CCALCULATING BOX DIMENSION WITH THE METHOD OF TYPES
ISTVÁN KOLOSSVÁRY
Abstract.
This paper presents a general procedure based on using the method of typesto calculate the box dimension of sets. The approach unifies and simplifies multiplebox counting arguments. In particular, we use it to generalize the formula for the boxdimension of self-affine carpets of Gatzouras–Lalley and of Barański type to their higherdimensional sponge analogues. In addition to a closed form, we also obtain a variationalformula which resembles the Ledrappier–Young formula for Hausdorff dimension. Introduction
The box dimension of a subset Λ of R d is defined as the limit dim B Λ = lim δ → log N δ (Λ) − log δ , where N δ (Λ) denotes the minimum number of d -dimensional boxes of sidelength δ neededto cover Λ . More precisely, one takes the lim inf and lim sup to get the lower and upperbox dimensions, respectively, but for all sets considered in this paper the limit exists. Themain aim of this paper is to provide a unified approach based on the ‘method of types’to calculate the box dimension. The effectiveness of the argument is demonstrated onvarious families of self-affine sponges in R d . Thanks to the flexibility of the method, onecan hope to apply it to more complicated constructions in the future and gain additionalinsight as to when does the Hausdorff and box dimension of a set differ.The outline of the general argument goes as follows. Assume that at scale δ we are givena collection B δ of sets of diameter δ that is a cover of Λ with cardinality B δ = N δ (Λ) .The first step is to partition B δ into type classes according to some rule. Let T δ denotethe set of all possible types and T ∗ δ be the class with the most elements. Then T ∗ δ ≤ N δ (Λ) ≤ T ∗ δ · T δ . (1.1)If T δ = o ( δ − ) and T ∗ δ has lower and upper bounds such that after taking logarithm,dividing by − log δ and letting δ → we get the same limit for the lower and upperbound, then the growth rate of T ∗ δ essentially determines dim B Λ . We refer to T ∗ δ as the dominant box counting class at scale δ and the type it corresponds to as the dominantbox counting type . The optimal δ -cover of all sets considered here have a clear symbolicrepresentation which allows us to apply the method of types with proper adaptations.The method of types is an elementary tool to give good estimates for the number ofsequences of a given length with prescribed digit frequencies where the digits come from afinite alphabet. It has roots dating back to works of Boltzmann, Hoeffding, Sanov or Shan-non to name a few. It was later systematically developed to study discrete memorylesssystems in information theory and has since found applications in for example hypothesistesting, combinatorics, or large deviations, see [7, 8] for some background. Mathematics Subject Classification.
Primary 28A80 Secondary 37C45
Key words and phrases. box dimension, method of types, self-affine sponge, Ledrappier–Young formula. a r X i v : . [ m a t h . M G ] F e b ALCULATING BOX DIMENSION WITH THE METHOD OF TYPES 2
Let us recall the basic notions and facts from the method of types that we will use.Let I = { , . . . , N } be the finite alphabet and Σ = I N be the set of all infinite sequences i = ( i i . . . ) . For any n ∈ N , we use the notation i | n = i . . . i n .The type of i at level n is the empirical vector τ n ( i ) = 1 n (cid:0) { ≤ (cid:96) ≤ n : i (cid:96) = 1 } , . . . , { ≤ (cid:96) ≤ n : i (cid:96) = N } (cid:1) , that is τ n ( i ) just tabulates the relative frequency of each symbol of I in i | n . The set ofall possible types at level n is T n = (cid:8) p : there exists i ∈ Σ such that p = τ n ( i ) (cid:9) . Let P N denote all probability vectors p = ( p , . . . , p N ) . Observe that as n → ∞ the set T n becomes dense in P N . A cylinder set is defined as [ i . . . i n ] = { j ∈ Σ : j | n = ( i . . . i n ) } .Then B n = { [ i . . . i n ] : ( i , . . . , i n ) ∈ I n } gives a partition of Σ . We simply identify theelements of B n with finite sequences ( i , . . . , i n ) . The type class of p ∈ T n is the set T n ( p ) = (cid:8) ( i , . . . , i n ) ∈ B n : τ n (( i , . . . , i n )) = p (cid:9) . Throughout, we will only use the following two simple facts from the method of types: T n ≤ ( n + 1) N (1.2)and ( n + 1) − N e nH ( p ) ≤ T n ( p ) ≤ e nH ( p ) (1.3)for every p ∈ T n , where H ( p ) = − (cid:80) i p i log p i is the entropy of the probability vector p . For a proof of these elementary facts, we refer to [10, Lemmas 2.1.2 and 2.1.8].Inequality (1.2) implies that it is indeed enough to consider the dominant box countingclass, while (1.3) ensures that we get matching lower and upper bounds for dim B Λ . Main contribution.
The idea of picking out classes of words from a code space in someoptimal way has been used before, however, the author is unaware of it being formalisedin such a general context previously to calculate the box dimension. The main result is todetermine the box dimension of Gatzouras–Lalley and of Barański sponges in arbitrarydimensions. The key technical contribution is to adapt (1.2) and (1.3) to more complicatedsettings where multi-dimensional types are used for sequences of varying lengths.
Structure of paper.
We begin by demonstrating the skeleton of the argument in thesimplest case of self-similar sets satisfying the open set condition which we later buildupon. Section 2 provides a brief overview of related literature on dimensions of self-affinesponges and states our main results, see Theorems 2.3 and 2.5. The proofs are presentedin Sections 3 and 4. In Section 5, we discuss possible generalizations of the approach andconnections with the Ledrappier–Young formula for the Hausdorff dimension.1.1.
Self-similar sets.
In general, an iterated function system (IFS) on R d is a finitefamily S = { S , . . . , S N } of contractions S i : R d → R d . The IFS determines a unique,non-empty compact set Λ , called the attractor , that satisfies the relation Λ = N (cid:91) i =1 S i (Λ) . In particular, if the maps are similarities, i.e. for every x, y ∈ R d (cid:107) S i ( x ) − S i ( y ) (cid:107) = λ i (cid:107) x − y (cid:107) , where < λ i < is the contraction ratio of S i , ALCULATING BOX DIMENSION WITH THE METHOD OF TYPES 3 then the IFS and its attractor are called self-similar . The IFS satisfies the open setcondition (OSC) if there exists a non-empty open set V such that S i ( V ) ⊆ V and S i ( V ) ∩ S j ( V ) = ∅ for i (cid:54) = j. (1.4)It is well-known that a self-similar set has equal Hausdorff and box dimension, moreover,if the OSC is also satisfied then the dimension is given by the Hutchinson formula, i.e.the unique solution s , often called the similarity dimension , to the equation N (cid:88) i =1 λ si = 1 . (1.5)We now sketch the argument for deriving the box dimension using the method of types.Let λ min := min i λ i , λ max := max i λ i and denote the Lyapunov-exponent with respectto p by χ ( p ) := − (cid:80) i p i log λ i . Throughout, we use the convention that a (cid:47) b if there isan independent constant C such that a ≤ Cb , similarly a (cid:39) b if a ≥ Cb and a ≈ b if a (cid:47) b and a (cid:39) b . The set of finite length words from the alphabet I = { , . . . , N } is denotedby Σ ∗ and the length of ı ∈ Σ ∗ is | ı | .On the symbolic space Σ , the δ -stopping of i ∈ Σ is the unique integer L δ ( i ) such that L δ ( i ) (cid:89) (cid:96) =1 λ (cid:96) ≤ δ < L δ ( i ) − (cid:89) (cid:96) =1 λ (cid:96) , i.e. L δ ( i ) ≈ log δ L δ ( i ) (cid:80) L δ ( i ) (cid:96) =1 log λ (cid:96) . (1.6)The symbolic δ -approximate ball containing i ∈ Σ is B δ ( i ) = (cid:8) j ∈ Σ : i | L δ ( i ) = j | L δ ( i ) (cid:9) , which we identify with the finite sequence ( i , i , . . . , i L δ ( i ) ) . The name comes from thefact that the image π ( B δ ( i )) on Λ has diameter ≈ δ , where π : Σ → Λ is the naturalprojection defined by π ( i ) = lim n →∞ S i ◦ S i ◦ . . . ◦ S i n (0) . The symbolic Moran-cover of Σ at scale δ is B δ = (cid:8) ı ∈ Σ ∗ : ( ∀ i ∈ [ ı ]) L δ ( i ) = | ı | (cid:9) . Itis straightforward that { [ ı ] } ı ∈B δ is a partition of Σ . Since π is surjective, the collection { π ( B δ ( ı )) } ı ∈B δ gives a δ -cover of Λ . Moreover, the OSC implies that N δ (Λ) ≈ B δ . As aresult, it is enough to work with the finite sequences ı ∈ B δ .Since L δ ( i ) depends on i , we adapt the method of types to handle sequences of differentlengths simultaneously. Similarly as before, the type of i ∈ Σ at scale δ is the empiricalvector τ δ ( i ) = 1 L δ ( i ) (cid:0) { ≤ (cid:96) ≤ L δ ( i ) : i (cid:96) = 1 } , . . . , { ≤ (cid:96) ≤ L δ ( i ) : i (cid:96) = N } (cid:1) . The set of all possible types at scale δ is T δ = (cid:8) p : there exists ı ∈ B δ such that p = τ δ ( ı ) (cid:9) and the type class of p ∈ T δ is the set T δ ( p ) = (cid:8) ı ∈ B δ : τ δ ( ı ) = p (cid:9) . For fixed p ∈ T δ , observe that within its type class it follows from (1.6) that L δ ( i ) ≈− log δ/χ ( p ) for all i such that i ∈ [ ı ] for some ı ∈ T δ ( p ) . Thus, (1.3) implies that (cid:18) − log δχ ( p ) + 1 (cid:19) − N δ − H ( p ) /χ ( p ) (cid:47) T δ ( p ) (cid:47) δ − H ( p ) /χ ( p ) . (1.7) ALCULATING BOX DIMENSION WITH THE METHOD OF TYPES 4
To bound T δ from above, note that log δ/ log λ min (cid:47) L δ ( i ) (cid:47) log δ/ log λ max . Thenfrom (1.2) the following crude upper bound follows T δ (cid:47) (cid:18) λ max − λ min (cid:19) log δ · (cid:18) log δ log λ max + 1 (cid:19) N . (1.8)Let p ∗ δ ∈ T δ denote the type for which H ( p ∗ δ ) /χ ( p ∗ δ ) = max p ∈T δ H ( p ) /χ ( p ) . Then com-bining (1.7) and (1.8) with (1.1), we obtain that δ − H ( p ∗ δ ) /χ ( p ∗ δ ) · O (cid:0) ( − log δ ) − N (cid:1) (cid:47) N δ (Λ) (cid:47) δ − H ( p ∗ δ ) /χ ( p ∗ δ ) · O (cid:0) ( − log δ ) N +1 (cid:1) . Since T δ becomes dense in P N as δ → , moreover, H ( p ) /χ ( p ) is continuous in p , weconclude that p ∗ δ → p ∗ as δ → , where p ∗ ∈ P N maximises H ( p ) /χ ( p ) over all p ∈ P N .Hence, dim B Λ = H ( p ∗ ) /χ ( p ∗ ) . To finish, a standard use of the Lagrange multipliersshows that p ∗ = ( λ s , . . . , λ sN ) . As a result, dim B Λ = s as claimed. Thus, p ∗ is thedominant box counting type and T δ ( p ∗ ) is the dominant box counting class in this case. Remark 1.1. If Λ is a homogeneous self-similar set, i.e. λ i ≡ λ , then χ ( p ) = − log λ forany p . Hence, the dominant box counting type is the uniform measure p ∗ = (1 /N, . . . , /N ) since it maximises H ( p ) with value log N , which implies that dim B Λ = s = log N/ ( − log λ ) . Main results about self-affine sponges
The main application of the method of types in this paper is to determine the boxdimension of self-affine sponges in R d of Gatzouras–Lalley and of Barański type. A self-affine set is the attractor of an IFS in which all maps have the form S i ( x ) = A i x + t i ,where A i is a contracting non-singular d × d matrix and t i ∈ R d .Loosely speaking, sponges are referred to as higher dimensional analogues of self-affinecarpet-like constructions on the plane. The key features of these constructions is theirexcessive alignment of cylinders and defining diagonal matrices. The significance of thesecarpets is that they provide explicit examples for which the various notions of dimensionare different. They are part of a very small family of exceptions, since the box andHausdorff dimensions of self-affine sets coincide in a ‘typical’ sense [11, 12] in R d and alsoin a more explicit sense [3, 22] in R .Self-affine carpets were first studied independently by Bedford [6] and McMullen [27].Their construction was generalised by Gatzouras and Lalley [21] and later by Barański [1].The various dimensions of these basic models are well understood. Most of these resultshave been generalised in different directions on the plane to constructions with over-laps [20, 26, 29], to constructions using lower triangular matrices [2, 26] or to more general‘box-like’ constructions [16, 17]. Figure 1 shows different carpet-like constructions withincreasing complexity. In each case, the shaded rectangles or parallelograms are the im-ages of [0 , under the maps of the defining IFS. The attractor is obtained by repeatedlyapplying the maps to the remaining shaded areas ad infinitum.Much less is known, however, about the dimension theory of self-affine sponges. In thesimplest case of a Bedford–McMullen (also referred to as Sierpiński) sponge, its Hausdorffand box dimensions were obtained by Kenyon and Peres [25], while its lower and Assouaddimensions by Fraser and Howroyd [18]. Feng and Hu [15] relaxed the separation conditionin case of the Hausdorff and box dimension. Perhaps the paper with the most impact is dueto Das and Simmons [9], who by calculating the Hausdorff dimension of Gatzouras–Lalleyand Barański sponges gave the first example of a set which does not have a shift invariantmeasure of maximal Hausdorff dimension, thus resolving a long standing open problem in ALCULATING BOX DIMENSION WITH THE METHOD OF TYPES 5
Figure 1.
Different carpet-like constructions. From left to right:Bedford–McMullen carpet, Barański carpet, box-like construction, trian-gular Gatzaouras–Lalley carpet with overlaps.dynamical systems. The closest related result is a recent work of Fraser and Jurga [19],where they obtain results about the box dimension of certain sponges in R generated bygeneralised permutation matrices, which contain the Gatzouras–Lalley sponges but notthe Barański type. We continue with the formal definitions and state our main results.2.1. Gatzouras–Lalley sponges.
The definition is slightly technical and needs somenotation. We begin by defining the collection of index sets I , I , . . . , I d as follows:(1) Fix an integer N ≥ and let I := { , . . . , N } ;(2) For each i ∈ I fix N ( i ) ∈ N and let I ( i ) := { , . . . , N ( i ) } , moreover, I := (cid:91) i ∈I (cid:91) i ∈I ( i ) ( i , i ); (3) Continue inductively for ≤ n ≤ d : given I n − , fix N ( i ) ∈ N for each i ∈ I n − .Let I ( i ) := { , . . . , N ( i ) } and finally I n := (cid:91) i ∈I n − (cid:91) i n ∈I ( i ) ( i, i n ) . We extensively use projections. To denote the projection of i = ( i , . . . , i n ) ∈ I n (where ≤ n ≤ d ) to its first (cid:96) ≤ n coordinates, we use the notation i ( (cid:96) ) := ( i , . . . , i (cid:96) ) . The same notation is extended to vectors v and subsets V of R n : v ( (cid:96) ) = ( v , . . . , v (cid:96) ) and V ( (cid:96) ) = (cid:83) v ∈ V v ( (cid:96) ) .We can now introduce the IFSs S , . . . , S d , where S n = { S i : R n → R n } i ∈I n is definedas S i ( x ) = A i x + t i = λ ( i (1) ) 0 . . . λ ( i ( n ) ) · x + t ( i (1) ) ... t ( i ( n ) ) . The (cid:96) -th coordinate of S i ( x ) only depends on the first (cid:96) coordinates of i . We assume that < λ ( i ) < for every ≤ n ≤ d and i ∈ I n . Without loss of generality we assume thatthe t i are chosen so that S i ([0 , n ) ⊂ [0 , n . The attractor of S n is the unique, non-emptycompact set Λ n satisfying the relation Λ n = (cid:91) i ∈I n S i (Λ n ) . ALCULATING BOX DIMENSION WITH THE METHOD OF TYPES 6
Definition 2.1.
The attractor Λ d of S d is a Gatzouras–Lalley (GL) sponge in R d if S i ((0 , d ) ∩ S j ((0 , d ) = ∅ for every i (cid:54) = j ∈ I d (2.1) and < λ ( i ( d ) ) < λ ( i ( d − ) < . . . < λ ( i (1) ) < for every i ∈ I d . (2.2) We call condition (2.1) the cuboidal open set condition (COSC) and condition (2.2) isthe coordinate ordering condition . Remark 2.2. (1) Observe that Λ ( (cid:96) ) n = Λ (cid:96) for any (cid:96) ≤ n ≤ d . In addition, if Λ d is a GL-sponge in R d , then Λ ( n ) d is a GL-sponge in R n for every ≤ n ≤ d .(2) The COSC, introduced in [19] , is the higher dimensional analogue of the rectan-gular OSC defined in [16] . The name for (2.2) is taken from [9] .(3) Condition (2.2) could be assumed for any permutation of the coordinates, as longas the permutation is the same for all i ∈ I d . We chose this to simplify notation. Theorem 2.3.
Let Λ d be a Gatzouras–Lalley sponge in R d . Then dim B Λ d = s d , where the numbers s ≤ s ≤ . . . ≤ s d are defined as the unique solutions to the equations (cid:88) i ∈I ( λ ( i )) s = 1 and (cid:88) i ∈I n (cid:0) λ ( i (1) ) (cid:1) s · n (cid:89) (cid:96) =2 (cid:0) λ ( i ( (cid:96) ) ) (cid:1) s (cid:96) − s (cid:96) − = 1 for n = 2 , . . . , d. (2.3)The equations in (2.3) naturally define probability vectors p ∗ , . . . , p ∗ d . These vectorsdefine a multi-dimensional type class, see Section 3.2, and the proof reveals that this typeclass is the dominant box counting class.The theorem in two dimensions was first proved by Gatzouras and Lalley [21], and for d = 3 it follows from a more general result of Fraser and Jurga [19]. Their arguments arecompletely different and they are completely different from the proof presented here.Besides the closed form for s d given by (2.3), we also obtain a variational formula, seeProposition 4.1 and Lemma 4.3 for details. In particular, in two dimensions, dim B Λ = max p ∈P I , p ∈P I H ( p ) χ ( p ) + (cid:18) − χ ( p ) χ ( p ) (cid:19) H ( p ) χ ( p ) , (2.4)where P I and P I denote the set of probability vectors on I and I , respectively, andthe Lyapunov exponents are χ n ( p m ) = − (cid:80) i ∈I m p m ( i ) log λ ( i ( n ) ) for ≤ n ≤ m ≤ . Theformula resembles the Ledrappier–Young formula for Hausdorff dimension, see Section 5for a detailed discussion. For the three-dimensional analogue of this formula see (4.5).2.2. Barański sponges.
The notation is slightly simpler in this case. For ≤ n ≤ d ,the index set I n := { , . . . , N n } defines the base IFS F n in coordinate n by F n := (cid:8) f n,i ( x ) = λ n ( i ) · x + t n,i (cid:9) i ∈I n , where t n,i = i − (cid:88) (cid:96) =1 λ n ( (cid:96) ) . The choice of t n,i implies that each F n satisfies the OSC (1.4) with V = (0 , . Thealphabet is a subset I ⊆ (cid:81) dn =1 I n and an element of it is i = ( i , . . . , i d ) . For a subset D ⊂ { , . . . , d } let Π( i ; D ) := ( i (cid:96) ) (cid:96) ∈ D , i.e. the coordinates of i whose indices belong to D .Similarly, Π( I ; D ) := { Π( i ; D ) : i ∈ I} . ALCULATING BOX DIMENSION WITH THE METHOD OF TYPES 7
Definition 2.4.
The IFS S = { S i } i ∈I is of Barański type if S i ( x ) = (cid:0) f ,i ( x ) , . . . , f d,i d ( x d ) (cid:1) . The attractor
Λ = (cid:83) i ∈I S i (Λ) is a Barański sponge in R d . To state the result in this case, we let
Sym( { , . . . , d } ) denote the symmetric group onthe set of coordinates { , . . . , d } and denote a permutation by σ = (cid:18) · · · dσ σ · · · σ d (cid:19) ∈ Sym( { , . . . , d } ) . (2.5) Theorem 2.5.
Let Λ d be a Barański sponge in R d . Then dim B Λ d = max σ ∈ Sym( { ,...,d } ) s d ( σ ) , where for a fixed σ = ( σ , . . . , σ d ) ∈ Sym( { , . . . d } ) the numbers s ( σ ) ≤ s ( σ ) ≤ . . . ≤ s d ( σ ) are defined as the unique solutions to the equations (cid:88) i ∈ Π( I ; { σ } ) ( λ σ ( i )) s ( σ ) = 1 and (cid:88) ( i ...,i n ) ∈ Π( I ; { σ ,...,σ n } ) (cid:0) λ σ ( i ) (cid:1) s ( σ ) · n (cid:89) (cid:96) =2 (cid:0) λ σ (cid:96) ( i (cid:96) ) (cid:1) s (cid:96) ( σ ) − s (cid:96) − ( σ ) = 1 for n = 2 , . . . , d. Essentially the theorem states that for every possible ordering of the coordinates, onehas to calculate the numbers s ( σ ) , . . . , s d ( σ ) like in the GL case and then take a max-imum. The reason why all orderings are considered is because the coordinate orderingcondition (2.2) is not assumed for Barański sponges. The theorem in two dimensions wasfirst proved in [1], but the proof is different from the one presented here. Remark 2.6.
The packing dimension of every Gatzouras–Lalley or Barański sponge isequal to its box dimension, since Λ is compact and every open set intersecting Λ containsa bi-Lipschitz image of Λ , see [13, Corollary 3.9] . Preliminaries
This section introduces approximate cubes and multi-dimensional types. Here we con-centrate on Gatzouras–Lalley sponges. The slight modifications for Barański sponges arediscussed in Section 4.2.3.1.
Approximate cubes.
The natural generalization of approximate squares used ex-tensively in covering arguments for self-affine carpets on the plane are approximate cubesin higher dimensions. The δ -stopping of i ∈ Σ in the n -th coordinate (for n = 1 , . . . , d ) isthe unique integer L δ ( i , n ) such that L δ ( i ,n ) (cid:89) (cid:96) =1 λ (cid:0) i ( n ) (cid:96) (cid:1) ≤ δ < L δ ( i ,n ) − (cid:89) (cid:96) =1 λ (cid:0) i ( n ) (cid:96) (cid:1) . (3.1)Also let L δ ( i , d + 1) := 0 . The symbolic δ -approximate cube containing i ∈ Σ is B δ ( i ) = (cid:8) j ∈ Σ : i ( n ) | L δ ( i , n ) = j ( n ) | L δ ( i , n ) for every n = 1 , . . . , d (cid:9) . It is easy to see that for i (cid:54) = j ∈ Σ , either B δ ( i ) = B δ ( j ) or B δ ( i ) ∩ B δ ( j ) = ∅ . Hence, the setof approximate cubes B δ defines a partition of Σ . To make a distinction, for each element B δ ( i ) of the partition B δ we choose an ˆ ı ∈ B δ ( i ) to ‘represent’ it and write B δ ( ˆ ı ) ∈ B δ . ALCULATING BOX DIMENSION WITH THE METHOD OF TYPES 8
Since we assume the COSC (2.1), the image of two elements B δ ( ˆ ı ) (cid:54) = B δ ( ˆ ) ∈ B δ by thenatural projection π on Λ can only intersect on their boundary, so we obtain a cover of Λ for which B δ = N δ (Λ) . As a result, it is enough to work with the set of symbolicapproximate cubes B δ .3.2. Multidimensional types.
In order to introduce multidimensional types, first ob-serve that every approximate cube B δ ( i ) can be uniquely identified with the finite sequence (cid:8) i ( d )1 , . . . , i ( d ) L δ ( i ,d ) (cid:124) (cid:123)(cid:122) (cid:125) ∈ ( I d ) Lδ ( i ,d ) ; i ( d − L δ ( i ,d )+1 , . . . , i ( d − L δ ( i ,d − (cid:124) (cid:123)(cid:122) (cid:125) ∈ ( I d − ) Lδ ( i ,d − − Lδ ( i ,d ) ; . . . ; i (1) L δ ( i , , . . . , i (1) L δ ( i , (cid:124) (cid:123)(cid:122) (cid:125) ∈ ( I ) Lδ ( i , − Lδ ( i , (cid:9) . (3.2)This identification is indeed one-to-one because the coordinate ordering condition (2.2)implies that L δ ( i , d ) < L δ ( i , d − < . . . < L δ ( i , for every i ∈ Σ .In this setting, the type of i ∈ Σ at scale δ is the I d + I d − + . . . + I dimensionalempirical vector τ δ ( i ) = (cid:0) τ δ ( i , d ) ; τ δ ( i , d −
1) ; . . . ; τ δ ( i , (cid:1) , where for ≤ n ≤ dτ δ ( i , n ) = 1 L δ ( i , n ) − L δ ( i , n + 1) (cid:16) (cid:8) L δ ( i , n + 1) + 1 ≤ (cid:96) ≤ L δ ( i , n ) : i ( n ) (cid:96) = j (cid:9)(cid:17) j ∈I n . Note that τ δ ( i , n ) is an I n dimensional probability vector. The set of all possible typesat scale δ is T δ = (cid:8) P = ( p d ; p d − ; . . . ; p ) : there exists B δ ( ˆ ı ) ∈ B δ such that P = τ δ ( ˆ ı ) (cid:9) , and the type class of P ∈ T δ is the set T δ ( P ) = (cid:8) B δ ( ˆ ı ) ∈ B δ : τ δ ( ˆ ı ) = P (cid:9) . Let p m be a probability vector on I m . For ≤ n ≤ m , we denote the Lyapunov exponent by χ n ( p m ) := − (cid:88) i ∈I m p m ( i ) log λ ( i ( n ) ) . Lemma 3.1.
Fix a type P = ( p d ; p d − ; . . . ; p ) ∈ T δ . For every ≤ n ≤ d there exists aconstant C ( d ) n ( P ) depending on P only through χ (cid:96) ( p m ) for n ≤ (cid:96) ≤ m ≤ d such that L δ ( ˆ ı , n ) − L δ ( ˆ ı , n + 1) ≈ − C ( d ) n ( P ) · log δ, where ˆ ı ∈ Σ is such that τ δ ( ˆ ı ) = P . Moreover, (cid:80) dm = n +1 C ( d ) m ( P ) · χ n +1 ( p m ) = 1 for every ≤ n ≤ d − .Proof. For each ≤ n ≤ d and fixed P = ( p d ; p d − ; . . . ; p ) ∈ T δ , it follows that log δ ≈ d (cid:88) m = n (cid:0) L δ ( ˆ ı , m ) − L δ ( ˆ ı , m + 1) (cid:1) L δ ( ˆ ı , m ) − L δ ( ˆ ı , m + 1) L δ ( ˆ ı ,m ) (cid:88) (cid:96) = L δ ( ˆ ı ,m +1)+1 log λ (cid:0) i ( n ) (cid:96) (cid:1) = − d (cid:88) m = n (cid:0) L δ ( ˆ ı , m ) − L δ ( ˆ ı , m + 1) (cid:1) · χ n ( p m ) . In particular, for n = d (recall L δ ( ˆ ı , d + 1) = 0 by definition), log δ ≈ − L δ ( ˆ ı , d ) · χ d ( p d ) ,giving C ( d ) d ( P ) = 1 /χ d ( p d ) . In the next step for n = d − , L δ ( ˆ ı , d − − L δ ( ˆ ı , d ) ≈ − (cid:0) log δ + L δ ( ˆ ı , d ) · χ d − ( p d ) (cid:1) χ d − ( p d − ) = (cid:18) − χ d − ( p d ) χ d ( p d ) (cid:19) − log δχ d − ( p d − ) (cid:124) (cid:123)(cid:122) (cid:125) =: − C ( d ) d − ( P ) · log δ . ALCULATING BOX DIMENSION WITH THE METHOD OF TYPES 9
The argument continues by induction as n decreases further. After rearranging, L δ ( ˆ ı , n ) − L δ ( ˆ ı , n + 1) ≈ − χ n ( p n ) (cid:18) log δ + d (cid:88) m = n +1 (cid:0) L δ ( ˆ ı , m ) − L δ ( ˆ ı , m + 1) (cid:1) · χ n ( p m ) (cid:19) = (cid:18) − d (cid:88) m = n +1 C ( d ) m ( P ) · χ n ( p m ) (cid:19) − log δχ n ( p n ) =: − C ( d ) n ( P ) · log δ. (3.3)The final assertion follows simply by applying the definition of C ( d ) n +1 ( P ) : d (cid:88) m = n +1 C ( d ) m ( P ) · χ n +1 ( p m ) = C ( d ) n +1 ( P ) · χ n +1 ( p n +1 ) + d (cid:88) m = n +2 C ( d ) m ( P ) · χ n +1 ( p m )= (cid:18) − d (cid:88) m = n +2 C ( d ) m ( P ) · χ n +1 ( p m ) (cid:19) χ n +1 ( p n +1 ) χ n +1 ( p n +1 ) + d (cid:88) m = n +2 C ( d ) m ( P ) · χ n +1 ( p m ) = 1 . (cid:3) Proof of Theorems 2.3 and 2.5
We begin with the proof of Theorem 2.3 and then show what adjustments need to bemade to the argument to prove Theorem 2.5.4.1.
Proof of Theorem 2.3.
Let P ,...,d denote the set of all I d + I d − + . . . + I dimensional vectors P = ( p d ; p d − ; . . . ; p ) , where each p n is a probability vector on I n .We are ready to state our variational formula for dim B Λ d . Proposition 4.1.
Let Λ d be a Gatzouras–Lalley sponge in R d . Then dim B Λ d = max P ∈P ,...,d d (cid:88) n =1 C ( d ) n ( P ) · H ( p n ) , (4.1) where C ( d ) n ( P ) is defined in Lemma 3.1.Proof. The main step is to apply the method of types to the multi-dimensional type P ∈ T δ . For any type P ∈ T δ , we repeatedly use (1.3) for each p d , p d − , . . . , p to get that exp (cid:34) d (cid:88) n =1 ( L δ ( ˆ ı , n ) − L δ ( ˆ ı , n + 1)) H ( p n ) (cid:35) · d (cid:89) n =1 (cid:0) L δ ( ˆ ı , n ) − L δ ( ˆ ı , n + 1) + 1 (cid:1) − I n (cid:47) T δ ( P ) (cid:47) exp (cid:34) d (cid:88) n =1 ( L δ ( ˆ ı , n ) − L δ ( ˆ ı , n + 1)) H ( p n ) (cid:35) , where ˆ ı ∈ Σ is such that τ δ ( ˆ ı ) = P . From Lemma 3.1 it follows that δ − (cid:80) dn =1 C ( d ) n ( P ) · H ( p n ) · ( − log δ ) − (cid:80) dn =1 I n (cid:47) T δ ( P ) (cid:47) δ − (cid:80) dn =1 C ( d ) n ( P ) · H ( p n ) . (4.2)On the other hand, we can give a crude upper bound for T δ in a similar fashion byrepeating the argument in the self-similar case (1.8) for each coordinate ≤ n ≤ d : T δ ≤ d (cid:89) n =1 (cid:0) L δ ( ˆ ı , n ) − L δ ( ˆ ı , n + 1) + 1 (cid:1) I n · max B δ ( ˆ ı ) ∈B δ ( L δ ( ˆ ı , n ) − L δ ( ˆ ı , n + 1)) . (4.3) ALCULATING BOX DIMENSION WITH THE METHOD OF TYPES 10
Again by Lemma 3.1, the right hand side in (4.3) is o ( δ − ) . It follows from (4.2) that thedominant box counting type P ∗ δ = ( p ∗ δ,d ; p ∗ δ,d − ; . . . ; p ∗ δ, ) ∈ T δ maximises the expression (cid:80) dn =1 C ( d ) n ( P ) · H ( p n ) . Thus, combining (4.2) and (4.3) with (1.1) implies that ( − log δ ) · d (cid:88) n =1 C ( d ) n ( P ∗ δ ) · H ( p ∗ n,δ ) − ε (cid:47) log N δ (Λ) (cid:47) ( − log δ ) · d (cid:88) n =1 C ( d ) n ( P ∗ δ ) · H ( p ∗ n,δ ) + ε, where the error term ε = o ( − log δ ) . As a result, we obtain the variational formula dim B Λ d = lim δ → d (cid:88) n =1 C ( d ) n ( P ∗ δ ) · H ( p ∗ n,δ ) . As δ → , the set of types T δ becomes dense in the set P ,...,d . The compactness of P ,...,d and the continuity of C ( d ) n ( P ) and H ( p n ) implies that the dominant box counting type P ∗ δ tends to the limiting dominant type P ∗ = ( p ∗ d ; p ∗ d − ; . . . ; p ∗ ) ∈ P ,...,d , which satisfies d (cid:88) n =1 C ( d ) n ( P ∗ ) · H ( p ∗ n ) = max P ∈P ,...,d d (cid:88) n =1 C ( d ) n ( P ) · H ( p n ) = dim B Λ d . (4.4) (cid:3) Remark 4.2.
It is possible to express the formula in (4.1) in terms of Lyapunov expo-nents. In particular, for d = 2 one obtains the formula already presented in (2.4) . For d = 3 , slightly more work shows that the expression to be maximised is H ( p ) χ ( p ) + (cid:18) − χ ( p ) χ ( p ) (cid:19) H ( p ) χ ( p ) + (cid:20) − χ ( p ) χ ( p ) − (cid:18) − χ ( p ) χ ( p ) (cid:19) χ ( p ) χ ( p ) (cid:21) H ( p ) χ ( p ) . (4.5) We think of it as a Ledrappier–Young like formula for the box dimension, see Section 5 forfurther discussion. For d > the calculations get increasingly involved and cumbersome. The next lemma characterises the limiting dominant type P ∗ = ( p ∗ d ; p ∗ d − ; . . . ; p ∗ ) andconcludes the proof of Theorem 2.3. Lemma 4.3.
Let Λ d be a Gatzouras–Lalley sponge in R d . Then the maximum in (4.1) is uniquely attained by the type P ∗ = ( p ∗ d ; p ∗ d − ; . . . ; p ∗ ) , where the probability vectors p ∗ n = ( p ∗ n ( i )) i ∈I n are defined by p ∗ ( i ) = ( λ ( i )) s and p ∗ n ( i ) = (cid:0) λ ( i (1) ) (cid:1) s · n (cid:89) (cid:96) =2 (cid:0) λ ( i ( (cid:96) ) ) (cid:1) s (cid:96) − s (cid:96) − for n = 2 , . . . , n, where s ≤ s ≤ . . . ≤ s d were introduced in (2.3) . Moreover, dim B Λ d = s d . Proof.
We start by showing that dim B Λ d = s d . Immediate calculations yield that H ( p ∗ n ) = n (cid:88) (cid:96) =1 ( s (cid:96) − s (cid:96) − ) · χ (cid:96) ( p ∗ n ) , where we define s := 0 . Substituting this into (4.4), we see that dim B Λ d equals d (cid:88) n =1 C ( d ) n ( P ∗ ) · n (cid:88) (cid:96) =1 ( s (cid:96) − s (cid:96) − ) · χ (cid:96) ( p ∗ n ) = d (cid:88) (cid:96) =1 ( s (cid:96) − s (cid:96) − ) d (cid:88) n = (cid:96) C ( d ) n ( P ∗ ) · χ (cid:96) ( p ∗ n )= s d · C ( d ) d ( P ∗ ) · χ d ( p ∗ d ) + d − (cid:88) (cid:96) =1 s (cid:96) (cid:16) C ( d ) (cid:96) ( P ∗ ) · χ (cid:96) ( p ∗ (cid:96) ) + d (cid:88) n = (cid:96) +1 C ( d ) n ( P ∗ ) (cid:0) χ (cid:96) ( p ∗ n ) − χ (cid:96) +1 ( p ∗ n ) (cid:1)(cid:124) (cid:123)(cid:122) (cid:125) =: A (cid:96) (cid:17) . ALCULATING BOX DIMENSION WITH THE METHOD OF TYPES 11
Since C ( d ) d ( P ∗ ) = 1 /χ d ( p ∗ d ) , it remains to show that A (cid:96) = 0 for every ≤ (cid:96) ≤ d − . Usingthe definition of C ( d ) (cid:96) ( P ∗ ) from (3.3), it follows that A (cid:96) = (cid:18) − d (cid:88) m = (cid:96) +1 C ( d ) m ( P ∗ ) · χ (cid:96) ( p ∗ m ) (cid:19) χ (cid:96) ( p ∗ (cid:96) ) χ (cid:96) ( p ∗ (cid:96) ) + d (cid:88) n = (cid:96) +1 C ( d ) n ( P ∗ ) (cid:0) χ (cid:96) ( p ∗ n ) − χ (cid:96) +1 ( p ∗ n ) (cid:1) = 0 , where the final equality follows from Lemma 3.1.The maximising type P ∗ is obtained by repeated use of the Lagrange-multipliers method.For brevity, let t ( p d , p d − , . . . , p ) := (cid:80) dn =1 C ( d ) n ( P ) · H ( p n ) . Note that t ( p d , p d − , . . . , p ) depends on p only through the first term. More specifically, by (3.3), C ( d )1 ( P ) · H ( p ) = (cid:18) − d (cid:88) m =2 C ( d ) m ( P ) · χ ( p m ) (cid:19) H ( p ) χ ( p ) . The term in parenthesis is independent of p , furthermore, as mentioned already in Sec-tion 1.1, the quotient H ( p ) /χ ( p ) is maximised precisely by p ∗ with value s .The next step is to observe that t ( p d , . . . , p , p ∗ ) depends on p only through the firsttwo terms. More specifically writing out these two terms, by (3.3), (cid:18) − d (cid:88) m =3 C ( d ) m ( P ) · χ ( p m ) (cid:19) · s + (cid:18) − d (cid:88) m =3 C ( d ) m ( P ) · χ ( p m ) (cid:19) H ( p ) − χ ( p ) · s χ ( p ) . The two terms in parenthesis are independent of p , moreover, another use of the Lagrange-multipliers method shows that the quotient depending on p is maximised by p ∗ with value s − s . In general, at the n -th step one applies the Lagrange-multipliers method to theterm in t ( p d , . . . , p n , p ∗ n − , . . . , p ∗ ) that depends on p n . This concludes the proof. (cid:3) Proof of Theorem 2.5.
Without the coordinate ordering condition (2.2), the studyof Barański sponges is usually much more technical than the Gatzouras–Lalley case. How-ever, for our box counting argument only one extra natural step is required.The δ -stopping of i = ( i i . . . ) ∈ Σ in the n -th coordinate (for n = 1 , . . . , d ) is the sameas in (3.1) with the slightly modified notation: L δ ( i ,n ) (cid:89) (cid:96) =1 λ n (cid:0) i (cid:96),n (cid:1) ≤ δ < L δ ( i ,n ) − (cid:89) (cid:96) =1 λ n (cid:0) i (cid:96),n (cid:1) , where i (cid:96),n denotes the n -th coordinate of i (cid:96) . The symbolic δ -approximate cube containing i ∈ Σ is the same as before: B δ ( i ) = (cid:8) j ∈ Σ : i (cid:96),n = j (cid:96),n for every (cid:96) = 1 , . . . , L δ ( i , n ) and n = 1 , . . . , d (cid:9) . Also, the approximate cubes partition Σ , and their images by the natural projection π give an optimal δ -cover of the attractor. Without the coordinate ordering condition, wedo not know how the L δ ( i , n ) compare to each other for a specific B δ ( i ) like we did in (3.2)for the Gatzouras–Lalley case. Therefore, we sort the approximate cubes first.Recall, Sym( { , . . . , d } ) denotes the symmetric group on the set of coordinates { , . . . , d } and the notation for a permutation σ from (2.5). We say that a δ -approximate cube B δ ( i ) is σ -ordered if L δ ( i , σ d ) ≤ L δ ( i , σ d − ) ≤ . . . ≤ L δ ( i , σ ) . Potentially B δ ( i ) can be σ -ordered for different permutations if the δ -stopping is equalin multiple coordinates, but we will see in a moment that this is never a dominant boxcounting class. Let B δ ( σ ) denote the set of σ -ordered δ -approximate cubes. ALCULATING BOX DIMENSION WITH THE METHOD OF TYPES 12
For a fixed σ ∈ Sym( { , . . . , d } ) at every scale δ , the δ -stoppings within B δ ( σ ) areordered the same way, hence, we can identify B δ ( i ) ∈ B δ ( σ ) with the sequence (cid:8) i ,σ d , . . . , i L δ ( i ,σ d ) ,σ d ; i L δ ( i ,σ d )+1 ,σ d − , . . . , i L δ ( i ,σ d − ) ,σ d − ; . . . ; i L δ ( i ,σ )+1 ,σ , . . . , i L δ ( i ,σ ) ,σ (cid:9) , where a block is empty whenever L δ ( i , σ n ) = L δ ( i , σ n +1 ) . The type for an i ∈ B δ ( σ ) hasthe form τ δ ( i ) = (cid:0) τ δ ( i , σ d ) ; τ δ ( i , σ d − ) ; . . . ; τ δ ( i , σ ) (cid:1) , where τ δ ( i , σ n ) is equal to L δ ( i , σ n ) − L δ ( i , σ n +1 ) (cid:16) (cid:8) L δ ( i , σ n +1 ) + 1 ≤ (cid:96) ≤ L δ ( i , σ n ) : Π( i (cid:96) , { σ n } ) = j (cid:9)(cid:17) j ∈ Π( I ; { σ n } ) for ≤ n ≤ d , where { σ n } = { σ , . . . , σ n } . If L δ ( i , σ n ) = L δ ( i , σ n +1 ) , then the correspond-ing C ( d ) n ( τ δ ( i )) = 0 . Hence, from (4.1) of Proposition 4.1 it follows that such a type cannever be a dominant box counting type. Moreover, the number of different types with atleast one empty block is certainly bounded from above by o ( δ − ) . Therefore, from thepoint of view of determining the box dimension, we can simply discard the approximatecubes in these type classes.As a result, for any fixed σ ∈ Sym( { , . . . , d } ) , we are essentially back in the GLcase and can repeat the same argument. Within each B δ ( σ ) there is a dominant boxcounting type P ∗ δ ( σ ) = ( p ∗ δ,d ; p ∗ δ,d − ; . . . ; p ∗ δ, ) which consists of probability vectors p ∗ δ,n onthe index set Π( I ; { σ n } ) . As δ → , these vectors p ∗ δ,n converge to the ones defined bythe equations in Theorem 2.5. This is the limiting dominant type P ∗ ( σ ) which satisfies (cid:80) dn =1 C ( d ) n ( P ∗ ( σ )) · H ( p ∗ n ( σ )) = s d ( σ ) . Thus, T δ ( P ∗ δ ( σ )) ≈ δ − s d ( σ )+ o (1) . Since there arejust d ! different σ -orderings, we conclude that dim B Λ d = max σ ∈ Sym( { ,...,d } ) s d ( σ ) .5. Further discussion
This section provides some additional context to the results.First consider Bedford–McMullen (or Sierpiński) sponges. They are special cases ofGL sponges because the diagonal matrices A i defining the maps S i are all the same andindependent of i . Let > λ > λ > . . . > λ d > denote the diagonal entries. Similarlyto the homogeneous self-similar case, recall Remark 1.1, the δ -stoppings are independentof i and L δ ( i , n ) ≈ log δ/ log λ n for ≤ n ≤ d . Hence, C ( d ) n ( P ) = 1 / log λ n − / log λ n +1 regardless of P . Thus, Proposition 4.1 implies that all we need to maximise in (4.1) is H ( p n ) which is equal to log I n (attained by the uniform vector on the set I n ). This isthe formula obtained by Kenyon and Peres [25].Another setup to which the method can be applied to is if we consider GL carpets intwo dimensions defined by lower triangular matrices instead of diagonal matrices [2, 26].In this case the image of [0 , under any map of the IFS is a parallelogram with twovertical sides parallel with the y -axis. A simple lemma [26, Lemma 1.3] states that theslope of the iterates of these parallelograms remain uniformly bounded. Hence, there isa uniform constant C (depending only on the IFS) such that the image by π of any δ -approximate square on Λ can be covered by at most C squares of diameter δ . As a result, B δ ≈ N δ (Λ) still holds, so the box dimension remains unchanged.Our variational formula (2.4) also provides a very clear argument for one of the necessaryand sufficient conditions for the Hausdorff and box dimensions of GL carpets to agree.Gatzouras and Lalley [21] proved that the Hausdorff dimension satisfies the variationalformula dim H Λ = max p ∈P I H ( p ) χ ( p ) + (cid:18) − χ ( p ) χ ( p ) (cid:19) H ( q p ) χ ( q p ) , (5.1) ALCULATING BOX DIMENSION WITH THE METHOD OF TYPES 13 where q p = ( q , . . . , q I ) denotes the probability vector on I defined by q i = (cid:80) j ∈I ( i ) p ( i,j ) .Comparing this with (2.4), we immediately see that dim H Λ = dim B Λ ⇐⇒ q p ∗ = p ∗ ⇐⇒ (cid:88) j ∈I ( i ) (cid:0) λ ( i, j ) (cid:1) s − s = 1 for every i ∈ I . This is referred to as the uniform fibre case in the literature. The main result of Dasand Simmons [9] is that the variational formula (5.1) does not necessarily hold in higherdimensions. Instead, one needs to consider a wider class of measures, called pseudo-Bernoulli measures , which are not invariant.The expression being maximised in (5.1) is a special case of the
Ledrappier–Youngformula which holds in much higher generality for measures on self-affine sets [4, 5, 14, 15]and has been a key technical tool in recent advancements in the dimension theory of self-affine sets and measures, see [3, 22, 28, 30] to name a few. In light of our result, it isnatural to ask the following.
Question 5.1.
Does a Ledrappier–Young like formula (2.4) hold more generally for thebox dimension of self-affine sets on the plane? What about higher dimensions?
For three dimensions, the formula would be to maximise the expression in (4.5). Thegeneral argument itself is very flexible. If the optimal δ -cover of a set has a clear symbolicrepresentation, then by defining a proper space of types it seems plausible to apply themethod. The Barański case shows that some “orientation” of the boxes also plays a role.Overlapping systems could be particularly interesting to study from this vantage point.This is because it is still an open problem whether the box dimension of self-affine setsalways exists, regardless of overlaps. It does not exist for all sub -self-affine sets introducedin [24], see the very recent example of Jurga [23]. Moreover, for self-similar sets there isthe folklore conjecture that the only reason why its (box) dimension can drop below itssimilarity dimension (1.5) is if the system has exact overlaps. Acknowledgment.
The author was supported by a
Leverhulme Trust Research ProjectGrant (RPG-2019-034).
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István Kolossváry,University of St Andrews, School of Mathematics and Statistics,St Andrews, KY16 9SS, Scotland
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