On the dimensions of a family of overlapping self-affine carpets
OOn the dimensions of a family of overlapping self-affinecarpets
Jonathan M. Fraser and Pablo Shmerkin September 25, 2018
Dedicated to the memory of Dave Broomhead. School of Mathematics, The University of Manchester, Manchester, M13 9PL, UK.E-mail: [email protected] Department of Mathematics and Statistics, Torcuato Di Tella University, Av. Figueroa Alcorta7350, Buenos Aires, Argentina. E-mail: [email protected]
Abstract
We consider the dimensions of a family of self-affine sets related to the Bedford-McMullencarpets. In particular, we fix a Bedford-McMullen system and then randomise the translationvectors with the stipulation that the column structure is preserved. As such, we maintain oneof the key features in the Bedford-McMullen set up in that alignment causes the dimensionsto drop from the affinity dimension. We compute the Hausdorff, packing and box dimensionsoutside of a small set of exceptional translations, and also for some explicit translations evenin the presence of overlapping. Our results rely on, and can be seen as a partial extension of,M. Hochman’s recent work on the dimensions of self-similar sets and measures.
Mathematics Subject Classification
Key words and phrases : Self-affine carpet, Hausdorff dimension, packing dimension, boxdimension, overlaps
The dimension theory of self-affine sets has attracted a great deal of attention in the literatureover the past 30 years. There are two key starting points which have led to two thriving andcomplementary strands of research. The ‘generic case’ studies general self-affine sets by randomisingthe translation vectors in the defining iterated function system in an appropriate way and thenmaking almost sure statements about the corresponding attractors. This approach began withFalconer’s seminal paper [7] in 1988, which introduced the affinity dimension as a sure upper boundfor the upper box dimension of any self-affine set and if the translation vectors are randomisedand the norms of the defining matrices are strictly smaller than 1/2, then the Hausdorff, box andpacking dimensions of the attractor are all almost surely equal to the affinity dimension. Somearticles following this approach are [27, 8, 26, 19, 6, 17]. In contrast, the ‘specific approach’focuses on special classes of self-affine sets designed in a way to facilitate calculations and allows sure statements to be made about the attractors. This began with the work of Bedford [4] andMcMullen [23] from 1985 which introduced self-affine carpets and computed their Hausdorff andbox dimensions. Of particular note is that these values are typically different and strictly lessthan the affinity dimension. This second approach was further developed in [20, 1, 9, 11] amongothers. This paper has two main purposes. On one hand, we blend the two approaches in anatural context. We begin with a Bedford-McMullen carpet and then randomise the translations1 a r X i v : . [ m a t h . D S ] F e b hilst maintaining the key structural feature: the column alignment. This will be elaborated onin the following section. On the other hand, we wanted to illustrate how a recent breakthroughof Hochman [14] on the dimensions of self-similar sets and measures can also be applied to obtainanalogous results for self-affine sets. We obtain formulae for the Hausdorff, box and packingdimensions valid outside of a small set of parameters, with two points of interest being that thevalues of the dimension are typically different from each other and from the affinity dimension,and our class contains many overlapping self-affine sets. Fix positive integers n > m > m × n grid. The gridrectangles can now be labelled in a natural way as D = { ( i, j ) : i = 1 , . . . , m, j = 1 , . . . , n } .Choose a non-empty subset D ⊆ D and for each ( i, j ) ∈ D , let S ( i,j ) denote the affine contractionwhich maps the unit square onto the rectangle indexed by ( i, j ) defined by S ( i,j ) ( x, y ) = (cid:0) x/m + ( i − /m, y/n + ( j − /n (cid:1) . Together, the maps { S ( i,j ) } ( i,j ) ∈ D form an iterated function system (IFS) and it is well-known thatthere exists a unique non-empty compact set F satisfying F = (cid:91) ( i,j ) ∈ D S ( i,j ) ( F ) . This set F is called the attractor of the IFS and this class of attractors was first studied in themid-eighties independently by Bedford [4] and McMullen [23], who each gave a formula for theHausdorff and box dimensions. Such sets are now known as Bedford-McMullen carpets. We wishto consider the following generalisation. For a given Bedford-McMullen system, we randomise thehorizontal translates, whilst keeping the column structure intact, i.e., we always assume that iftwo rectangles are in the same column initially, then they are translated horizontally by the sameamount. See Figures 1 and 2. The advantage of this approach is that because we keep somealignment in the construction, even though we randomise the system, the ‘typical’ dimensionsare still exceptional (we will see that in fact they are the same as the dimensions of the originalsystem). Thus we provide a smoothly parametrised family of potentially overlapping self-affinecarpets whose dimensions are strictly less than the affinity dimension.Figure 1: A generating pattern for a Bedford-McMullen carpet (on the left) and a translation ofthe columns into an overlapping pattern (on the right). In this case, m = 4 and n = 6.More formally, let D = { i ∈ { , . . . , m } : ( i, j ) ∈ D for some j } be the projection of D onto thefirst co-ordinate. To each i ∈ D we associate a ‘random translation’ t i ∈ [0 , − /m ] and for agiven set of translates t = ( t i ) i ∈ D ∈ [0 , − /m ] D we define a new IFS consisting of the maps S ( i,j ) ,t ( x, y ) = ( x/m, y/n ) + ( t i , ( j − /n )and denote the attractor, which of course depends on t , by F t . In the case where t i = ( i − /m for all i ∈ D , then we recover the original Bedford-McMullen system. The restriction that t i ∈ [0 , − /m ]is meant to ensure the attractor is a subset of the unit square, and it is not essential.2e now wish to make statements about the dimensions of F t in terms of the parameters t ∈ [0 , − /m ] D .Figure 2: Four self-affine carpets based on the same Bedford-McMullen system, but with thecolumns translated in different ways. The original Bedford-McMullen carpet is on the top left. Ineach case m = 3 and n = 4. In this section we state our main results on the dimensions of the self-affine sets described inthe previous section. We write dim H , dim B and dim P to denote the Hausdorff, box and packingdimensions respectively. Recall that an IFS { S , . . . , S k } is said to have an exact overlap if thesemigroup generated by the S i is not free. We write N i = |{ j = 1 , . . . , n : ( i, j ) ∈ D }| for thenumber of chosen rectangles in the i th column. Theorem 2.1.
There exists a set E ⊂ [0 , − /m ] D of Hausdorff and packing dimension | D | − (in particular of zero | D | -dimensional Lebesgue measure) such that dim H F t = log (cid:80) mi =1 N log m/ log ni log m if t ∈ [0 , − /m ] D \ E, dim H F t < log (cid:80) mi =1 N log m/ log ni log m if t ∈ E. Moreover, if t is algebraic and the IFS { x/m + t i } i ∈ D does not have an exact overlap, then t / ∈ E . We will prove Theorem 2.1 in Section 5. The main idea, which was inspired by results of Jordan[16] and Jordan and Pollicott [18], is to use Marstrand’s slicing theorem to bound the dimensionfrom below by the sum of the dimension of the projection, and the dimension of a typical slice; seeSection 5 for further discussion. 3 heorem 2.2.
There exists a set E ⊂ [0 , − /m ] D of Hausdorff and packing dimension | D | − (in particular of zero | D | -dimensional Lebesgue measure) such that dim B F t = dim P F t = log | D | log m + log | D | / | D | log n if t ∈ [0 , − /m ] D \ E, dim B F t = dim P F t < log | D | log m + log | D | / | D | log n if t ∈ E. Moreover, if t is algebraic and the IFS { x/m + t i } i ∈ D does not have an exact overlap, then t / ∈ E . We will prove Theorem 2.2 in Section 6. Note that we say t = { t i } i ∈ D is algebraic if all ofthe t i are algebraic. A generalisation of our main results is discussed in Section 7.1 below. Wenote that the exceptional set E in both cases is contained in the set for which “super exponentialconcentration of cylinders” occurs in the vertical projection; see Section 4 below. However, wecannot guarantee that the exceptional set E is precisely the same in both theorems.We underline that the dimensions appearing in Theorems 2.1 and 2.2 are the same as thedimensions of the original carpet as proved by Bedford and McMullen. We recall that (for theunperturbed carpets) the box counting dimension is obtained by covering each rectangle in the n -th stage of the construction by the same number of disks of the appropriate size, independentlyof each other. When allowing covers by disks of different sizes, it is usually more efficient to coverlarge collections of parallel rectangles by a single disk, leading to the expression for Hausdorffdimension (which also has a variational interpretation as the supremum of Hausdorff dimensionsof Bernoulli measures for the natural Markov partition of the carpet). Our results suggest thatthis geometric picture typically persists when the columns are allowed to overlap.We finish this section by commenting on the relation between these results and previous work onthe subject. The “hybrid” approach to the dimension of self-affine sets was undertaken previouslyin [18, 2]. These papers give a formula, valid for Lebesgue almost all parameters, for certain(different) families of parametrized self-affine carpets. A point in common with our model isthat these typical dimensions are strictly less than the affinity dimension. However, in [18] onlyHausdorff dimension was considered, while the results of [2] are for box-counting dimension andinvolve non-overlapping self-affine sets. Also, because we use Hochman’s recent results (which werenot available to the authors of [18, 2]), we obtain far better information about the exceptional set. Most of the proofs in the subsequent sections are symbolic in nature, and thus rely more on thecombinatorics of the symbolic spaces D N and D N than the geometry of the corresponding fractals.In Section 4 we review the results of Hochman which will allow us to pass from the symbolicinformation back to the geometric setting. In this section we briefly summarise some notation wewill use throughout the rest of the paper.Define the vertical projection π : [0 , → [0 ,
1] by π ( x, y ) = x . As is often the case for self-affine carpets, this projection will play a key role. For i ∈ D and t ∈ [0 , − /m ] D , we will denote S i,t ( x ) = m x + t i . Note that the IFS { S i,t } i ∈ D generates the projection πF t .Given λ = ( λ , . . . , λ k ) = (( i , j ) , . . . , ( i k , j k )) ∈ D k , and ρ = ( i , . . . , i k ) ∈ D k , we denote S λ,t = S ( i ,j ) ,t ◦ · · · ◦ S ( i k ,j k ) ,t ,S ρ,t = S i ,t ◦ · · · ◦ S i k ,t . Rather than define all the notions of dimension we are interested in, we simply refer the readerto [5, Chapters 2-3]. The key properties of Hausdorff and packing dimensions which we need will bediscussed in the following sections when required. We now recall the definition of box dimension.The lower and upper box dimensions of a bounded set F ⊆ R n are defined bydim B F = lim inf r → log N r ( F ) − log r and dim B F = lim sup r → log N r ( F ) − log r , respectively, where N r ( F ) is the smallest number of sets required for a r -cover of F . Here an r -cover of F is a finite (or countable) collection of open sets { U k } k with the property that the4iameter of each set | U k | (cid:54) r and F ⊆ ∪ k U k . If dim B F = dim B F , then we call the common valuethe box dimension of F and denote it by dim B F . It is useful to note that we can replace N r withseveral different definitions all based on covering or packing the set at scale r , see [5, Section 3.1].For example, it can be the number of cubes in an r -grid which intersect F . In this section we recall a recent result of Hochman [14] on the dimensions of self-similar measures.We consider only the special case of homogeneous self-similar measures, which is all we will require.To this end, consider an affine IFS of the form I = { S i ( x ) = ax + t i } i ∈ A , where A is a finite indexset and a ∈ (0 ,
1) is a fixed contraction rate and the maps S i are assumed to act on R . Definition 4.1.
We say that the IFS I = { S i ( x ) = ax + t i } i ∈ A has super-exponential concentra-tion of cylinders if − log ∆ k /k → ∞ (with the convention log 0 = −∞ ), where ∆ k = min ρ (cid:54) = ρ (cid:48) ∈ A k | S ρ (0) − S ρ (cid:48) (0) | , and, as usual, S ρ ( x ) = S i ◦ · · · ◦ S i k if ρ = ( i , . . . , i k ) ∈ A k . In other words, I has super-exponential concentration of cylinders if the distance betweencylinders of level k coming from different codes decreases faster than any power as a function of k . The only known mechanism by which super-exponential concentration of cylinders can occur isthe presence of exact overlaps; by definition, this means that ∆ k = 0 for some k . One observationthat will be useful later is that if I does not have super-exponential concentration of cylinders,then the same is true for any IFS which is obtained by first iterating all the maps in I a fixednumber of times, and then dropping some of the maps.The following is the key result of Hochman that we will require. Recall that the Hausdorffdimension of a Borel probability measure ν is defined bydim H ν = inf { dim H F : F is a Borel set with ν ( F ) = 1 } . Theorem 4.2 ([14, Theorem 1.1]) . Suppose the IFS I = { S i ( x ) = ax + t i } i ∈ A does not havesuper-exponential concentration of cylinders. Let p = ( p i ) i ∈ A be a probability vector, and let ν be the self-similar measure associated to the IFS I and the vector p , that is, the unique Borelprobability measure satisfying ν = (cid:88) i ∈ A p i ν ◦ S − i , Then dim H ν = min (cid:18) (cid:80) i ∈ A p i log p i log a , (cid:19) . In particular, if F is the invariant set for the IFS I , that is, the only nonempty compact setsatisfying F = (cid:83) i ∈ A S i ( F ) , then dim H F = min (cid:18) log | A | log(1 /a ) , (cid:19) . The next result, which also follows from Hochman’s work, tells us that super-exponential con-centration of cylinders is a rare phenomenon - in a quantitative sense and in some special cases,as rare as exact overlaps.
Proposition 4.3.
Let A be a finite index set and fix a ∈ (0 , / .1. The family of ( t i ) i ∈ A such that the IFS I = { ax + t i } i ∈ A has super-exponential concentrationof cylinders has Hausdorff and packing dimension | A | − .2. If a and t i , ( i ∈ A ) , are all algebraic, then the IFS I = { ax + t i } i ∈ A has super-exponentialconcentration of cylinders if and only if there is an exact overlap, that is, if and only if ∆ k = 0 for some k . roof. Let E denote the set of parameters in a box [ − M, M ] | A | for which there is super-exponentialconcentration of cylinders. If i (cid:54) = j ∈ A , then E contains the piece of hyperplane { t i = t j } ∩ [ − M, M ] | A | , so dim H ( E ) (cid:62) | A | −
1. The proof of the upper bound is similar to the proof of [14,Theorem 1.8] (in fact simpler because of the linearity of the projection map) and also follows fromresults in [15], but we give the proof for completeness. Given a set of translations t = ( t i ) i ∈ A , wewrite the maps in the corresponding IFS as S i,t = ax + t i to emphasise dependence on t . Giventwo sequences ρ = ( i , . . . , i n ) , ρ (cid:48) = ( j , . . . , j n ) ∈ A n , let ∆ ρ,ρ (cid:48) : [ − M, M ] | A | → R be the map∆ ρ,ρ (cid:48) ( t ) = S ρ,t (0) − S ρ (cid:48) ,t (0) , where S ρ,t , S ρ (cid:48) ,t are the compositions of maps coming from the IFS { S i,t = ax + t i } i ∈ A . It followsfrom the definition of super-exponential concentration that E = (cid:92) ε> ∞ (cid:91) N =1 (cid:92) n>N (cid:91) ρ (cid:54) = ρ (cid:48) ∈ A n ∆ − ρ,ρ (cid:48) ( − ε n , ε n ) . Since t → S ρ,t (0) is linear, then so is ∆ ρ,ρ (cid:48) for any ρ, ρ (cid:48) ∈ A n . If we write ∆ ρ,ρ (cid:48) ( t ) = (cid:80) (cid:96) ∈ A c (cid:96) t (cid:96) ,then the coefficients c (cid:96) are given by c (cid:96) = (cid:88) k ∈{ ,...,n } : i k = (cid:96) a k − − (cid:88) k ∈{ ,...,n } : j k = (cid:96) a k − . Hence | c (cid:96) | (cid:54) / (1 − a ) for all (cid:96) and, if ρ (cid:54) = ρ (cid:48) , there is (cid:96) such that | c (cid:96) | (cid:62) a n (1 − a/ (1 − a ));this is positive since a ∈ (0 , / ρ, ρ (cid:48) ∈ A n ,the set ∆ − ρ,ρ (cid:48) ( − ε n , ε n ) is contained in the C a ( ε/a ) n -neighborhood of the hyperplane ∆ ρ,ρ (cid:48) = 0,where C a is a constant independent of n and ε . Hence ∪ ρ (cid:54) = ρ (cid:48) ∈ A n ∆ − ρ,ρ (cid:48) ( − ε n , ε n ) can be covered by C a, | A | ,M | A | n ( a/ε ) n ( | A |− balls of radius ( ε/a ) n , where C a, | A | ,M is a constant depending only on a, | A | and M , and in particular, not on n or ε . It follows that for N (cid:62) B (cid:92) n>N (cid:91) ρ (cid:54) = ρ (cid:48) ∈ A n ∆ − ρ,ρ (cid:48) ( − ε n , ε n ) (cid:54) | A | − | A | log( a/ε ) . We recall that one characterization of the packing dimension of a set E is dim P ( E ) =inf { sup i dim B ( E i ) : E ⊆ ∪ i E i } , see [5, Proposition 3.8]. Therefore the above implies thatdim P E (cid:54) lim ε (cid:38) dim P ∞ (cid:91) N =1 (cid:92) n>N (cid:91) ρ (cid:54) = ρ (cid:48) ∈ A n ∆ − ρ,ρ (cid:48) ( − ε n , ε n ) (cid:54) lim ε (cid:38) (cid:18) | A | − | A | log( a/ε ) (cid:19) = | A | − , which yields the first claim since M was arbitrary.The second part of the proposition is [14, Theorem 1.5]. In this section we prove Theorem 2.1, which gives the Hausdorff dimension of F t outside of a smallexceptional set of t . The proof relies on Marstrand’s slice theorem and being able to control twothings: the Hausdorff dimension of a particular self-similar measure supported on π ( F t ); and, forthis measure, the almost sure Hausdorff dimension of the vertical slices through points in π ( F t ).We borrow the slicing idea from the works of Jordan [16] and Jordan and Pollicott [18], where theHausdorff dimension of overlapping Sierpi´nski gaskets and carpets was considered. In particular,Jordan and Pollicott [18, Section 6.2] find the Hausdorff dimension of certain overlapping carpetsof Bedford-McMullen type, for almost all values of the parameter in a certain interval; however, intheir work the parameter determines the contraction ratios, while we work with fixed contractionsand vary the translations. 6hroughout this section let s = log (cid:80) mi =1 N log m/ log ni log m be the target Hausdorff dimension.Let µ be the McMullen measure on the symbolic space D N , i.e., the Bernoulli measure withweights p ( i,j ) = N log m/ log n − i /m s and let µ = µ ◦ π − be the natural projection of µ onto D N , where in a slight abuse of notationwe let π denote projection onto the first coordinate in both the symbolic and geometric spaces. Inparticular, µ is a Bernoulli measure with weights p i = N log m/ log ni /m s . Let Π t denote the natural coding map from D N to F t and Π t denote the natural coding map from D N to π ( F t ), the projection of F t onto the horizontal axis. Note that µ ◦ Π − t is nothing elsethan the self-similar measure for the IFS { S i,t } i ∈ D with weights ( p i ) i ∈ D . The following is thenimmediate from Theorem 4.2 and Proposition 4.3. Lemma 5.1.
Suppose m (cid:62) . Let E be the set of parameters t ∈ [0 , − /m ] D such that the IFS { S i,t } i ∈ D has super-exponential concentration of cylinders. Then E has Hausdorff and packingdimension | D | − . Moreover, if t ∈ [0 , − /m ] D \ E , then dim H (cid:0) µ ◦ Π − t (cid:1) = − (cid:80) i ∈ D p i log p i log m (5.1) Furthermore, if t is algebraic and the IFS { S i,t } i ∈ D does not have an exact overlap, then t / ∈ E . For x ∈ [0 ,
1] let L x = { ( x, y ) : y ∈ R } be the vertical line through the point ( x, Lemma 5.2.
For any t ∈ [0 , − /m ] D we have dim H L x ∩ F t (cid:62) (cid:88) i ∈ D p i log N i log n for µ ◦ Π − t almost all x ∈ π ( F t ) .Proof. Let x = Π t ( i ) ∈ π ( F t ) for some i = ( i , i , . . . ) ∈ D N . It is straightforward to see that L x ∩ F t contains the set (cid:92) k ∈ N (cid:91) j :( i ,j ) ∈ D ... j k :( i k ,j k ) ∈ D S ( i ,j ) ,t ◦ · · · ◦ S ( i k ,j k ) ,t (cid:0) [0 , (cid:1) ∩ L x which is a particular realisation of a 1-variable random self-similar set in the sense of Barnsley-Hutchinson-Stenflo (see e.g. [3]) where the deterministic IFSs used are the natural IFSs of similari-ties induced by the columns in our construction. A realisation of a 1-variable random constructionscorresponds to a particular infinite sequence over the set of deterministic IFSs and the above ex-ample is given by the sequence i . We can apply the dimension results in [3, Section 4] with weights { p i } to obtain that such 1-variable random self-similar sets have dimension (cid:88) i ∈ D p i log N i log n for µ almost all i ∈ D N , which completes the proof.7e will use the following version of Marstrand’s slice theorem, which follows from, for example,[5, Corollary 7.12]. Lemma 5.3.
Let F ⊆ R and let ν be a Borel probability measure with support in R . If dim H ( F ∩ L x ) (cid:62) s for ν almost all x , then dim H F (cid:62) s + dim H ν . We can now complete the proof of Theorem 2.1.
Proof.
In McMullen’s original proof, the calculation of the upper bound of the Hausdorff dimensionof F t is performed using covers by approximate squares on an appropriately defined symbolic space.Since these covers cover each symbolic ‘column’ independently, the upper bound continues to holdwhen projecting to the actual fractal even in the presence of overlaps. Thus dim H ( F t ) (cid:54) s for all t ∈ [0 , − /m ] D .Now suppose t i = t i for some distinct i , i ∈ D , i.e. we have an exact column overlap.Symbolically, this corresponds to replacing two columns with N i , N i rectangles by a single columnwith N (cid:48) (cid:54) N i + N i rectangles. Then in this case we get an upper bounddim H ( F t ) (cid:54) log (cid:16) ( N i + N i ) log m/ log n + (cid:80) i ∈{ ,...,m }\{ i ,i } N log m/ log ni (cid:17) log m < s, using that ( N i + N i ) γ < N γi + N γi for γ ∈ (0 , H ( E ) (cid:62) | D | − H ( F t ). The case m = 2 is not very interesting,as the systems obtained for any t (cid:54) = t are affinely conjugated to each other, and hence havethe same dimensions as the original carpet. Hence from now on we assume m (cid:62)
3. In thissituation the exceptional set E in the theorem can be taken to be precisely E , where E is the( | D | − t ∈ [0 , − /m ] D \ E . It is enough to show thatdim H ( F t ) (cid:62) s . Lemmas 5.1-5.2 and Marstrand’s slice theorem (Lemma 5.3) combine to yielddim H F t (cid:62) − (cid:80) i ∈ D p i log p i log m + (cid:88) i ∈ D p i log N i log n = (cid:88) i ∈ D p i (cid:18) log N i log n − log p i log m (cid:19) = (cid:88) i ∈ D p i s by the definition of p i = s as required. In this section we prove Theorem 2.2 which gives the packing and box dimensions of F t outsideof a small set of exceptional t . This proof is rather more complicated, but perhaps less elegant,than the Hausdorff dimension case given in the previous section. The reason for this is that we donot have a useful analogue of Marstrand’s slice theorem and we do not have an analogue of theMcMullen measure, i.e., a Bernoulli measure with full packing dimension.The box (and packing) dimension of F t depends on three things: the dimension of the projection π ( F t ); the number of maps in the IFS; and how much ‘separation’ there is in the construction. Inorder to find t which give rise to maximal box dimension, these three things have to be controlled,and optimised, simultaneously. Our strategy is somewhat involved and so we briefly describe ithere before we begin the proof. First we apply Hochman’s results to control the dimension of π ( F t ). For a fixed t which maximises dim B π ( F t ), the defining IFS has the correct projectiondimension and enough maps but not enough separation and so we need to ‘approximate it fromwithin’ by finding a subsystem which has almost enough maps and enough separation to give thedesired result. We could find a subsystem of the projected IFS which gives the same projection8imension and guarantees separation, but this introduces a problem: there will be too few mapsin the induced IFS on the square if the original system did not have uniform vertical fibres (thesystem is said to have uniform vertical fibers if the numbers N i = |{ j : ( i, j ) ∈ D }| are constantover i ∈ D ). As such, we employ a technique similar to that used in [10] by finding a subsystemof the IFS on the square which has almost the correct number of mappings, but uniform verticalfibres. The issue now is that the projected dimension may be too small, but we can neverthelessfind a subsystem of the projected IFS with the same (albeit too small) dimension which guaranteesseparation in the induced IFS on the square. Instead of treating the induced subsystem as an IFSin its own right, we consider images of the original overlapping self-affine set by these maps. Thismeans that when we come to cover the, now disjoint, images, we are covering a subset of F t withthe correct projection dimension and, because we have uniform fibres, enough maps.Throughout this section let s = log | D | log m and s = log | D | log m + log | D | / | D | log n . In particular, s is the target almost sure box dimension of F t and s is the target almost sure boxdimension of the relevant projection F t , which plays a key role. We will prove Theorem 2.2 inthe box dimension case and note that, since each F t is compact and has the property that everyopen ball centered in F t contains a bi-Lipschitz image of F t , dim P F t = dim B F t for all t . For moredetails on this useful alternative formulation of packing dimension, see [5, Corollary 3.9].We note the following consequence of Theorem 4.2 and Proposition 4.3. Lemma 6.1.
Fix m (cid:62) . Let E be the set of parameters t ∈ [0 , − /m ] D such that the IFS { S i,t } i ∈ D has super-exponential concentration of cylinders. Then E has Hausdorff and packingdimension | D | − . Moreover, if t ∈ [0 , − /m ] D \ E , then dim H π ( F t ) = dim B π ( F t ) = log | D | log m = s Furthermore, if t is algebraic and the IFS { S i,t } i ∈ D does not have an exact overlap, then t / ∈ E . Let N = | D | , p = (1 /m ) s (1 /n ) s − s = 1 /N and, for k ∈ N , let θ ( k ) = (cid:88) i ∈ D (cid:98) pk (cid:99) = N (cid:98) k/N (cid:99) ∈ N . Note that k − N (cid:54) θ ( k ) (cid:54) k for all k ∈ N . Consider D θ ( k ) and let H k = (cid:110) λ = ( λ , . . . , λ θ ( k ) ) ∈ D θ ( k ) : for all ( i, j ) ∈ D, |{ n ∈ { , . . . , θ ( k ) } : λ n = ( i, j ) }| = (cid:98) pk (cid:99) (cid:111) . It is straightforward to see that | H k | = θ ( k )! (cid:81) i ∈ D (cid:98) pk (cid:99) ! = ( N (cid:98) k/N (cid:99) )!( (cid:98) k/N (cid:99) !) N (6.1)and the IFS { S λ,t } λ ∈ H k corresponding to H k has uniform vertical fibres. Define s k by s k = log | H k | k log n + s (cid:18) − log m log n (cid:19) Lemma 6.2.
We have s k → s as k → ∞ .Proof. We will use a version of Stirling’s approximation for the logarithm of large factorials. Thisstates that for all b ∈ N \ { } we have b log b − b (cid:54) log b ! (cid:54) b log b − b + log b. (6.2)9ote that s − s (cid:18) − log m log n (cid:19) = log N log n . Hence we need to show that log | H k | k → log N as k → ∞ . It is easy to see that | H k | (cid:54) N k for all k . For the opposite inequality, we estimate, for large enough k , log | H k | k = log θ ( k )! − (cid:80) i ∈ D log (cid:98) pk (cid:99) ! k (cid:62) θ ( k ) log θ ( k ) − θ ( k ) − (cid:80) i ∈ D (cid:16) (cid:98) pk (cid:99) log (cid:98) pk (cid:99) − (cid:98) pk (cid:99) + log (cid:98) pk (cid:99) (cid:17) k by (6.2)= θ ( k ) log θ ( k ) − (cid:80) i ∈ D (cid:98) pk (cid:99) log (cid:98) pk (cid:99) k − N log (cid:98) pk (cid:99) k (cid:62) θ ( k ) log θ ( k ) − log( pk ) (cid:80) i ∈ D (cid:98) pk (cid:99) k − N log (cid:98) pk (cid:99) k = θ ( k ) log ( θ ( k ) / ( pk )) k − N log (cid:98) pk (cid:99) k (cid:62) k − Nk log (cid:18) k − Npk (cid:19) − N log (cid:98) pk (cid:99) k → log N as k → ∞ , which completes the proof.Let ε ∈ (0 , s ) and fix k ∈ N large enough to guarantee that s k (cid:62) s − ε which we can do byLemma 6.2. Let H k = { ( i , . . . , i θ ( k ) ) : (cid:0) ( i , j ) , . . . , ( i θ ( k ) , j θ ( k ) ) (cid:1) ∈ H k for some j , . . . , j θ ( k ) } , and consider the IFS of similarities I k = { S i,t } i ∈ H k associated to H k . Since the original projectedIFS { S i,t } i ∈ D had no super-exponential concentration of cylinders by assumption, neither does I k ,and so, by Theorem 4.2, the attractor has Hausdorff and box dimension equal to the new similaritydimension, given by s k = log | H k | k log m . (6.3)The following lemma is a version of a standard result which allows one to approximate the di-mension of a self-similar set with overlaps by subsystems without overlaps. Recall that an IFS { S i } i ∈ A (cid:48) with attractor F satisfies the strong separation condition (SSC) if the S i ( F ) ∩ S i (cid:48) ( F ) = ∅ for distinct i, i (cid:48) ∈ A (cid:48) . If the SSC is satisfied it makes the IFS and corresponding attractor mucheasier to handle. Lemma 6.3.
Let { S i } i ∈ A be an IFS of similarities on [0 , , each with the same contraction ratio a ∈ (0 , , and with self-similar attractor F having Hausdorff and box dimension t and let ε > .There exists (cid:96) ∈ N such that for all (cid:96) (cid:62) (cid:96) there exists a subsystem corresponding to a subset A (cid:96) ⊆ A (cid:96) which satisfies the SSC and | A (cid:96) | (cid:62) − t a − (cid:96) ( t − ε ) . Before proving this lemma, we note that the (cid:96) appearing in A (cid:96) merely indicates dependence on (cid:96) , whereas the (cid:96) appearing in A (cid:96) indicates, as usual, that we consider words of length (cid:96) over A . Proof.
This follows easily from the Vitali covering lemma, which has been used to prove a similarresult previously, see, for example, [24]. We include the details for completeness. Let (cid:96) ∈ N and10onsider the set A l consisting of words of length l over A and the sets { S i ([0 , } i ∈ A (cid:96) . By theVitali covering lemma, we can extract a subset { S i ([0 , } i ∈ A (cid:96) for some A (cid:96) ⊆ A (cid:96) such that F ⊆ (cid:91) i ∈ A (cid:96) S i ([0 , ⊆ (cid:91) i ∈ A (cid:96) S i ([ − , { S i ([0 , } i ∈ A (cid:96) are pairwise disjoint subsets of [0 , A (cid:96) satisfies the SSC. It follows that N a (cid:96) ( F ) (cid:54) | A (cid:96) | . Moreover, the definition of boxdimension implies that for all ε > (cid:96) ∈ N such that for all (cid:96) (cid:62) (cid:96) , N a (cid:96) ( F ) (cid:62) (cid:0) a (cid:96) (cid:1) − ( t − ε ) which completes the proof.We can now complete the proof of Theorem 2.2. Proof.
The upper bound dim B F t (cid:54) s holds for all t ∈ [1 − /m ] D ; this follows from [11, Theorem2.4] which gave an upper bound for the upper box dimension of a class of self-affine carpets(which contains all of the sets F t ) in terms of the box dimensions of the orthogonal projectionswithout any separation conditions. For completeness, we sketch the argument in this case. Since F t = ∪ λ ∈ D (cid:96) S λ,t F t , we have N r ( F t ) (cid:54) (cid:88) λ ∈ D (cid:96) N r ( S λ,t F t ) . Let r = (1 /n ) (cid:96) . Since S λ,t maps the unit square to a rectangle of size (1 /m ) (cid:96) × r , it follows that N r ( F t ) (cid:54) C | D | (cid:96) N rm (cid:96) ( πF t )where C > r, l, m or n . But πF t is a self-similar set withsimilarity dimension s = log | D | / log m . As the upper box counting dimension is bounded aboveby the similarity dimension, for any ε > C ε > N ρ ( πF t ) (cid:54) C ε ρ s − ε for all ρ >
0. Applying this with ρ = rm (cid:96) and putting all estimates together yields the desired upperbound.Now, as in the proof of Theorem 2.1, pick t such that t i = t i for some distinct i , i ∈ D .This merges two columns and does not increase the total number of rectangles, so applying theupper bound to the resulting system we getdim B F t (cid:54) log( | D | − m + log | D | / ( | D | − n < s, since m < n . It follows that dim H E (cid:62) | D | − m = 2 is straightforward, so weassume that m (cid:62)
3. Again, in this setting the exceptional set E in the theorem can be taken to beprecisely E , where E is the ( | D |− t ∈ [0 , − /m ] D \ E .For this t we will prove that the lower box dimension is at least s , which completes the proof.We will apply Lemma 6.3 to the IFS of similarities I k corresponding to H k . In particular, thereexists (cid:96) ∈ N such that for all (cid:96) (cid:62) (cid:96) we may find a subset G k,(cid:96) ⊆ H (cid:96)k such that the system { S i,t } i ∈ G k,(cid:96) corresponding to G k,(cid:96) satisfies the SSC, and | G k,(cid:96) | (cid:62) − s k (1 /m ) − k(cid:96) ( s k − ε ) = 3 − s k (1 /m ) k(cid:96)ε | H k | (cid:96) (6.4)by (6.3). Fix such an (cid:96) (cid:62) (cid:96) and consider the set G k,(cid:96) = { (( i , j ) , . . . , ( i k(cid:96) , j k(cid:96) )) ∈ D k(cid:96) : ( i , . . . , i k(cid:96) ) ∈ G k,(cid:96) } and observe that, since H k had uniform vertical fibres, | G k,(cid:96) | = (cid:18) | H k || H k | (cid:19) (cid:96) | G k,(cid:96) | (cid:62) | H k | (cid:96) − s k (1 /m ) k(cid:96)ε (6.5)11y (6.4). Let r = (1 /n ) k(cid:96) and consider the set F := (cid:91) λ ∈ G k,(cid:96) S λ,t ( F t ) ⊆ F t . (Note that F depends on k, (cid:96) and t , but we do not display this dependence.) We will adopt the ρ -grid definition of N ρ ( · ). It follows immediately from the definition of box dimension that thereexists a constant C ε > ε such that for all ρ ∈ (0 ,
1] we have N ρ (cid:0) π ( F t ) (cid:1) (cid:62) C ε ρ − ( s − ε ) . (6.6)Notice that each set S λ,t ( F ) in the composition of F is contained in the rectangle S λ,t (cid:0) [0 , (cid:1) which has height r and base length (1 /m ) k(cid:96) . It follows that N r ( S i,t ( F t )) (cid:62) N r (1 /m ) − k(cid:96) (cid:0) π ( F t ) (cid:1) (cid:62) C ε (cid:32) (1 /m ) k(cid:96) r (cid:33) s − ε (6.7)by (6.6). Let U be any closed square of sidelength r . Since { S λ,t (cid:0) [0 , (cid:1) } λ ∈ G k,(cid:96) is a collection ofrectangles which can only intersect at the boundaries each with shortest side having length r , it isclear that U can intersect no more than 9 of the sets { S λ,t ( F t ) } λ ∈ G k,(cid:96) . It follows that (cid:88) λ ∈ G k,(cid:96) N r (cid:0) S λ,t ( F t ) (cid:1) (cid:54) N r (cid:32) (cid:91) λ ∈ G k,(cid:96) S λ,t ( F t ) (cid:33) (cid:54) N r ( F t ) . This yields N r ( F t ) (cid:62) (cid:88) λ ∈ G k,(cid:96) N r (cid:0) S i,t ( F ) (cid:1) (cid:62) | G k,(cid:96) | C ε (cid:32) (1 /m ) k(cid:96) r (cid:33) s − ε by (6.7) (cid:62) C ε − s k r − ( s k − ε ) | H k | (cid:96) (cid:16) (1 /m ) s (1 /n ) s k − s (cid:17) k(cid:96) by (6.5) (cid:62) C ε r − ( s k − ε ) (cid:32) | H k | (1 /m ) ks (1 /n ) k ( s k − s ) (cid:33) (cid:96) = C ε r − ( s k − ε ) by the definition of s k . This is valid for all (cid:96) (cid:62) (cid:96) and hencelim inf (cid:96) →∞ log N (1 /n ) k(cid:96) ( F t ) − log(1 /n ) k(cid:96) (cid:62) s k − ε (cid:62) s − ε. Fortunately, letting r tend to zero through the sequence (1 /n ) k(cid:96) as (cid:96) → ∞ is sufficient to give alower bound on the lower box dimension of F t , see [5, Section 3.1] and so, since ε can be madearbitrarily small, this yields dim B F t (cid:62) s as required. We note that the fact we used reciprocals of integers 1 /m and 1 /n as the principle contractions inthe defining system was not important. We could equally well have chosen arbitrary a, b ∈ (0 , / a > b . Moreover, we could allow different arrangements of the a × b rectangles in each fixedcolumn provided they do not overlap, i.e. they need not be integer multiples of b apart. See Figure3 for an example of a pattern of this more general type. Thus, we have the following result.12igure 3: A more general column pattern to which our results apply, and a concrete realization inwhich the columns overlap. Theorem 7.1.
Let < b < a (cid:54) / . Suppose there are numbers { w ij } (cid:54) i (cid:54) m, (cid:54) j (cid:54) N i for someintegers (cid:54) m (cid:54) /a, N i (cid:62) such that (cid:54) w ij (cid:54) − b and | w ij − w ij | (cid:62) b for all i, j, j (cid:54) = j .Given t ∈ R m , let F t be the attractor of the IFS { S ( i,j ) ,t } (cid:54) i (cid:54) m, (cid:54) j (cid:54) N i , where S ( i,j ) ,t ( x, y ) = ( ax, by ) + ( t i , w ij ) . Then for all t ∈ [0 , − a ] m such that the IFS { ax + t i } mi =1 does not have super exponential concen-tration of cylinders, we have dim H ( F t ) = log (cid:80) mi =1 N log a/ log bi log(1 /a ) , dim B ( F t ) = log m log(1 /a ) + log( (cid:80) mi =1 N i /m )log(1 /b ) . In particular, this holds for all t outside of an exceptional set E (depending only on a ) of Hausdorffand packing dimension m − .Moreover, if a is algebraic, then it also holds for all algebraic t such that the IFS { ax + t i } mi =1 does not have an exact overlap. We make some remarks on the assumptions of the above theorem. The restrictions w ij ∈ [0 , − b ]and t ∈ [0 , − a ] are not essential; they simply make sure that F t is a subset of the unit square, whichcan always be achieved by a change of coordinates. The hypothesis | w ij − w ij | (cid:62) b guaranteesthat the rectangles in each column are non-overlapping, and this is an obvious necessary conditionin general. The assumption m (cid:54) /a is meant to ensure that the similarity dimension of theprojected self-similar set (and measure) is at most 1, and we require a (cid:54) / m (cid:54) /a isnot a vacuous assumption. For the box dimension calculation, these are not essential restrictions:if m > /a and a ∈ (0 ,
1) is arbitrary, the proof goes through just by replacing log m/ log(1 /a ) by1 at the points where the dimension of the projection comes up (in the proofs of both the lowerand upper bound), to give the same result with the formula for the box dimension replaced by1 + log( a (cid:80) mi =1 N i )log(1 /b ) . For Hausdorff dimension, however, the result fails if a > /
2. Recall that a
Pisot number is analgebraic integer > < m = 2, 1 /a to be any Pisot number in (1 , b ∈ (0 , a ) and N = N = 1. Note that the translations do not play a role when m = 2 (as long asthe maps do not have the same fixed point, which is a co-dimension one phenomenon in parameterspace). In this case it was shown by Przytycki and Urba´nski in [25] that the Hausdorff dimensiondrops from the “expected” value, see also [26, Theorem 15] for a simpler proof using McMullen’smethod from [23]. In fact, the latter proof shows that the same phenomenon holds if 1 /a ∈ (1 , m (cid:62) N i . We note that the issue here is that a > /
2; it would beinteresting to understand the behaviour of Hausdorff dimension when m > /a but a < / .2 Final remarks There are various other directions in which this work could move. A further generalisation in thedirection of Theorem 7.1 would be to consider Lalley-Gatzouras type columns [20], which wouldallow for rectangles of varying heights and widths. We do not see any difficulty in extending ourarguments to cover this setting, but do not pursue it here to aid clarity of exposition. One couldalso consider random versions of the more general self-affine carpets considered by Bara´nski [1],Feng-Wang [9] or Fraser [11], however, in these cases our random model seems less natural as thedimension can depend on both principal projections, rather than just π .In this article we have focused on the case with “column alignment” where the dimension dropsfrom the affinity dimension, but Theorems 2.1, 2.2 and 7.1 hold also when each column has just onerectangle, i.e. there are no special alignments, and in this case the dimension formulas we obtaincoincide with the affinity dimension of the respective systems. Once one gives up the alignment, itmakes sense to consider arbitrary self-affine systems, including those for which there is no dominantdirection for all maps. We hope to address this situation in a forthcoming paper, leading to animprovement on Falconer’s classical theorem from [7] in the case of diagonal maps.Another interesting direction for further work would be to consider self-affine measures sup-ported on our carpets. Then one could ask if, for example, the Hausdorff dimension or L q -spectrumwas almost surely equal to the Hausdorff dimension or L q -spectrum when the columns do not over-lap. For the Hausdorff dimension of self-affine measures, the proof of Theorem 2.1 should applywith minor changes to yield an analogous result. On the other hand, L q -spectra behave more likebox counting dimension, and our methods clearly do not work here as we heavily relied on takingsubsystems, which does not work for measures, but only for sets.One could also look at different notions of dimension other than just the Hausdorff, packingand box dimensions considered here. For example, the Assouad dimension dim A , and its naturaldual the lower dimension dim L , have recently been gaining some attention in the literature onfractals and in particular overlapping self-similar sets [13] and self-affine carpets [22, 12]. Thedefinitions of these dimensions are quite technical and so we do not give them here, but ratherrefer the reader to the papers [21, 12]. One of the key properties of our construction is that thebox and Hausdorff dimensions can never be larger than the box and Hausdorff dimensions of theoriginal Bedford-McMullen carpet. We conclude this section by briefly pointing out via two simpleexamples that this is not the case for Assouad and lower dimension. This is based on the recentwork of Mackay [22] and Fraser [12] who computed these dimensions for certain classes of self-affinecarpets. Theorem 7.2 (Fraser, Mackay) . Let F be a standard Bedford-McMullen carpet. Then dim A F = log | D | log m + max i =1 ,...,m log N i log n and dim L F = log | D | log m + min i =1 ,...,m log N i log n . We will now use this theorem to provide examples showing that the Assouad and lower di-mension can increase from the original values upon translation of columns. The iterated functionsystems and their attractors will be given in the following figures. In all cases we choose m = 3and n = 4. 14igure 4: Two Bedford-McMullen IFSs with attractors F (left) and F (right). Note that we cantranslate the columns in the carpet on the left to obtain the carpet on the right.By Theorem 7.2, we havedim A F = 1 + log 2log 4 < log 2log 3 + 1 = dim A F . Figure 5: Two Bedford-McMullen IFSs with attractors F (left) and F (right). Note that we cantranslate the columns in the carpet on the left to obtain the carpet on the right.By Theorem 7.2, we havedim L F = 1 < log 2log 3 + 1 = dim L F . Throughout this paper we relied on being able to understand the dimension of the projection ontothe first coordinate, which is a self-similar subset of the unit interval, typically with overlaps. TheAssouad dimension and lower dimension also depend on this, however, the Assouad dimension ofa self-similar subset of the unit interval with overlaps does not necessarily equal the Hausdorffdimension. In [13], it was recently shown that in the cases when the Assouad dimension is strictlygreater than the Hausdorff dimension, then it is automatically equal to 1, no matter how smallthe Hausdorff dimension is. The lower dimension, on the other hand, always coincides with theHausdorff dimension, see [12, Theorem 2.11].
Acknowledgements
This work began whilst P.S. was visiting J.M.F. at the University of St Andrews. The work ofJ.M.F. was supported by the EPSRC grant EP/J013560/1 whilst at Warwick and an EPSRCdoctoral training grant whilst at St Andrews. P.S. acknowledges support from Project PICT2011-0436 (ANPCyT).
References [1] K. Bara´nski. Hausdorff dimension of the limit sets of some planar geometric constructions.
Adv. Math. , 210(1):215–245, 2007. 152] B. B´ar´any. Dimension of the generalized 4-corner set and its projections.
Ergodic TheoryDynam. Systems , 32(4):1190–1215, 2012.[3] M. Barnsley, J. E. Hutchinson, and ¨O. Stenflo. V -variable fractals: dimension results. ForumMath. , 24(3):445–470, 2012.[4] T. Bedford.
Crinkly curves, Markov partitions and box dimensions in self-similar sets . PhDthesis, University of Warwick, 1984.[5] K. Falconer.
Fractal geometry . John Wiley & Sons, Inc., Hoboken, NJ, second edition, 2003.Mathematical foundations and applications.[6] K. Falconer and J. Miao. Exceptional sets for self-affine fractals.
Math. Proc. CambridgePhilos. Soc. , 145(3):669–684, 2008.[7] K. J. Falconer. The Hausdorff dimension of self-affine fractals.
Math. Proc. Cambridge Philos.Soc. , 103(2):339–350, 1988.[8] K. J. Falconer. Generalized dimensions of measures on self-affine sets.
Nonlinearity , 12(4):877–891, 1999.[9] D.-J. Feng and Y. Wang. A class of self-affine sets and self-affine measures.
J. Fourier Anal.Appl. , 11(1):107–124, 2005.[10] A. Ferguson, T. Jordan, and P. Shmerkin. The Hausdorff dimension of the projections ofself-affine carpets.
Fund. Math. , 209(3):193–213, 2010.[11] J. M. Fraser. On the packing dimension of box-like self-affine sets in the plane.
Nonlinearity ,25(7):2075–2092, 2012.[12] J. M. Fraser. Assouad type dimensions and homogeneity of fractals.
Trans. Amer. Math. Soc. ,366(12):6687–6733, 2014.[13] J. M. Fraser, A. M. Henderson, E. J. Olson, and J. C. Robinson. On the as-souad dimension of self-similar sets with overlaps.
Preprint , 2014. Available at http://arxiv.org/abs/1404.1016v1 .[14] M. Hochman. On self-similar sets with overlaps and inverse theorems for entropy.
Ann. ofMath. , 180(2):773–822, 2014.[15] M. Hochman. On self-similar sets with overlaps and inverse theorems for entropy in higherdimensions.
Work in progress , 2014.[16] T. Jordan. Dimension of fat Sierpi´nski gaskets.
Real Anal. Exchange , 31(1):97–110, 2005/06.[17] T. Jordan and N. Jurga. Self-affine sets with non-compactly supported random perturbations.
Ann. Acad. Sci. Fenn. Math. , 39:771–785, 2014.[18] T. Jordan and M. Pollicott. Properties of measures supported on fat Sierpinski carpets.
Ergodic Theory Dynam. Systems , 26(3):739–754, 2006.[19] T. Jordan, M. Pollicott, and K. Simon. Hausdorff dimension for randomly perturbed selfaffine attractors.
Comm. Math. Phys. , 270(2):519–544, 2007.[20] S. P. Lalley and D. Gatzouras. Hausdorff and box dimensions of certain self-affine fractals.
Indiana Univ. Math. J. , 41(2):533–568, 1992.[21] D. G. Larman. A new theory of dimension,
Proc. London Math. Soc. (3) , , (1967), 178–192.[22] J. M. Mackay. Assouad dimension of self-affine carpets. Conform. Geom. Dyn. , 15:177–187,2011.[23] C. McMullen. The Hausdorff dimension of general Sierpi´nski carpets.
Nagoya Math. J. , 96:1–9,1984. 1624] T. Orponen. On the distance sets of self-similar sets.
Nonlinearity , 25(6):1919–1929, 2012.[25] F. Przytycki and M. Urba´nski. On the Hausdorff dimension of some fractal sets.
Studia Math. ,93(2):155–186, 1989.[26] P. Shmerkin. Overlapping self-affine sets.
Indiana Univ. Math. J. , 55(4):1291–1331, 2006.[27] B. Solomyak. Measure and dimension for some fractal families.