ ( A ) σ ⌈ s ⌉ ( A ) s −⌊ s ⌋ if 0 ≤ s ≤ d, | det A | sd if s ≥ d for all real d × d matrices A . The inequality ϕ s ( AB ) ≤ ϕ s ( A ) ϕ s ( B ) is valid for all s , A and B and was originally noted in [10]. If ( A , . . . , A N ) ∈ GL d ( R ) N is giventhen for each s ≥ ϕ s -pressure of ( A , . . . , A N ) to be the quantity P ϕ s ( A , . . . , A N ) := lim n →∞ n log N X i ,...,i n =1 ϕ s ( A i · · · A i n )which is well-defined by subadditivity. The function s P ϕ s ( A , . . . , A N ) is con-tinuous with respect to s for fixed ( A , . . . , A N ) ∈ GL d ( R ) N . When ( A , . . . , A N ) ∈ GL d ( R ) N is fixed and has the property that max i k A i k < R d ,the function s P ϕ s ( A , . . . , A N ) has a unique zero which we call the affinity di-mension of ( A , . . . , A N ). If ( T , . . . , T N ) is an iterated function system of the form T i x = A i x + v i where ( A , . . . , A N ) ∈ GL d ( R ) N , we define the affinity dimensionof ( T , . . . , T N ) to be dim aff ( A , . . . , A N ).The affinity dimension is always an upper bound for the box dimension of theattractor of ( T , . . . , T N ), see [10]. If µ is an ergodic σ -invariant measure on Σ N thenwe define its Lyapunov dimension to be the unique zero of the function [0 , ∞ ) → R defined by s h ( µ ) + lim n →∞ n Z log ϕ s ( A x · · · A x n ) dµ [( x k ) ∞ k =1 ] . The Hausdorff dimension of Π ∗ µ is always bounded above by the Lyapunov dimen-sion of µ , which is bounded above by the affinity dimension of ( A , . . . , A N ), see N AFFINE IFS WITH TWO MEASURES OF MAXIMAL DIMENSION 3 [19] and [18, § µ is a ϕ s -equilibriumstate for ( A , . . . , A N ) if it satisfies P ϕ s ( A , . . . , A N ) = h ( µ ) + lim n →∞ n Z log ϕ s ( A x · · · A x n ) dµ [( x k ) ∞ k =1 ] , and a K¨aenm¨aki measure if it is a ϕ s -equilibrium state with s := dim aff ( A , . . . , A N ).For every ( A , . . . , A N ) ∈ GL d ( R ) N and s ≥ ϕ s -equilibrium state for ( A , . . . , A N ), a point which we discuss in more detail in § aff ( A , . . . , A N ).In certain highly degenerate cases it is possible for the Hausdorff dimension of theattractor of an iterated function system to exceed the dimension of every invariantmeasure Π ∗ µ supported on it, and even to exceed the supremum of the dimensions ofsuch measures: see [7]. However, in generic cases the attractor of an affine iteratedfunction system has Hausdorff dimension equal to the affinity dimension [1, 10, 12],and for generic affine iterated functions it is also the case that every K¨aenm¨akimeasure µ on Σ N projects to a measure Π ∗ µ on the attractor which has dimensionequal to the affinity dimension [12, 18] and is fully supported on the attractor[4]. 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 interestto ask how many measures of the form Π ∗ µ may achieve this maximal dimensionvalue. Since any convex combination of measures with maximal dimension will alsohave maximal dimension, we ask specifically how many pairwise mutually singular measures of the form Π ∗ µ may have dimension equal to that of the attractor, where µ is shift-invariant. In generic cases this is equivalent to asking how many ergodicK¨aenm¨aki measures a given iterated function system may have. This latter questionwas first raised by A. K¨aenmaki [19] and is the subject of the present article.Let us say that ( A , . . . , A N ) ∈ GL d ( R ) N is reducible if there exists a nonzeroproper subspace V of R d such that A i V = V for every i = 1 , . . . , N , and otherwiseis irreducible . We also say that ( A , . . . , A N ) is strongly irreducible if there doesnot exist a finite collection V , . . . , V m of nonzero proper subspaces V j such that A i (cid:0) ∪ mj =1 V j (cid:1) = ∪ mj =1 V j for every i . We extend the notions of irreducibility andstrong irreducibility to subsets of GL d ( R ) in the obvious fashion. It is not difficult tosee that a subset of GL d ( R ) is (strongly) irreducible if and only if the subsemigroupof GL d ( R ) which it generates is (strongly) irreducible. We will say that an affineiterated function system ( T , . . . , T N ) is (strongly) irreducible if it has the form T i x = A i x + v i where ( A , . . . , A N ) is (strongly) irreducible.It is easy to show that every ( A , . . . , A N ) ∈ GL d ( R ) N has a unique ϕ s -equilibriumstate when s ≥ d . There exist reducible tuples ( A , . . . , A N ) ∈ GL d ( R ) N whichhave as many as ( d − ⌊ s ⌋ ) (cid:0) d ⌊ s ⌋ (cid:1) = ⌈ s ⌉ (cid:0) d ⌈ s ⌉ (cid:1) mutually singular ϕ s -equilibrium states(see [20]) and it is believed that this is the maximum possible number of mutu-ally singular ϕ s -equilibrium states for any tuple ( A , . . . , A N ) ∈ GL d ( R ) N . Thisnumber is known to be a sharp upper bound for the number of mutually singular ϕ s -equilibrium states in dimensions up to four [23] and for simultaneously uppertriangularisable tuples [20], but in the general case the best upper bound which hasbeen obtained so far for the number of mutually singular ϕ s -equilibrium states is (cid:0) d ⌊ s ⌋ (cid:1)(cid:0) d ⌈ s ⌉ (cid:1) , see [4]. When s ∈ (0 , d ) ∩ Z the maximum possible number of mutuallysingular ϕ s -equilibrium states can be shown to equal (cid:0) ds (cid:1) using the techniques of IAN D. MORRIS AND CAGRI SERT [13, 20] although this result does not seem to have been explicitly stated in theliterature.If ( A , . . . , A N ) ∈ GL d ( R ) N is irreducible, it was shown by D.-J. Feng and A.K¨aenm¨aki in [13] that ( A , . . . , A N ) has a unique ϕ s -equilibrium state for all s ∈ (0 , s ∈ [ d − , d ). In particularif ( T , . . . , T N ) is an irreducible affine iterated function system acting on R thenit has a unique K¨aenm¨aki measure. It was shown by the first named author andA. K¨aenm¨aki in [20] that in three dimensions strong irreducibility is sufficient forthe uniqueness of ϕ s -equilibrium states (and hence of K¨aenm¨aki measures) butirreducibility is not. A criterion for uniqueness of ϕ s -equilibrium states in terms ofirreducibility and strong irreducibility of successive exterior powers was also givenin that article, and is discussed further in § §
9] together with the results of [24] one may show that theexample(1) A := , A :=
00 0 is irreducible with dim aff ( A , A ) ∈ (1 ,
2) and has exactly two ergodic K¨aenm¨akimeasures.These examples leave open the question of whether or not strong irreducibilityis sufficient for the uniqueness of ϕ s -equilibrium states and K¨aenm¨aki measures indimensions higher than three. The purpose of this article is to show that in fourdimensions strong irreducibility does not suffice for the uniqueness of ϕ s -equilibriumstates. We give the following example: Theorem 1.
Let α , α be nonzero real numbers such that | α | 6 = | α | and let θ ∈ R \ π Z . Let ( A , A ) ∈ GL ( R ) where A = (cid:18) α α (cid:19) ⊗ (cid:18) cos θ − sin θ sin θ cos θ (cid:19) = α cos θ − α sin θ α sin θ α cos θ α cos θ − α sin θ α sin θ α cos θ and A = (cid:18) cos θ − sin θ sin θ cos θ (cid:19) ⊗ (cid:18) α α (cid:19) = α cos θ − α sin θ α cos θ − α sin θα sin θ α cos θ α sin θ α cos θ . Then ( A , A ) is strongly irreducible and for every s ∈ (1 , there exist exactly twodistinct ergodic ϕ s -equilibrium states for ( A , A ) . These equilibrium states are bothfully supported on Σ N . Here the symbol A ⊗ B represents the Kronecker product of the two matrices A and B , which is a standard mechanism for representing the tensor product of twolinear maps in terms of their matrices; for a more detailed description see § N AFFINE IFS WITH TWO MEASURES OF MAXIMAL DIMENSION 5
Theorem 1 arises as a special case of a substantially more general result whosestatement requires some additional notation and definitions; we postpone the state-ment of this more general theorem until the following section. The precise choiceof the two matrices in Theorem 1 incorporates some arbitrary elements so as tofacilitate the proof of Theorem 2 below. In fact, we will see in the remarks fol-lowing Theorem 3 below that almost any contracting pair of matrices of the form A := B ⊗ B , A := B ⊗ B with B , B ∈ GL ( R ) would suffice just as well.Theorem 1 implies the existence of strongly irreducible affine iterated functionsystems in four dimensions where there exists more than one fully-supported mea-sure on the attractor with maximal dimension: Theorem 2.
Let ( B , B , B , B ) = ( A , A , A , A ) ∈ GL ( R ) where A and A are as defined in Theorem 1 with < α < α < √ , α α > , and arbitrary θ ∈ R \ π Z . Then there exists ( v , . . . , v ) ∈ ( R ) such that the iterated functionsystem defined by T i x := B i x + v i satisfies the strong separation condition, has < dim aff ( B , . . . , B ) < , and admits two mutually singular invariant measures m := Π ∗ µ , m := Π ∗ µ with Hausdorff dimension equal to dim aff ( B , . . . , B ) ,each of which is fully supported on the attractor. Examples of affine iterated function systems with two fully-supported mea-sures of maximal dimension were previously constructed in two dimensions by A.K¨aenm¨aki and M. Vilppolainen [22, Example 6.2] and by J. Barral and D.-J. Feng[2]; in these examples the linear parts of the affinities are given by diagonal matrices,and in particular these examples are not irreducible.Theorem 1 may be considered to have the following heuristic implication forthe investigation of affine iterated function systems. Works which attempt to provevery general statements about the thermodynamic formalism of affine iterated func-tion systems – that is, assuming only invertibility and perhaps irreducibility of thelinear parts of the system – can encounter the problem that the number of distinct ϕ s -equilibrium states may in general be very large, forcing any complete mathe-matical argument to deteriorate into a branching investigation of sub-cases arisingfrom the families of different ϕ s -equilibrium states which may exist for a singleiterated function system. (Indeed, in the articles [4] and [25] this phenomenon isresponsible for most of the length of the proofs of the main results; in those specialcases where a unique ϕ s -equilibrium state exists the proofs of the results of botharticles can be made an order of magnitude shorter.) It is therefore natural to askwhether a simple, testable general condition can be imposed on an affine IFS whichforces the ϕ s -equilibrium state to be unique, allowing simpler and more economicalarguments to be applied without any very substantial loss of generality. While itwas shown in [20] that the Zariski density of the semigroup generated by the linearparts is sufficient for the uniqueness of the ϕ s -equilibrium state (see also [17] for aclosely related result) this condition is arguably the strongest possible condition ofan algebraic nature and it is reasonable to ask whether a weaker condition such asstrong irreducibility might instead be sufficient. The results of this article demon-strate that this is not the case and strongly suggest that proofs which incorporatethe consideration of multiple inequivalent ϕ s -equilibrium states are likely to re-main a feature of the literature in situations where statements assuming a weakercondition than Zariski density are proved. IAN D. MORRIS AND CAGRI SERT More general examples
Theorem 1 is obtained as a special case of a more general construction which wenow describe. We first establish some necessary notation and definitions. We recallthat PGL d ( R ) denotes the quotient of GL d ( R ) by the subgroup consisting of allscalar multiples of the identity matrix. For an element g ∈ GL d ( R ) we denote by ¯ g the corresponding equivalence class ¯ g ∈ PGL d ( R ). For the purpose of exposition,let G denote either of the groups SL d ( R ), GL d ( R ) and PGL d ( R ). These groupsare linear algebraic groups : each G can be realised as the set of common zeros ofan ideal of polynomials with real coefficients in k variables for some k ∈ N . TheZariski topology on each such group G is defined by declaring the closed subsets of G to be the sets of common zeros of collections of polynomials in k variables. Thistopology does not depend on the choice of the embedding in the space R k and it iscoarser than the standard topology on G , which we refer to as the analytic topology .For example, PGL d ( R ) is connected with respect to the Zariski topology, whereasfor even d it has two connected components with respect to the analytic topology,one corresponding to linear maps with positive determinant (which is equal toPSL d ( R )) and one corresponding to linear maps with negative determinant. A set Z ⊆ G is called Zariski dense in G if it is a dense subset of G with respect to theZariski topology in the usual sense; this is equivalent to the stipulation that everypolynomial function G → R which is identically zero on Z is also identically zeroon G . The Zariski closure of any subsemigroup of G is a Lie group with finitelymany connected components. In particular a subsemigroup of G fails to be Zariskidense if and only if it is contained in a proper Lie subgroup of G which has finitelymany connected components.If N ≥ word is any finite sequence i =( i k ) nk =1 ∈ { , . . . , N } n . We define the length of the word i = ( i k ) nk =1 to be n anddenote the length of any word i by | i | . We denote the set of all words by Σ ∗ N .If ( A , . . . , A N ) ∈ GL d ( R ) N is also understood then we define A i := A i · · · A i n for every i = ( i k ) nk =1 ∈ Σ ∗ N . If ι : { , . . . , N } → { , . . . , N } is a permutation then ι naturally extends to a map ι : Σ ∗ N → Σ ∗ N defined by ι [( i k ) nk =1 ] := ( ι ( i k )) Nk =1 .Clearly ι thus defined induces a permutation of the set { i ∈ Σ ∗ N : | i | = n } for each n ≥ Theorem 3.
Let d, N ≥ , let ( B , . . . , B N ) ∈ GL d ( R ) N and suppose that thegroup generated by the projective linear maps ¯ B i is Zariski dense in PGL d ( R ) . Let ι be a permutation of { , . . . , N } such that:(i) There does not exist h ∈ PGL d ( R ) such that for every i ∈ Σ ∗ N , we have ¯ B i = h ¯ B ι ( i ) h − .(ii) There does not exist h ∈ PGL d ( R ) such that for every i ∈ Σ ∗ N , we have ¯ B i = h ( ¯ B − ι ( i ) ) ⊤ h − .Define an N -tuple ( A , . . . , A N ) ∈ GL d ( R ) N by A i := B i ⊗ B ι ( i ) for each i =1 , . . . , N . Then ( A , . . . , A N ) is strongly irreducible and for every s ∈ (1 , ∪ [ d − , d − there exist exactly two distinct ergodic ϕ s -equilibrium states for ( A , . . . , A N ) , both of which are both fully supported on Σ N . We make the following remarks:
N AFFINE IFS WITH TWO MEASURES OF MAXIMAL DIMENSION 7
Remark 2.1.
1. Provided that the permutation ι is non-trivial, the set of tuples ( B , . . . , B N ) ∈ GL d ( R ) N that do not satisfy the assumptions of Theorem 3 is con-tained in a null set with respect to the Haar measure on GL d ( R ) N . When ι isnon-trivial the assumptions ( i ) and ( ii ) hold for an open dense subset of GL d ( R ) N .2. In practice, it is easy and usually sufficient to verify the assumptions directly onthe generating tuple ( B , . . . , B N ) , that is, by considering words i of length one inthe hypotheses (i) and (ii).3. Since the permutation ι necessarily satisfies ι m = id for some natural num-ber m ≤ N ! it is not hard to see that if there exists an h satisfying ( i ) then thecorresponding power h m of h must commute with every element of the semigroupgenerated by B , . . . , B N . By Zariski density it follows that h m belongs to the centreof PGL d ( R ) , which is the trivial group. We conclude that necessarily h m = id forsome integer m not greater than N ! . Similarly if ( ii ) holds then the same observa-tion applies to ( h − ) ⊤ in place of h . If we define B := (cid:18) α α (cid:19) , B := (cid:18) cos θ − sin θ sin θ cos θ (cid:19) where α , α and θ are as in Theorem 1 and let ι : { , } → { , } be given by ι (1) :=2, ι (2) := 1 then clearly the pair ( A , A ) defined by Theorem 3 corresponds tothat considered in Theorem 1. Obviously the assumptions ( i ) and ( ii ) are satisfied.Since θ = π/ B and B is a non-elementary subgroup ofSL ( R ) and hence is Zariski dense in PGL ( R ). In particular Theorem 1 followsfrom Theorem 3.We believe that it should be possible to extend the method of Theorem 3 so asto construct examples in dimension d > s ∈ (1 , d −
1) thereare multiple distinct ϕ s -equilibrium states. In those cases the number of ergodicequilibrium states will in general be much larger than 2. Such a generalisationof Theorem 3 would be likely to need additional hypotheses on the ordering ofproducts of singular values of the matrices B i in order to make the comparison ofdifferent families of equilibrium states practical. Since the purpose of this article issimply to demonstrate that strong irreducibility is compatible with the existence ofmultiple ϕ s -equilibrium states we do not pursue the problem of optimising Theorem3 in this manner.The proof of Theorem 3 is presented as follows. In the following section werecall some necessary concepts from linear algebra, thermodynamic formalism andthe theory of linear algebraic groups. In § A , . . . , A N )and establish some related algebraic facts which will be used later; in § §
3. In § § Preliminaries
Linear algebra.
For the remainder of the article k · k will denote either theEuclidean norm defined by the standard inner product or a specified inner product,
IAN D. MORRIS AND CAGRI SERT or the operator norm on matrices defined by such a Euclidean norm. If A ∈ GL d ( R )and B ∈ GL d ( R ) are represented by the matrices A = a · · · a d ... . . . ... a d · · · a d d , B = b · · · b d ... . . . ... b d · · · b d d , then their Kronecker product may be understood to be the linear map A ⊗ B ∈ GL d d ( R ) with matrix given by A ⊗ B = a B · · · a d B ... . . . ... a d B · · · a d d B . This construction satisfies the identities ( A ⊗ B )( A ⊗ B ) = ( A A ) ⊗ ( B B )and A ⊤ ⊗ B ⊤ = ( A ⊗ B ) ⊤ for all A , A , A ∈ GL d ( R ) and B , B , B ∈ GL d ( R ).The identity ( A ⊗ B ) − = ( A − ⊗ B − ) follows from the first of these two identities.If α , . . . , α d are the eigenvalues of A and α ′ , . . . , α ′ d the eigenvalues of B thenthe eigenvalues of A ⊗ B are precisely the d d products α i α ′ j with 1 ≤ i ≤ d and1 ≤ j ≤ d . Combining these observations it follows that the singular values of A ⊗ B are the products σ i ( A ) σ j ( B ) such that 1 ≤ i ≤ d and 1 ≤ j ≤ d and inparticular k A ⊗ B k = σ ( A ⊗ B ) = σ ( A ) σ ( B ) = k A k·k B k for all A ∈ GL d ( R ) and B ∈ GL d ( R ). Proofs of these identities may be found in [15, § A ⊗ B may be understood algebraically as the matrix representation ofthe tensor product of the linear maps A and B , but this interpretation will not beneeded explicitly in the present work.For each k = 1 , . . . , d the k th exterior power of R d , denoted ∧ k R d , is a (cid:0) dk (cid:1) -dimensional real vector space spanned by the set of all vectors of the form v ∧ v ∧· · · ∧ v k where v , . . . , v k ∈ R d , where the symbol “ ∧ ” is subject to the identities λ ( v ∧ v ∧ · · · ∧ v k ) + ( v ′ ∧ v ∧ · · · ∧ v k ) = ( λv + v ′ ) ∧ v ∧ · · · ∧ v k ,v ∧ v ∧ · · · ∧ v k = ( − i +1 v i ∧ v ∧ · · · ∧ v i − ∧ v ∧ v i +1 ∧ · · · ∧ v k for all v , . . . , v k , v ′ ∈ R d , λ ∈ R and i = 1 , . . . , k . If u , . . . , u d is any basis for R d then the vectors u i ∧ · · · ∧ u i k such that 1 ≤ i < · · · < i k ≤ d form a basis for ∧ k R d . The standard inner product h· , ·i on R d induces an inner product on ∧ k R d by h u ∧ · · · ∧ u k , v ∧ · · · ∧ v k i := det [ h u i , v i i ] ki,j =1 . If A ∈ GL d ( R ) then A induces a linear map A ∧ k on ∧ k R d by A ∧ k ( u ∧ · · · ∧ u k ) = Au ∧ · · · ∧ Au k . By considering appropriate bases it is easy to see that if α , . . . , α d are the eigenvalues of A then the eigenvalues of A ∧ k are the numbers α i · · · α i k suchthat 1 ≤ i < · · · < i k ≤ d . The identity ( A ⊤ ) ∧ k = ( A ∧ k ) ⊤ follows directly from thedefinition of the inner product on ∧ k R d , and combining these observations we seethat the singular values of A ∧ k are precisely the products σ i ( A ) · · · σ i k ( A ) such that1 ≤ i < · · · < i k ≤ d . In particular we have k ∧ k A k ≡ σ ( A ∧ k ) ≡ σ ( A ) · · · σ k ( A ).The significance of exterior powers to the present article arises from the followingidentity: if A ∈ GL d ( R ) and 0 ≤ s ≤ d , then(2) ϕ s ( A ) = (cid:13)(cid:13)(cid:13) A ∧⌊ s ⌋ (cid:13)(cid:13)(cid:13) ⌊ s ⌋− s (cid:13)(cid:13)(cid:13) A ∧⌈ s ⌉ (cid:13)(cid:13)(cid:13) s −⌊ s ⌋ by the identity previously remarked. N AFFINE IFS WITH TWO MEASURES OF MAXIMAL DIMENSION 9
Finally, given A ∈ GL d ( R ) and i = 1 , . . . , d we let λ i ( A ) denote the modulusof the i th largest of the eigenvalues of A . As an easy consequence of the formula k A ∧ k k = σ ( A ) · · · σ k ( A ) together with Gelfand’s formula, we have(3) lim n →∞ σ i ( A n ) n = λ i ( A )for every A ∈ GL d ( R ).3.2. Linear algebraic groups.
Given a representation ρ : Γ → GL d ( R ) we shallsay that ρ is strongly irreducible (resp. irreducible) if its image is strongly irreducible(resp. irreducible) in the sense previously defined. Given a subgroup G of GL d ( R )with finitely many connected components which is closed in the analytic topology,it is not difficult to see that G acts strongly irreducibly if and only if the connectedcomponent of the identity (in the analytic topology) acts irreducibly. When G is alinear algebraic group, i.e. a closed subgroup of GL d ( R ) with respect to the Zariskitopology, it automatically has finitely many connected components (for both theanalytic and Zariski topologies) and the previous assertion remains true for theconnected component of G with respect to the Zariski topology. We also mentionthat every linear algebraic group is a Lie group, and that the Zariski closure of asemigroup is itself a linear algebraic group. We lastly observe that a subset S ofGL d ( R ) remains strongly irreducible/irreducible if each element g ∈ S is replacedby some nonzero real scalar multiple cg , where c ∈ R ∗ is allowed to depend on g . Accordingly we may without ambiguity speak of a subset of PGL d ( R ) as beingstrongly irreducible or irreducible.In the proof of Theorem 3 we will use a special case of a result of Y. Benoiston the properties of the limit cone of a semigroup in a reductive linear algebraicgroup. For these notions, as well as the following statement, we refer the reader to[3, Th´eor`eme 1.4] (see also [26, Proposition 1.3] and [5, Theorem 1.4]). We observethat given a projective linear transformation γ ∈ PGL d ( R ), the ratios λ i λ i +1 ( γ ) ofspecified pairs of absolute eigenvalues of γ are well-defined as the ratio λ i ( g ) λ i +1 ( g ) forany g ∈ GL d ( R ) with ¯ g = γ . The result which we require is the following: Proposition 3.1.
Let Γ be a Zariski-dense subsemigroup of PGL d ( R ) × PGL d ( R ) .Then there exists ( γ , γ ) ∈ Γ such that λ λ ( γ ) = λ λ ( γ ) . Thermodynamic formalism. If N ≥ N := { , . . . , N } N which we equip with the infinite product topology. This topological space iscompact and metrisable. We define the shift transformation σ : Σ N → Σ N by σ [( x k ) ∞ k =1 ] := ( x k +1 ) ∞ k =1 which is a continuous surjection. We let M σ denote theset of all σ -invariant Borel probability measures on Σ N equipped with the weak-*topology, which is the smallest topology such that µ R f dµ is continuous forevery f ∈ C (Σ N ). With respect to this topology M σ is a nonempty, compact,metrisable topological space.As was described earlier we will say that a word is any finite sequence i =( i k ) nk =1 ∈ { , . . . , N } n . We define the length of the word i = ( i k ) nk =1 to be n anddenote the length of any word i by | i | . When N is understood we denote theset of all words by Σ ∗ N . If x = ( x k ) ∞ k =1 ∈ Σ N then we define x | n to be the word( x k ) nk =1 ∈ Σ ∗ N . If i ∈ Σ ∗ N then we define the corresponding cylinder set to be the set [ i ] := { x ∈ Σ N : x | n = i } . The set of all cylinder sets is a basis for the topologyof Σ N . If i = ( i k ) nk =1 , j = ( j k ) mk =1 ∈ Σ ∗ N are arbitrary words then we define theirconcatenation ij in the obvious fashion: it is the word ( ℓ k ) n + mk =1 such that ℓ k = i k for 1 ≤ k ≤ n and ℓ k = j k − n for n + 1 ≤ k ≤ n + m . If ( A , . . . , A N ) ∈ GL d ( R ) N isunderstood then we define A i := A i · · · A i n for every i = ( i k ) nk =1 ∈ Σ ∗ N .We will find it convenient in proofs to appeal to more general notions of pressureand equilibrium state than those defined in the introduction. If N ≥ potential is any function Φ : Σ ∗ N → (0 , + ∞ ). We will say thata potential is submultiplicative if it has the property Φ( ij ) ≤ Φ( i )Φ( j ) for all i , j ∈ Σ ∗ N . All potentials considered in this article will be submultiplicative. If Φis a submultiplicative potential then the sequence of functions Φ n : Σ N → (0 , + ∞ )defined by Φ n ( x ) := Φ( x | n ) satisfies the submultiplicativity relation Φ n + m ( x ) ≤ Φ n ( σ m x )Φ m ( x ) for all n, m ≥ x ∈ Σ N . Each Φ n is continuous since itdepends only on finitely many co-ordinates. For every µ ∈ M σ we defineΛ(Φ , µ ) := lim n →∞ n Z log Φ( x | n ) dµ ( x ) = inf n ≥ n Z log Φ( x | n ) dµ ( x )which is well-defined by subadditivity. By the subadditive ergodic theorem, if µ ∈ M σ is ergodic then we have lim n →∞ n log Φ( x | n ) = Λ(Φ , µ ) for µ -a.e. x ∈ Σ N .If Φ is a submultiplicative potential then we define its pressure to be the quantity P (Φ) := lim n →∞ n log X | i | = n Φ( i )which is well-defined by subadditivity. By the subadditive variational principle ofD.-J. Feng, Y.-L. Cao and W. Huang we have P (Φ) = sup µ ∈M σ [ h ( µ ) + Λ(Φ , µ )] , see [6, Theorem 1.1]). Since the map µ R log Φ( x | n ) dµ ( x ) is continuous for each n ≥ µ h ( µ ) is upper semi-continuous, the map µ h ( µ )+Λ(Φ , µ )is upper semi-continuous. In particular the supremum above is always attained. Wecall a measure which attains this supremum an equilibrium state for Φ.If ( A , . . . , A N ) ∈ GL d ( R ) and s ≥ s : Σ N → (0 , + ∞ ) by Φ s ( i ) := ϕ s ( A i ). Clearly in this case P (Φ s ) = P ϕ s ( A , . . . , A N ) and the notion of equilibrium state for Φ s coincides with thenotion of ϕ s -equilibrium state for ( A , . . . , A N ) introduced in the introduction.Our mechanism for studying equilibrium states in this article will be the followingresult which was given as [4, Corollary 2.2]: Theorem 4.
Let ℓ ≥ and N ≥ . For each j = 1 , . . . , ℓ let d j ≥ and β j > , andlet ( A ( j )1 , . . . , A ( j ) N ) ∈ GL d j ( R ) N be strongly irreducible. Define a submultiplicativepotential Φ : Σ ∗ N → (0 , + ∞ ) by Φ( i ) := ℓ Y j =1 (cid:13)(cid:13)(cid:13) A ( j ) i (cid:13)(cid:13)(cid:13) β j for all i ∈ Σ ∗ N . Then there exists a unique equilibrium state µ for Φ . It is ergodic,and there exists a constant C > such that C − Φ( i ) ≤ e | i | P (Φ) µ ([ i ]) ≤ C Φ( i ) for every i ∈ Σ ∗ N . N AFFINE IFS WITH TWO MEASURES OF MAXIMAL DIMENSION 11 Proof of Theorem 3: the algebraic part
In this section we prove that the tuple ( A , . . . , A N ) considered in Theorem 3 isstrongly irreducible and also show that ( B ∧ , . . . , B ∧ N ) is strongly irreducible, whichwill be needed later in the proof. Specifically we prove the following statement: Proposition 4.1.
Let ( B , . . . , B N ) ∈ GL d ( R ) N and ι be as in Theorem 3. Then ( B ⊗ B ι (1) , . . . , B N ⊗ B ι ( N ) ) is strongly irreducible, and for each k = 1 , . . . , d thetuple ( B ∧ k , . . . , B ∧ kN ) is strongly irreducible. This result will follow by combining the various lemmas given subsequently. Webegin with recalling some general results.
Lemma 4.2.
Let G be a linear algebraic group and suppose that π : G → GL( V ) is a strongly irreducible representation. If Γ is a Zariski-dense subsemigroup of G then π | Γ is a strongly irreducible representation.Proof. By a previous remark on irreducibility of the (Zariski) connected component G o ( § π | Γ ∩ G o is an irreducible representation. Thesemigroup Γ ∩ G o is clearly Zariski dense in G o and the latter acts irreduciblyon V . Since the property of preserving a subspace can be expressed in terms ofpolynomial equations, the same is true of Γ ∩ G o by direct appeal to the definitionof Zariski topology. (cid:3) We also need the following classical fact (for a proof see e.g. [8]):
Lemma 4.3.
Let φ : PGL d ( R ) → PGL d ( R ) be a Lie group automorphism. Then φ has one of the following two forms: either there exists x ∈ PGL d ( R ) such that φ ( g ) = xgx − for all g ∈ PGL d ( R ) , or there exists x ∈ PGL d ( R ) such that φ ( g ) = x ( g ⊤ ) − x − for every g ∈ PGL d ( R ) . In combination with Lemma 4.2 and Lemma 4.6 below, the following lemmaproves the first statement of Proposition 4.1.
Lemma 4.4.
Let ( B , . . . , B N ) ∈ GL d ( R ) N and ι be as in Theorem 3. Then thesubsemigroup of PGL d ( R ) × PGL d ( R ) generated by { ( ¯ B i , ¯ B ι ( i ) ) : 1 ≤ i ≤ N } isZariski dense in PGL d ( R ) × PGL d ( R ) .Proof. Let Γ denote the subsemigroup of PGL d ( R ) × PGL d ( R ) generated by theset { ( g i , g ι ( i ) ) : 1 ≤ i ≤ N } . Recall that the Zariski closure of Γ, which we denoteby G , is a linear algebraic group. Let π , π : PGL d ( R ) × PGL d ( R ) → PGL d ( R )denote the projections onto the first and second co-ordinates respectively. Sincethe subsemigroup of PGL d ( R ) generated by g , . . . , g N includes a Zariski densesubsemigroup of PGL d ( R ) it follows that π ( G ) = π ( G ) = PGL d ( R ). Define H := ker π ⊆ { id } × PGL d ( R )and H := ker π ⊆ PGL d ( R ) × { id } and write H = { id } × N and H = N × { id } . Obviously H and H are normalsubgroups of G and are closed in the Zariski topology. Using the surjectivity of theprojections π and π it is not difficult to deduce that N and N are also normalsubgroups of PGL d ( R ). Consider the map G → (PGL d ( R ) /N ) × (PGL d ( R ) /N )defined by ( g, h ) ( gN , hN ). By Goursat’s lemma the image of this map is thegraph of a linear group isomorphism φ : PGL d ( R ) /N → PGL d ( R ) /N . Our objective is to show that N = N = PGL d ( R ); once this has been shown itwill follow immediately that G contains the groups { id }× PGL d ( R ) and PGL d ( R ) ×{ id } , and hence must equal PGL d ( R ) × PGL d ( R ). But the only Zariski-closednormal subgroups of PGL d ( R ) are { id } and PGL d ( R ) itself. Together with theexistence of the isomorphism φ this implies that necessarily N = N since the twopossible quotients of PGL d ( R ) by its normal subgroups are trivially non-isomorphic.We must therefore eliminate the possibility that N = N = { id } .Suppose for a contradiction that N = N = { id } . In this case we have G = { ( g, φ ( g )) : g ∈ PGL d ( R ) } where φ : PGL d ( R ) → PGL d ( R ) is an isomorphism oflinear algebraic groups, and in particular is a Lie group isomorphism. By Lemma4.3, either there exists x ∈ PGL d ( R ) such that φ ( g ) = xgx − for all g ∈ PGL d ( R ),or there exists x ∈ PGL d ( R ) such that φ ( g ) = x ( g ⊤ ) − x − for all g ∈ PGL d ( R );but hypothesis (i) of Theorem 3 excludes the first possibility and hypothesis (ii)excludes the second. The proof is complete. (cid:3) In the following two expository results, we note two standard facts and providebrief a proof for the convenience of the readers.
Lemma 4.5.
Let ≤ k ≤ d . Then the representation π : GL d ( R ) → GL( ∧ k R d ) defined by π ( g ) := g ∧ k is strongly irreducible.Proof. It suffices to consider the restriction of π to SL d ( R ) and by connectednessof SL d ( R ), it suffices to show that this restriction of π is irreducible. To see this,note that any irreducible non-trivial SL d ( R )-submodule (i.e. SL d ( R )-invariant non-trivial subspace) W of ∧ k R d is a direct sum of irreducible A -submodules, where A is the diagonal subgroup of SL d ( R ). But any irreducible A -submodule of ∧ d R d isgiven by R ( e i ∧ · · · ∧ e i k ) where e i ’s is the canonical basis of R d . Since SL d ( R ) actstransitively on these pure wedge vectors, we have W = ∧ k R d proving the claim. (cid:3) One similarly deduces the following
Lemma 4.6.
Let d ≥ . Then the representation π : GL d ( R ) × GL d ( R ) → GL d ( R ) defined by π ( g, h ) := g ⊗ h is strongly irreducible. (cid:3) Proof of Theorem 3: the analytic part
Fix N , d , ι , ( B , . . . , B N ) ∈ GL d ( R ) N , ( A , . . . , A N ) ∈ GL d ( R ) N and s ∈ (1 , ∪ [ d − , d −
1) as in the statement of Theorem 3. We have A i := B i ⊗ B ι ( i ) for every i = 1 , . . . , N and it is clear that ι ( ij ) = ι ( i ) ι ( j ) and A i = B i ⊗ B ι ( i ) for every i , j ∈ Σ ∗ N . We claim that without loss of generality we may make theadditional assumption 1 < s ≤
2. To prove this claim we adapt an argument from[23, § s ∈ [ d − , d −
1) let us define ( B ′ , . . . , B ′ N ) ∈ GL d ( R ) N by B ′ i := | det B i | d − s (cid:0) B − i (cid:1) ⊤ , define a tuple ( A ′ , . . . , A ′ N ) by A ′ i := B ′ i ⊗ B ′ ι ( i ) = | det A i | d − s (cid:0) A − i (cid:1) ⊤ N AFFINE IFS WITH TWO MEASURES OF MAXIMAL DIMENSION 13 and define s ′ := d − s ∈ (1 , B ′ , . . . , B ′ N ), ( A ′ , . . . , A ′ N ) and s ′ . Furthermore we have ϕ s ′ ( A ′ i ) = σ ( A ′ i ) σ ( A ′ i ) s ′ − = σ (cid:16) | det A i | s ′ (cid:0) A − i (cid:1) ⊤ (cid:17) σ (cid:16) | det A i | s ′ (cid:0) A − i (cid:1) ⊤ (cid:17) s ′ − = | det A i | σ (cid:0) A − i (cid:1) σ (cid:0) A − i (cid:1) s ′ − = | det A i | σ d ( A i ) − σ d − ( A i ) s − d = σ ( A i ) · · · σ d − ( A i ) s − ( d − = ϕ s ( A i )for all i ∈ Σ ∗ N , which implies that the ϕ s -equilibrium states of ( A , . . . , A N ) areprecisely the ϕ s ′ -equilibrium states of ( A ′ , . . . , A ′ N ). Since 1 < s ′ ≤ < s ≤ < s ≤ (1) , Φ (2) : Σ ∗ N → (0 , + ∞ ) byΦ( i ) := ϕ s ( A i ) = σ ( A i ) σ ( A i ) s − = k A i k − s (cid:13)(cid:13) A ∧ i (cid:13)(cid:13) s − , Φ (1) ( i ) := σ ( B i ) s σ ( B ι ( i ) ) σ ( B ι ( i ) ) s − = k B i k s (cid:13)(cid:13) B ι ( i ) (cid:13)(cid:13) − s (cid:13)(cid:13)(cid:13) B ∧ ι ( i ) (cid:13)(cid:13)(cid:13) s − , Φ (2) ( i ) := Φ (1) ( ι ( i )) = σ ( B ι ( i ) ) s σ ( B i ) σ ( B i ) s − = (cid:13)(cid:13) B ι ( i ) (cid:13)(cid:13) s k B i k − s (cid:13)(cid:13) B ∧ i (cid:13)(cid:13) s − . It is clear that each is a submultiplicative potential. By Proposition 4.1, the tuple( B ∧ k , . . . , B ∧ kN ) is strongly irreducible for each k = 1 , . . . , d − B ∧ kι (1) , . . . , B ∧ kι ( N ) ). Hence the conditions of Theorem 4 are met by Φ (1) and byΦ (2) and each has a unique equilibrium state. We denote these equilibrium statesrespectively by µ and µ .We observed in § A, B ∈ GL d ( R ) the singular values of A ⊗ B are precisely the numbers σ i ( A ) σ j ( B ) where 1 ≤ i, j ≤ d . In particular the largestsingular value is σ ( A ) σ ( B ) and the second-largest is necessarily either σ ( A ) σ ( B )or σ ( A ) σ ( B ). This simple observation implies the identityΦ( i ) = σ ( A i ) σ ( A i ) s − (4) = σ (cid:0) B i ⊗ B ι ( i ) (cid:1) σ (cid:0) B i ⊗ B ι ( i ) (cid:1) s − = σ ( B i ) σ ( B ι ( i ) ) max (cid:8) σ ( B i ) s − σ ( B ι ( i ) ) s − , σ ( B ι ( i ) ) s − σ ( B i ) s − (cid:9) = max n Φ (1) ( i ) , Φ (2) ( i ) o which is the fundamental observation around which the whole of Theorem 3 is built.We claim that P (Φ) = P (Φ (1) ) = P (Φ (2) ). Since for every i ∈ Σ ∗ N we have X | i | = n Φ s ( i ) = X | i | = n max n Φ (1) ( i ) , Φ (2) ( i ) o ≤ X | i | = n Φ (1) ( i ) + X | i | = n Φ (2) ( i ) and X | i | = n Φ s ( i ) = X | i | = n max n Φ (1) ( i ) , Φ (2) ( i ) o ≥ X | i | = n Φ (1) ( i ) + X | i | = n Φ (2) ( i ) it follows by direct consideration of the definition of the pressure that P (Φ s ) =max { P (Φ (1) ) , P (Φ (2) ) } . To prove the claim it is therefore sufficient to show that P (Φ (1) ) = P (Φ (2) ). But for every n ≥ X | i | = n Φ (1) ( i ) = X | i | = n Φ (2) ( ι ( i )) = X | i | = n Φ (2) ( i )where the first equation follows from the definition of Φ (2) and the second fromthe fact that ι : { , . . . , N } n → { , . . . , N } n is a bijection. The equation P (Φ (1) ) = P (Φ (2) ) follows directly and the claim is proved.We now claim that the measures µ and µ are precisely the ergodic equilibriumstates of Φ, which is to say the ergodic ϕ s -equilibrium states of ( A , . . . , A N ). Tosee this suppose that µ ∈ M σ is an arbitrary ergodic measure on Σ N . By thesubadditive ergodic theorem we havelim n →∞ n log Φ s ( x | n ) = Λ (Φ s , µ ) , lim n →∞ n log Φ (1) ( x | n ) = Λ (cid:16) Φ (1) , µ (cid:17) , lim n →∞ n log Φ (2) ( x | n ) = Λ (cid:16) Φ (2) , µ (cid:17) for µ -a.e. x ∈ Σ N . In particular for any such x we haveΛ(Φ s , µ ) = lim n →∞ n log Φ s ( x | n )= lim n →∞ n log max n Φ (1) ( x | n ) , Φ (2) ( x | n ) o = max (cid:26) lim n →∞ n log Φ (1) ( x | n ) , lim n →∞ n log Φ (2) ( x | n ) (cid:27) = max n Λ(Φ (1) , µ ) , Λ(Φ (2) , µ ) o where we have used (4) in the second equation. We have shown that Λ(Φ s , µ ) =max { Λ(Φ (1) , µ ) , Λ(Φ (2) , µ ) } for every ergodic measure µ . Hence if µ is an ergodicequilibrium state of Φ (1) then P (Φ (1) ) = P (Φ s ) ≥ h ( µ ) + Λ(Φ s , µ ) ≥ h ( µ ) + Λ(Φ (1) , µ ) = P (Φ (1) )where the first inequality follows from the subadditive variational principle. Itfollows that P (Φ s ) = h ( µ ) + Λ(Φ s , µ ) and therefore µ is an equilibrium state of Φ s .Similarly if µ is an ergodic equilibrium state of Φ (2) then it is an equilibrium stateof Φ s . On the other hand if µ is an ergodic equilibrium state of Φ s then eitherΛ(Φ s , µ ) = Λ(Φ (1) , µ ) so that P (Φ (1) ) = P (Φ s ) = h ( µ ) + Λ(Φ s , µ ) = h ( µ ) + Λ(Φ (1) , µ )and µ is an equilibrium state of Φ (1) , or Λ(Φ s , µ ) = Λ(Φ (2) , µ ) so that P (Φ (2) ) = P (Φ s ) = h ( µ ) + Λ(Φ s , µ ) = h ( µ ) + Λ(Φ (2) , µ )and µ is an equilibrium state of Φ (2) . This proves the claim.We have shown that the ergodic ϕ s -equilibrium states of ( A , . . . , A N ) are pre-cisely µ and µ , so to complete the proof of the theorem it remains only to showthat these two measures are distinct. By Theorem 4 there exists C > C − Φ (1) ( i ) ≤ e | i | P (Φ s ) µ ([ i ]) = e | i | P (Φ (1) ) µ ([ i ]) ≤ C Φ (1) ( i ) N AFFINE IFS WITH TWO MEASURES OF MAXIMAL DIMENSION 15 and C − Φ (2) ( i ) ≤ e | i | P (Φ s ) µ ([ i ]) = e | i | P (Φ (2) ) µ ([ i ]) ≤ C Φ (2) ( i )for all i ∈ Σ ∗ N . (This shows in particular that both measures are fully supported onΣ N , since it implies that every cylinder set has nonzero measure and since cylindersets form a basis for the topology of Σ N .) If it were the case that µ = µ thenthese inequalities would imply the relation(5) C − ≤ Φ (1) ( i )Φ (2) ( i ) ≤ C for all i ∈ Σ ∗ .By Lemma 4.4 the subsemigroup of PGL d ( R ) × PGL d ( R ) generated by the pairs( ¯ B , ¯ B ι (1) ) , . . . , ( ¯ B N , ¯ B ι ( N ) ) is Zariski dense in PGL d ( R ) × PGL d ( R ), so by Propo-sition 3.1 there exists i ∈ Σ ∗ N such that λ ( B i ) /λ ( B i ) = λ ( B ι ( i ) ) /λ ( B ι ( i ) ).If we have µ = µ then applying (5) to i := i n now yields C − ≤ σ (cid:0) B n i (cid:1) σ (cid:16) B nι ( i ) (cid:17) σ (cid:16) B nι ( i ) (cid:17) σ (cid:0) B n i (cid:1) s − ≤ C for all n ≥
1. Since s − = 0 it follows thatlim n →∞ σ (cid:0) B n i (cid:1) σ (cid:16) B nι ( i ) (cid:17) σ (cid:16) B nι ( i ) (cid:17) σ (cid:0) B n i (cid:1) n = 1 , but since clearlylim n →∞ σ (cid:0) B n i (cid:1) σ (cid:16) B nι ( i ) (cid:17) σ (cid:16) B nι ( i ) (cid:17) σ (cid:0) B n i (cid:1) n = lim n →∞ σ (cid:0) B n i (cid:1) /σ (cid:0) B n i (cid:1) σ (cid:16) B nι ( i ) (cid:17) /σ (cid:16) B nι ( i ) (cid:17) n = λ ( B i ) /λ ( B i ) λ ( B ι ( i ) ) /λ ( B ι ( i ) ) = 1using (3), this is impossible. We conclude that µ and µ must be distinct, and thetheorem is proved.6. Two further perspectives on Theorem 3
Readers of this article may be aware of the following sufficient condition for theuniqueness of equilibrium states of submultiplicative potentials: a submultiplicativepotential Φ : Σ ∗ N → (0 , + ∞ ) is called quasi-multiplicative if there exist δ > n ≥ | k |≤ n Φ( ikj ) ≥ δ Φ( i )Φ( j )for all i , j ∈ Σ ∗ N . Every quasi-multiplicative submultiplicative potential has aunique equilibrium state (see for example [11, 21]). Theorem 3 therefore impliesthat the potential Φ s ( i ) := ϕ s ( A i ) fails to be quasi-multiplicative for 1 < s < A , A are as defined in the statement of that theorem.This failure of quasi-multiplicativity can be demonstrated directly in the follow-ing manner. For simplicity of exposition we suppose in this section that α > α >
0. Let us consider the second exterior powers of A and A . In the basis e ∧ e , e ∧ e , e ∧ e − e ∧ e , e ∧ e , e ∧ e , e ∧ e + e ∧ e for ∧ R the matrix of A ∧ is α α α α α α cos θ α α sin θ − α α cos θ sin θ α α sin θ α α cos θ α α cos θ sin θ α α cos θ sin θ − α α cos θ sin θ α λ (cos θ − sin θ ) and that of A ∧ is α α cos θ α α sin θ − α α cos θ sin θ α α sin θ α α cos θ α α cos θ sin θ α α cos θ sin θ − α α cos θ sin θ α λ (cos θ − sin θ ) 0 0 00 0 0 α α
00 0 0 0 0 α α . In particular if we define B := α α
00 0 α α ,B := α α cos θ α α sin θ − α α cos θ sin θα α sin θ α α cos θ α α cos θ sin θα α cos θ sin θ − α α cos θ sin θ α λ (cos θ − sin θ ) , then we have A ∧ = (cid:18) B B (cid:19) , A ∧ = (cid:18) B B (cid:19) in the aforementioned basis. The eigenvalues of A are α e iθ , α e − iθ , α e iθ and α e − iθ . The eigenvalues of A ∧ are the products of pairs of distinct eigenvalues of A and hence are α α e iθ , α α e − iθ , α α (with multiplicity two), α and α .Since B obviously has eigenvalues α , α α and α it follows that the remainingeigenvalues of A ∧ pertain to B , and in particular every eigenvalue of B hasabsolute value α α . Thus λ ( B ) = α and λ ( B ) = α α . In particular if i isthe word consisting of n ones and j the word consisting of n twos, the linear map A ∧ i A ∧ k A ∧ j has the form (cid:18) B n B n (cid:19) (cid:18) B k D k (cid:19) (cid:18) B n B n (cid:19) . Since k B n k = α n and k B n k ≃ α n α n the norm of this product is necessarilybounded above by approximately α n α n max {k B k k , k D k k} ≪ α n = k A ∧ i k ·k A ∧ j k .Thus the failure of quasi-multiplicativity of ϕ , and more broadly of ϕ s when1 < s <
3, can be seen to arise from the splitting of ∧ R into two invariantthree-dimensional subspaces on which the actions of A ∧ and A ∧ are substantiallydifferent.However, this description of the mechanism of Theorem 1 in terms of bare-hands algebraic computation is unsatisfying insofar as it lacks any reference to the a priori more geometrically relevant action on R . We therefore offer the following N AFFINE IFS WITH TWO MEASURES OF MAXIMAL DIMENSION 17 more geometric explanation of the failure of quasi-multiplicativity for the matricesdefined in Theorem 3. Let us define A -invariant subspaces of R by U := ab : a, b ∈ R , U := ab : a, b ∈ R and A -invariant subspaces of R by V := a b : a, b ∈ R , V := a b : a, b ∈ R . We observe that the two larger singular values of A n , both being equal to α n , arisefrom its action on U , whereas the two smaller singular values are both equal to α n and arise from the invariant subspace U . Similarly the two largest singularvalues of A n arise from its action on V and the two smaller singular values fromits action on V . In order to find a word k such that the first two singular valuesof A n A k A n are both approximately α n , therefore, the matrix A k would have totranspose the subspace V into a position where its angle with U is bounded awayfrom perpendicularity by an a priori amount, and in particular where it does notintersect U ⊥ = U . But this is impossible: if X , X ∈ GL ( R ) are arbitrarymatrices then ( X ⊗ X ) V necessarily intersects U nontrivially. To see this let P denote the 2 × V = ( I ⊗ P ) R and U = ker( P ⊗ I ). If ( X ⊗ X ) V did not intersect U then the image of ( X ⊗ X )( I ⊗ P ) would not intersect the kernel of P ⊗ I and thematrices ( P ⊗ I )( X ⊗ X )( I ⊗ P ) = P X ⊗ X P and ( X ⊗ X )( I ⊗ P ) = X ⊗ X P would have the same rank; but the first matrix has rank one and the second hasrank two, since the rank of the Kronecker product of two matrices is equal to theproduct of their ranks. This argument moreover shows that U ∩ ( X ⊗ X ) V hasdimension precisely 1. Thus no element of GL ( R ) ⊗ GL ( R ), and in particular noelement of the group generated by A and A , can move V into a position whereits intersection with U is anything other than one-dimensional.7. Proof of Theorem 2
Suppose that ( A , . . . , A ) ∈ GL ( R ) satisfies the hypotheses of Theorem 3.Each A i is the Kronecker product of a matrix with singular values α and α anda matrix with singular values 1 and 1. Hence each A i has singular values α , α , α and α . Since for every B ∈ GL ( R ) we have σ ( B ) ≥ ( σ ( B ) σ ( B ) σ ( B ) σ ( B )) = | det B | it follows that for each n ≥ X | i | = n ϕ ( A i ) ≥ X | i | = n | det A i | = X | i | = n ( α α ) n = 4 n ( α α ) n which implies that P ϕ ( A , . . . , A ) ≥
12 log (16 α α ) > and therefore dim aff ( A , . . . , A ) >
1. For each n ≥ X | i | = n ϕ ( A i ) ≤ X i =1 ϕ ( A i ) ! n = (cid:0) α (cid:1) n < (cid:16) q (cid:17) n where we have made use of the submultiplicativity property ϕ ( AB ) ≤ ϕ ( A ) ϕ ( B )in the first inequality. Hence P ϕ ( A , . . . , A ) ≤ α ) <
21 + q < aff ( A , . . . , A ) ∈ (1 ,
2) as claimed. For the remainder of the proof define s := dim aff ( A , . . . , A ). By Theorem 3 there exist precisely two distinct ergodic ϕ s -equilibrium states µ , µ for ( A , . . . , A ) and these measures have Lyapunovdimension equal to dim aff ( A , . . . , A ) and are fully supported on Σ .Consider now the iterated function system defined by T i x := A i x + v i for all x ∈ R , where ( v , . . . , v ) ∈ ( R ) is to be determined. We claim that the set of all( v , . . . , v ) ∈ ( R ) such that ( T , . . . , T ) satisfies the strong separation conditionhas positive Lebesgue measure. To do this we will show that the set of all suchtuples ( v , . . . , v ) contains a nonempty open set. Define v := √ , v := − √ , v := − √ , v := − − √ and observe that every two distinct vectors v i , v j are separated by a Euclideandistance of 2. Define X ⊂ R to be the closed origin-centred Euclidean ball ofradius 1 + q . For each i = 1 , . . . , v i is a subset of X , and these subsets do not intersect one another. If wedefine T i x := A i x + v i for all x ∈ R and i = 1 , . . . , ≤ i ≤ k A i k = α < / (1 + q ), each of the sets T i X is contained in the open Euclidean ballof radius 1 and centre v i . Since these balls are pairwise disjoint, the sets T i X are pairwise disjoint subsets of X and therefore ( T , . . . , T ) satisfies the strongseparation condition. It is clear that for every ( v ′ , . . . , v ′ ) sufficiently close to( v , . . . , v ) the four images of X are again contained in the open Euclidean ballsof radius 1 and centre v i , so the strong separation condition remains satisfied forany ( v ′ , . . . , v ′ ) sufficiently close to ( v , . . . , v ). The claim is proved.We may now prove the theorem. Sincemax ≤ i ≤ k A i k = α <
11 + q < , by [18, Theorem 1.9] for Lebesgue a.e. ( v , . . . , v ) ∈ ( R ) the measures m :=Π ∗ µ and m := Π ∗ µ both have dimension equal to their Lyapunov dimension,which is dim aff ( A , . . . , A ). It follows in particular that there is a positive-measureset of tuples ( v , . . . , v ) such that the strong separation condition is satisfiedand additionally m := Π ∗ µ and m := Π ∗ µ both have dimension equal todim aff ( A , . . . , A ). When ( T , . . . , T ) satisfies the strong separation condition we N AFFINE IFS WITH TWO MEASURES OF MAXIMAL DIMENSION 19 note that Π defines a homeomorphism from Σ to the attractor and therefore Π ∗ µ and Π ∗ µ are mutually singular if and only if µ and µ are; but these two mea-sures are distinct ergodic shift-invariant measures on Σ , and such measures areautomatically mutually singular. Since µ and µ are fully supported on Σ , Π ∗ µ and Π ∗ µ are fully supported on the attractor Π(Σ ). The proof is complete.8. Acknowledgements
The research of I.D. Morris was partially supported by the Leverhulme Trust(Research Project Grant RPG-2016-194). C.S. is supported by SNF grant 178958.The authors thank A. K¨aenm¨aki for several helpful bibliographical suggestions.
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I. D. Morris: Department of Mathematics, University of Surrey, Guildford, GU27XH, UK
E-mail address : [email protected] C. Sert: Department Mathematik, ETH Z¨urich, R¨amistrasse 101, 8092, Z¨urich, Switzer-land
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