Non-existence of the box dimension for dynamically invariant sets
aa r X i v : . [ m a t h . D S ] F e b NON-EXISTENCE OF THE BOX DIMENSION FOR DYNAMICALLYINVARIANT SETS
NATALIA JURGA
Abstract.
One of the key challenges in the dimension theory of smooth dynamical systemsis in establishing whether or not the Hausdorff, lower and upper box dimensions coincidefor invariant sets. For sets invariant under conformal dynamics, these three dimensionsalways coincide. On the other hand, considerable attention has been given to examplesof sets invariant under non-conformal dynamics whose Hausdorff and box dimensions donot coincide. These constructions exploit the fact that the Hausdorff and box dimensionsquantify size in fundamentally different ways, the former in terms of covers by sets of varyingdiameters and the latter in terms of covers by sets of fixed diameters. In this article weconstruct the first example of a dynamically invariant set with distinct lower and upper boxdimensions. Heuristically, this describes that if size is quantified in terms of covers by setsof equal diameters, a dynamically invariant set can appear bigger when viewed at certainresolutions than at others. Introduction
The dimension theory of dynamical systems is the study of the complexity of sets and measureswhich remain invariant under dynamics, from a dimension theoretic point of view. This branchof dynamical systems has its foundations in the seminal work of Bowen [7] on the dimension ofquasicircles and Ruelle [28] on the dimension of conformal repellers, and has since developedinto an independent field of research which continues to receive noteworthy attention in theliterature [1, 8, 10]. For an overview of this extensive field, see the monographs of Pesin [26]and Barreira [3] and the surveys [4, 9, 29].The most common ways of measuring the dimension of invariant sets are through the Haus-dorff and the lower and upper box dimensions, which quantify the complexity of the set inrelated but subtly distinct ways. Roughly speaking, the Hausdorff dimension measures howefficiently the set can be covered by sets of arbitrarily small size, whereas the lower and upperbox dimensions measure this in terms of covers by sets of uniform size, along the scales forwhich this can be done in the most and least efficient way, respectively. Given a subset E ofa separable metric space X , the lower and upper box dimensions are defined bydim B E = lim inf δ → log N δ ( E ) − log δ and dim B E = lim sup δ → log N δ ( E ) − log δ , respectively, where N δ ( E ) denotes the smallest number of sets of diameter δ > E . If the lower and upper box dimensions coincide we call the common value the boxdimension, written dim B , otherwise we say that the box dimension does not exist. The author was supported by an
EPSRC Standard Grant (EP/R015104/1). The author would like to expressher gratitude to Ian Morris and Jonathan Fraser, whose interesting comments and suggestions improved thepaper. The author also thanks Pablo Shmerkin, whose question stimulated this work.
For any subset E ⊆ X , dim H E ≤ dim B E ≤ dim B E (1)where dim H denotes the Hausdorff dimension. A priori each inequality may or may not bestrict. However, when E is invariant under a smooth mapping f , the additional structureimposed by the dynamical invariance of E means that certain properties of f can eitherforce some degree of homogeneity or, on the contrary, inhomogeneity across the set, forcingequalities or strict inequalities in (1) respectively. Characterising which properties of f implyor preclude equalities in (1) is one of the key challenges in dimension theory.A common feature in the dimension theory of smooth conformal dynamics is the coincidenceof the Hausdorff and lower and upper box dimensions for invariant sets. For example, inthe setting of smooth expanding maps, the following result pertains to a more general resultwhich was obtained independently by Gatzouras and Peres [16] and Barreira [2], generalisingprevious results of Falconer [12]. Theorem 1.1 ([2, 16]) . Suppose f : M → M is a C map of a Riemannian manifold M andthat Λ = f (Λ) is a compact set such that f − (Λ) ∩ U ⊂ Λ for some open neighbourhood U of Λ . Additionally, assume that • f is conformal : for each x ∈ M the derivative d x f is a scalar multiple of an isometry, • f is expanding on Λ : there exist constants C > , λ > such that for all x ∈ Λ and u in the tangent space T x M , k d x f n u k ≥ Cλ n k u k . Then for any compact set F = f ( F ) ⊂ Λ , dim B F = dim B F = dim H F. Similar results hold in the setting of smooth diffeomorphisms. For example, if f : M → M is a topologically transitive C diffeomorphism with a basic set Λ, and f is conformal on Λthen dim H Λ = dim B Λ = dim B Λ [2, 26] and an analogous statement holds for the dimensionsof the intersections of Λ with its local stable and unstable manifolds [25, 30].In contrast, in the realm of smooth non-conformal dynamical systems, coincidence of theHausdorff and box dimensions is no longer a universal trait of invariant sets. Indeed, ex-amples of invariant sets with distinct Hausdorff and box dimensions have attracted enormousattention [5, 18, 20, 23, 24, 27] and discussion in surveys [4, 9, 14]. This type of dimension gapresult exploits the fact that the Hausdorff dimension quantifies the size of the set in termsof covers by sets of varying diameters rather than fixed diameters which are used by the boxdimension. Indeed invariant sets of certain non-conformal dynamics will contain long, thinand well-aligned copies of itself, meaning that covering by sets of varying diameter is oftenmore efficient, inducing this type of dimension gap. However, surprisingly there seems to beno mention in the literature of the possibility of a dynamically invariant set with distinctlower and upper box dimensions . Our main result demonstrates the existence of such sets.
ON-EXISTENCE OF THE BOX DIMENSION FOR DYNAMICALLY INVARIANT SETS 3
Theorem 1.2.
There exist integers n > m ≥ and a compact subset of the torus F ⊂ T such that F is invariant, F = T ( F ) , under the expanding toral endomorphism T ( x, y ) = ( mx mod 1 , ny mod 1) and dim B F < dim B F. In particular, the box dimension of F does not exist. Since n > m , T is a non-conformal map. Well-known examples from the literature, suchas Bedford-McMullen carpets [14], demonstrate that equality of the Hausdorff and box di-mensions is not guaranteed in Theorem 1.1 if the assumption of conformality is dropped.Furthermore, Theorem 1.2 indicates that the lower and upper box dimensions need not co-incide either in Theorem 1.1 if the assumption of conformality is dropped. This is arguablya more striking type of dimension gap since, while it is easy to see that sets invariant undernon-conformal dynamics may cease to be homogenous in space, which is verified by the pos-sibility of distinct Hausdorff and box dimensions, one would expect the dynamical invarianceto at least force homogeneity in scale, but our result demonstrates that this too can fail.In particular Theorem 1.2 describes that, when measuring size in terms of covers by sets ofequal diameter, a dynamically invariant set can sometimes appear bigger and at other timesappear smaller depending on the “resolution” we are viewing it at. We highlight that ourconstruction is also significantly more involved than standard examples of invariant sets withdistinct Hausdorff and box dimensions, such as Bedford-McMullen carpets.The dynamics of T on the invariant set F , which will be constructed in §
2, has two keyfeatures which in conjunction induce distinct box dimensions. Firstly, the non-conformalityof T causes the box dimensions of F to be sensitive to the length of time it takes for an orbitof T to move from a subset A ⊂ F which is “entropy maximising” for the dynamics of T toa subset B which is “entropy maximising” for the dynamics of the projection x mx mod 1of T . Secondly the dynamics on F , which can be modelled by a topologically mixing codedsubshift [6] on an appropriate symbolic space, has the property that the length of time ittakes an orbit of T to move from A to B is highly dependent on how long the orbit hasspent in A . In particular, the dynamics fails to satisfy most forms of specification [19]. Theresolution at which F is viewed determines how long the orbits of points of interest (forthe dimension estimates at that particular resolution) spend in A , and combined with theproperties mentioned above this forces distinct box dimensions.Finally, we discuss some connections between Theorem 1.2 and the literature on self-affineand sub-self-affine sets. Let { S i : R d → R d } Ni =1 be a collection of affine contractions, i.e. S i ( · ) = A i ( · ) + t i for each 1 ≤ i ≤ N where A i ∈ GL ( d, R ) with Euclidean norm k A i k < t i ∈ R d . We call { S i } Ni =1 an affine iterated function system. A sub-self-affine set [17], isa non-empty, compact set E ⊂ R d such that E ⊆ N [ i =1 S i ( E ) . (2)If (2) is an equality then E is called a self-affine set , in particular every self-affine set isan example of a sub-self-affine set. Every affine iterated function system admits a uniqueself-affine set. However, there are infinitely many sub-self-affine sets which are not self-affine. NATALIA JURGA
Indeed, the unique self-affine set is the image of the full shift { , . . . , N } N under an appropri-ate projection induced from the family { S i } Ni =1 , whereas sub-self-affine sets are in one-to-onecorrespondence with the projections of subshifts of the full shift. Under suitable “separationconditions” on { S i } Ni =1 , any sub-self-affine set E satisfies f ( E ) ⊆ E for an appropriate piece-wise expanding map f given by the inverses of the contractions. The set F which will beconstructed in § T .The dimension theory of self-affine sets has been an active topic of research since the 1980sand substantial progress has been made in recent years. Sub-self-affine sets were introducedby K¨aenm¨aki and Vilppolainen [17] as natural analogues of sub-self-similar sets which werestudied earlier by Falconer [13]. It is known by the results of Falconer [11] and K¨aenm¨aki andVilppolainen [17] that the box dimension of a generic sub-self-affine sets exists, moreover thishas been verified for large explicit families of planar self-affine sets [1]. However, the followingquestion was open till now. Question 1.3.
Does the box dimension of every (sub-)self-affine set exist?The version of the above question for self-affine sets is a folklore open question within thefractal geometry community, to which the answer is widely conjectured to be affirmative.In contrast, a corollary of our main result is that the answer to Question 1.3 for generalsub-self-affine sets is negative.
Corollary 1.4.
There exist sub-self-affine sets whose box dimension does not exist.
Organisation of paper. In § F and its underlying subshift Σ andoffer some heuristic reasoning behind Theorem 1.2. § § § Construction of ( × m, × n ) -invariant set. Fix m = 2, n = 12. Let ∆ = { ( a, b ) : 1 ≤ a ≤ , ≤ b ≤ , a, b ∈ N } . For any ( a, b ) ∈ ∆define the contraction S ( a,b ) : [0 , → [0 , as S ( a,b ) ( x, y ) = (cid:18) x a − , y
12 + b − (cid:19) which are the partial inverses of T . If i , j ∈ ∆ N with i = j we let i ∧ j denote the longestcommon prefix to i and j , and denote its length by | i ∧ j | . We equip ∆ N with the metric d ( i , j ) = ( | i ∧ j | if i = j i = j The set F that satisfies Theorem 1.2 will be given by the projection of a set Σ ⊆ ∆ N underthe continuous and surjective (but not injective) coding map Π : ∆ N → [0 , given byΠ (( a , b )( a , b ) . . . ) := lim n →∞ S ( a ,b ) ··· ( a n ,b n ) (0)where S ( a ,b ) ··· ( a n ,b n ) denotes the composition S ( a ,b ) ◦ · · · ◦ S ( a n ,b n ) . ON-EXISTENCE OF THE BOX DIMENSION FOR DYNAMICALLY INVARIANT SETS 5
Let Ω = { (1 , i ) } i =3 . For each N ∈ N let Ω N denote words of length N with symbols in Ω,and Ω N the set of infinite sequences with symbols in Ω. Given any ( a, b ) ∈ ∆, ( a, b ) n denotesthe word ( a, b )( a, b ) · · · ( a, b ) of length n . Define C to be the collection of words C := { (1 , , (2 , } ∪ ∞ [ N =1 [ w ∈ Ω N { w (1 , N } and B := { uu u u . . . : u i ∈ C for all i ∈ N , u is a suffix of some word in C} . (3)Then we define the sequence space Σ = B . Equivalently B can be understood as the set ofall infinite sequences which label a one-sided infinite path on the directed graph G in Figure1. G is called the presentation of Σ. v w (1 , z ( w ∈ Ω z , z =13 N )(1 , ,
1) (1 , , , , , , , , , , , ,
12) (2 , Figure 1.
Left: the presentation G of Σ. The dashed loop indicates that foreach N ∈ N and w ∈ Ω N there is a path of length 2 · N which begins andends at v such that its sequence of labels reads w (1 , N . Right: Imagesof [0 , under S ( a,b ) , for each ( a, b ) that labels some edge in G . The darkercoloured rectangles correspond to S ( a,b ) ([0 , ) for ( a, b ) ∈ Ω.It is easy to check that σ (Σ) = Σ where σ : Σ → Σ denotes the left shift map. In particular, Σis an example of a coded subshift , meaning a subshift which can be expressed as the closure ofthe space of all infinite paths on a path-connected (possibly infinite) graph, which were firstintroduced by Blanchard and Hansel [6]. Note that whenever this graph is finite, its codedsubshift is necessarily sofic, and that any ( × m, × n )-invariant set which can be modelledby a sofic shift has a well-defined box dimension which can be explicitly computed [15, 18].Finally we set F = Π(Σ), noting that F = T ( F ) since σ (Σ) = Σ and Π ◦ σ = T ◦ Π.From this it is easy to see that F is a sub-self-affine set for the iterated function system { S ( a,b ) : ( a, b ) ∈ ∆ ∪ { (1 , , (2 , , (1 , }} .While of course it will be necessary to cover the entirety of F and obtain bounds on thesize of this cover at different scales, the proof of Theorem 1.2 will essentially boil down tothe asymptotic difference that emerges between (a) the size of the cover, by squares of side12 − N , of the intersection of F with the collection of rectangles { S i ([0 , ) : i ∈ Ω N } and The set of accumulation points Σ \ B will turn out to be unimportant for our analysis, but for the readersconvenience we provide a description of this set in (4). NATALIA JURGA (b) the size of the cover, by squares of side 12 − N − / , of the intersection of F with thecollection of rectangles { S i ([0 , ) : i ∈ Ω N − / } .Roughly speaking, F occupies a large proportion of the width of each rectangle S i ([0 , ) incase (a). Such a rectangle has width 2 − N and height 12 − N (which equals the sidelength ofsquares in the cover). For any i ∈ Ω N and j ∈ { (1 , , (2 , } N (log 12 / log 2 − , i (1 , N j con-stitutes a legal word in Σ and each S i (1 , N j ([0 , ) has width roughly 12 − N (which equalsthe sidelength of squares in the cover), therefore S i ([0 , ) requires roughly 2 N (log 12 / log 2 − squares to cover it. Importantly, this is a positive power of 12 N , which indicates “growth”in dimension.On the other hand, F occupies a very thin proportion of the width of each rectangle S i ([0 , )in case (b). Each such rectangle has width 2 − N − / and height 12 − N − / (which equalsthe sidelength of squares in this cover). Any i ∈ Σ which begins with a word in Ω N − / can be written as i = ij j for i ∈ Ω N − / , j = (1 , b ) · · · (1 , b N ) and some infinite word j ∈ Σ. In particular, any point in F ∩ S i ([0 , ) belongs to S ij ([0 , ) which has width lessthan − N − / . In particular, only one square of sidelength 12 − N − / is required to cover S i ([0 , ), meaning no further “growth” in dimension at this scale.2.1. Notation.
For any N ∈ N we let Σ N denote the subwords of sequences in Σ of length N . Finite words in S ∞ N =1 Σ N will be denoted in bold, using notation such as i or j whereasinfinite words in Σ will be denoted using typewriter notation such as i and j . For integers n ≥
1, and infinite sequences i = ( a , b )( a , b ) · · · , i | n denotes the truncation of i to itsfirst n symbols i | n = ( a , b ) · · · ( a n , b n ). The same notation is used for truncations of finitewords i = ( a , b ) · · · ( a m , b m ) to its first n symbols i | n = ( a , b ) · · · ( a n , b n ) when m ≥ n . Forany finite word i = ( a , b ) · · · ( a n , b n ), its length is denoted by | i | = n . Given any ( a, b ) ∈ ∆,( a, b ) ∞ denotes the infinite word ( a, b )( a, b ) . . . . For any finite word i we denote the cylinderset by [ i ] := { i ∈ Σ : i | n = i } . We let ∅ denote the empty word.To avoid profusion of constants, we write A . B if A ≤ cB for some universal constant c > A . ε B if A ≤ c ε B for all ε > c ε depends on ε . We write A & B if B . A and A ≈ B if both A . B and B . A , and define the notation A & ε B and A ≈ ε B analogously. 3. Entropy estimates
In this section we obtain estimates on the entropy of important subsets of Σ. Let G N be thewords in Σ N which label a path that starts and ends at the vertex v of the graph G in Figure1. Define h ( G ) := lim sup N →∞ N log G N where G N denotes the cardinality of G N . Lemma 3.1. h ( G ) ≤ log 4 .Proof. Fix N ∈ N . Given a word in G N , let c denote the number of symbols belonging to Ωand a denote the number of symbols belonging to { (1 , , (2 , } , noting that ON-EXISTENCE OF THE BOX DIMENSION FOR DYNAMICALLY INVARIANT SETS 7 (a) 2 c + a = N and(b) c = P ji =1 n i for some integers n , . . . n j .Fix 0 ≤ a ≤ N and let S c be the set of possible ways that c = N − a can be written as anordered sum c = P ji =1 n i . By ordered sum, we mean that if ( n ′ , . . . , n ′ j ) is a permutation of( n , . . . , n j ) such that ( n ′ , . . . , n ′ j ) = ( n , . . . , n j ) then P ji =1 n i is considered a distinct wayof writing c as a sum of powers of 13. We begin by showing that S c grows sub-exponentiallyin c . By [21], the number of ways that c can be written as c = m M + · · · + m k M k with M < · · · < M k and m , . . . , m k ∈ N grows subexponentially in c . Therefore the claim willbe completed by showing that for fixed m , . . . , m k and M , . . . , M k , the number of distinctways in which c = m M + · · · + m k M k can be rewritten as an ordered sum c = P ji =1 n i (i.e. where j = m + · · · + m k and each n i ∈ { M , . . . , M k } ) is also sub-exponential in c .Observe that j ≤ c (eg. consider writing c = 13 · c when c is a multiple of 13), and k ≤ log c log 13 since the M i are strictly increasing. The number of ways of reordering c = m M + · · · + m k M k as an ordered sum c = P ji =1 n i is (cid:18) m + · · · + m k m (cid:19)(cid:18) m + · · · + m k m (cid:19) · · · (cid:18) m k − + m k m k (cid:19) = ( m + · · · + m k )! m ! · · · m k ! ≤ ( m + · · · + m k ) k = j k ≤ (cid:16) c (cid:17) log c log 13 which is clearly sub-exponential in c thereby completing the proof of the claim.Now let us return to considering a word in G N . Following each substring of symbols from Ω,there is a tail of the same length consisting of (1 , a symbols from { (1 , , (2 , } caneither be placed directly after any of these tails or at the beginning of the word. Therefore as-suming that the string contains c = N − a symbols from Ω in blocks of lengths 13 n , . . . , n j (so c = P ji =1 n i ) it follows that there are (cid:0) a + jj (cid:1) ways in which the a symbols from { (1 , , (2 , } can be distributed. Bounding this above by the central binomial term and using the bounds (cid:0) KK (cid:1) ≤ K and j ≤ c we obtain (cid:0) a + jj (cid:1) ≤ a + N − a · hence for all ε > G N ≤ N X a =0 S N − a a + N − a · N − a a . ε N X a =0 a + N − a · N − a a e ε ( N − a )2 . ε (4 e ε ) N where in the third inequality we have used that for sufficiently small ε , · e ε >
1. Thiscompletes the proof of the lemma. (cid:3)
Let I N be the words in Σ N which label a path that ends at v in the graph G in Figure 1.Clearly G N ⊆ I N . Denoting I ∗ = S ∞ N =1 I N and Ω ∗ = S ∞ N =1 Ω N observe thatΣ \ B = { u w : u ∈ I ∗ ∪ ∅ , w ∈ Ω N } ∪ { w (1 , ∞ : w ∈ Ω ∗ ∪ ∅} . (4) NATALIA JURGA
Define h ( I ) = lim sup N →∞ N log I N . Lemma 3.2. h ( I ) ≤ log 4 . Proof.
Fix N ∈ N . Note that any word in I N \ G N is either of the form(a) (1 , z g for g ∈ G N − z or(b) w (1 , z g for z = 13 k for some k ∈ N , w ∈ Ω w where 0 < w < z and g ∈ G N − z − w .Fix any ε >
0. The number of words of the form (a) is N X z =1 G N − z . ε e N ( h ( G )+ ε ) = (4 e ε ) N . The number of words of the form (b) is X z =13 k
In this section, we introduce the sequences of scales which will be used for the lower and upperbox dimension estimates, and prove Theorem 1.2. We also show how the proof of Theorem1.2 can be used to construct an infinitely generated self-affine set whose box dimension doesnot exist.Let δ >
0. We let k ( δ ) denote the unique positive integer satisfying 12 − k ( δ ) ≤ δ < − k ( δ ) and l ( δ ) denote the unique positive integer satisfying 2 − l ( δ ) ≤ δ < − l ( δ ) , noting that k ( δ ) < l ( δ )for sufficiently small δ . By definition l ( δ ) = ⌈ − log δ log 2 ⌉ and k ( δ ) = ⌈ − log δ log 12 ⌉ . ON-EXISTENCE OF THE BOX DIMENSION FOR DYNAMICALLY INVARIANT SETS 9
For i ∈ Σ k and l > k define M ( i , l ) = π ( j ∈ Σ l : j | k = i ) . (5)Our general covering strategy at each scale δ can now be described as follows. For each i ∈ Σ k ( δ ) observe that S i ([0 , ) is a rectangle of height k ( δ ) ≈ δ . In particular N δ (Π(Σ)) ≈ P i ∈ Σ k ( δ ) N δ (Π([ i ])). Notice that for each j ∈ Σ l ( δ ) , S j ([0 , ) has width l ( δ ) ≈ δ . Thereforefor each i ∈ Σ k ( δ ) we cover each projected cylinder Π([ i ]) independently by considering howmany level l ( δ ) columns contain part of the set Π(Σ) inside Π([ i ]). Since by definition thenumber of such columns is given by M ( i , l ( δ )) we obtain N δ (Π(Σ)) ≈ X i ∈ Σ k ( δ ) N δ (Π([ i ])) ≈ X i ∈ Σ k ( δ ) M ( i , l ( δ )) . Define the null sequence { δ N } N ∈ N by δ N = N , noting that k ( δ N ) = 13 N and l ( δ N ) = ⌈ N log 12log 2 ⌉ . Also define the null sequence { δ ′ N } N ∈ N by δ ′ N = N − , noting that k ( δ ′ N ) = ⌈ N − ⌉ and l ( δ ′ N ) = ⌈ N − log 12log 2 ⌉ .In this section we will prove thatlim sup N →∞ log N δ N (Π(Σ)) − log δ N > lim inf N →∞ log N δ ′ N (Π(Σ)) − log δ ′ N . (6)Theorem 1.2 will follow from (6) since it implies that dim B Π(Σ) > dim B Π(Σ).
Lemma 4.1 (Scales with large dimension) . lim sup N →∞ log N δ N (Π(Σ)) − log δ N ≥ log 10log 12 + log 2 (cid:18) − (cid:19) . Proof.
For all w ∈ Ω k ( δ N ) and u ∈ { (1 , , (2 , } l ( δ N ) − k ( δ N ) , w (1 , k ( δ N ) u ∈ Σ l ( δ N ) . Inparticular for any w ∈ Ω k ( δ N ) , M ( w , l ( δ N )) = 2 l ( δ N ) − k ( δ N ) ≈ ( log 12log 2 − N , (7)noting that log 12log 2 >
2. Hence N δ N (Π(Σ)) ≥ N δ N [ w ∈ Ω k ( δN ) Π([ w ]) ≈ X w ∈ Ω k ( δN ) N δ N (Π([ w ])) ≈ X w ∈ Ω k ( δN ) M ( w , l ( δ N )) ≈ N ( log 12log 2 − N . Hence for some uniform constant c > N δ N (Π(Σ)) − log δ N ≥ N log 1013 N log 12 + 13 N ( log 12log 2 −
2) log 213 N log 12 + log c − N log 12= log 10log 12 + log 2 (cid:18) − (cid:19) + log c − N log 12 . The result follows by letting N → ∞ . (cid:3) Lemma 4.2 (Scales with small dimension) . lim inf N →∞ log N δ ′ N (Π(Σ)) − log δ ′ N ≤ √ log 10 + (1 − √ ) log 4log 12 + log 2 − √ log 12 ! . Proof.
Let ε >
0. Recall that for all N ∈ N , − log δ ′ N = 13 N − log 12, k ( δ ′ N ) = ⌈ N − ⌉ and l ( δ ′ N ) = ⌈ N − log 12log 2 ⌉ . Any word i ∈ Σ k ( δ ′ N ) has one of the following forms:(a) i = u for u ∈ I k ( δ ′ N ) ,(b) i = uw for u ∈ I u , w ∈ Ω w where u + w = k ( δ ′ N ),(c) i = w for w ∈ Ω k ( δ ′ N ) ,(d) i = w (1 , z for w ∈ Ω w where w + z = k ( δ ′ N ),(e) i = uw (1 , z for u ∈ I u and w ∈ Ω w where u + w + z = k ( δ ′ N ) and z ≤ w .Let Y a ⊂ Σ k ( δ ′ N ) be the set of words which have the form (a) and let X a ⊂ Σ be the subset { i ∈ Σ : i | k ( δ ′ N ) ∈ Y a } . Define X b , X c , X d , X e and Y b , Y c , Y d , Y e analogously. We note thatthese sets are not all mutually exclusive, for example Y a ∩ Y e = ∅ , but this will not affect ourbounds. Upper bound on N δ ′ N (Π( X a )) . For any j ∈ { (1 , , (2 , } l ( δ ′ N ) − k ( δ ′ N ) and u ∈ I k ( δ ′ N ) , uj ∈ Σ l ( δ ′ N ) . Therefore for each u ∈ I k ( δ ′ N ) , M ( u , l ( δ ′ N )) = 2 l ( δ ′ N ) − k ( δ ′ N ) ≈ N − ( log 12log 2 − . (8)Hence N δ ′ N (Π( X a )) ≈ X u ∈ Y a N δ ′ N (Π([ u ])) ≈ X u ∈I k ( δ ′ N ) M ( u , l ( δ ′ N )) . ε (4 e ε ) N − N − ( log 12log 2 − by Lemma 3.2 and (8). Since ε > − log δ ′ N = 13 N − log 12, wededuce that lim inf N →∞ log N δ ′ N (Π( X a )) − log δ ′ N ≤ log 4log 12 + log 2 (cid:18) − (cid:19) . (9) Upper bound on N δ ′ N (Π( X c )) . Suppose i ∈ X c so i | k ( δ ′ N ) = w ∈ Ω k ( δ ′ N ) . By definition of Σ,either i ∈ Ω N or i begins with u (1 , z for some u ∈ Ω ∗ where | u | ≥ k ( δ ′ N ) = ⌈ N − ⌉ and ON-EXISTENCE OF THE BOX DIMENSION FOR DYNAMICALLY INVARIANT SETS 11 z ≥ N . For N sufficiently large z + | u | ≥ N + 13 N − > N − > ⌈ log 12log 2 13 N − ⌉ = l ( δ ′ N ) . In particular for any w ∈ Ω k ( δ ′ N ) , M ( w , l ( δ ′ N )) = 1 . (10)By (10) N δ ′ N (Π( X c )) ≈ X w ∈ Y c N δ ′ N (Π([ w ])) ≈ X w ∈ Ω k ( δ ′ N ) M ( w , l ( δ ′ N )) = 10 k ( δ ′ N ) ≈ N − . Therefore since − log δ ′ N = 13 N − log 12,lim inf N →∞ log N δ ′ N (Π( X c )) − log δ ′ N ≤ log 10log 12 . (11) Upper bound on N δ ′ N (Π( X d )) . For x > T ( x ) denote the smallest power of 13 whichis greater than or equal to x . Suppose i ∈ X d , so that i | k ( δ ′ N ) = w (1 , z for w ∈ Ω w where w + z = k ( δ ′ N ). Either i = w (1 , ∞ or i begins with w (1 , z ′ j for some j ∈ Σ \ { (1 , } and z ′ ≥ T (max { w, z } ) = T (max { w, k ( δ ′ N ) − w } ) = 13 N where the final equality is because for sufficiently large N max { w, k ( δ ′ N ) − w } ≥ k ( δ ′ N )2 = ⌈ N − ⌉ > N − . Moreover, for sufficiently large Nw + z ′ ≥ N > ⌈ N − log 12log 2 ⌉ = l ( δ ′ N ) . In particular, for any w (1 , z ∈ Y d , M (cid:0) w (1 , z , l ( δ ′ N ) (cid:1) = 1 . (12)By (12) N δ ′ N (Π( X d )) ≈ X i ∈ Y d N δ ′ N (Π([ i ])) ≈ k ( δ ′ N ) − X w =1 X w ∈ Ω w M (cid:16) w (1 , k ( δ ′ N ) − w , l ( δ ′ N ) (cid:17) . ε (10 e ε ) k ( δ ′ N ) ≈ (10 e ε ) N − . Since ε > − log δ ′ N = 13 N − log 12,lim inf N →∞ log N δ ′ N (Π( X d )) − log δ ′ N ≤ log 10log 12 . (13) Upper bound on N δ ′ N (Π( X b )) . Suppose i ∈ X b , so i | k ( δ ′ N ) = uw for u ∈ I u , w ∈ Ω w where u + w = k ( δ ′ N ). Either i = u j where j ∈ Ω N or i begins with uv (1 , z where z = | v | = T ( | v | ) ≥ T ( w ). In particular, for any uw ∈ Y b M ( uw , l ( δ ′ N )) ≤ l ( δ ′ N ) − k ( δ ′ N )+ w − ·T ( w ) . (14)By (14) and Lemma 3.2 N δ ′ N (Π( X b )) ≈ X i ∈ Y b N δ ′ N (Π([ i ])) ≈ k ( δ ′ N ) − X w =1 X u ∈I k ( δ ′ N ) − w X w ∈ Ω w M ( uw , l ( δ ′ N )) . ε k ( δ ′ N ) − X w =1 w (4 e ε ) k ( δ ′ N ) − w l ( δ ′ N ) − k ( δ ′ N )+ w − T ( w ) ≤ N − X w =1 (cid:18) · e ε · (cid:19) w (4 e ε ) k ( δ ′ N ) l ( δ ′ N ) − k ( δ ′ N ) + (15) k ( δ ′ N ) − X w =13 N − +1 (cid:18) · e ε · √ (cid:19) w (4 e ε ) k ( δ ′ N ) l ( δ ′ N ) − k ( δ ′ N ) (16)where in the first sum (15) we have used the trivial lower bound T ( x ) ≥ x and in the secondsum (16) we have used that for all 13 N − + 1 ≤ x ≤ k ( δ ′ N ) − ⌈ N − / ⌉ − √ x ≤ √ N − − ≤ N = T ( x ) . For sufficiently small ε >
0, the first sum (15) can be bounded above by N − X w =1 (cid:18) · e ε · (cid:19) w (4 e ε ) k ( δ ′ N ) l ( δ ′ N ) − k ( δ ′ N ) . ε N − (4 e ε ) k ( δ ′ N ) − N − l ( δ ′ N ) − k ( δ ′ N ) − N − . For sufficiently small ε >
0, 10 · e ε · √ = 5 e ε √ < k ( δ ′ N ) − X w =13 N − +1 (cid:18) · e ε · √ (cid:19) w (4 e ε ) k ( δ ′ N ) l ( δ ′ N ) − k ( δ ′ N ) . ε N − (4 e ε ) k ( δ ′ N ) − N − l ( δ ′ N ) − k ( δ ′ N )+13 N − − · N − < N − (4 e ε ) k ( δ ′ N ) − N − l ( δ ′ N ) − k ( δ ′ N ) − N − . In particular N δ ′ N (Π( X b )) . ε N − (4 e ε ) k ( δ ′ N ) − N − l ( δ ′ N ) − k ( δ ′ N ) − N − ≈ N − (4 e ε ) N − − N − ( log 12log 2 − N − − N − . ON-EXISTENCE OF THE BOX DIMENSION FOR DYNAMICALLY INVARIANT SETS 13
Since ε > − log δ ′ N = 13 N − log 12,lim inf N →∞ log N δ ′ N (Π( X b )) − log δ ′ N ≤ √ log 10 + (1 − √ ) log 4log 12 + log 2 − √ log 12 ! . (17) Upper bound on N δ ′ N (Π( X e )) . If uw (1 , z ∈ Y e with | w | = w and | u | = u then since u + w ≤ k ( δ ′ N ) = ⌈ N − ⌉ we have l ( δ ′ N ) − w − u ≥ l ( δ ′ N ) − ⌈ N − ⌉ > l ( δ ′ N ) − ⌈ log 12log 2 13 N − ⌉ = 0 . In particular M ( uw (1 , z , l ( δ ′ N )) = 2 l ( δ ′ N ) − w − u . (18)By (18) and Lemma 3.2, N δ ′ N (Π( X e )) ≈ X i ∈ Y e N δ ′ N (Π([ i ])) ≈ X w =13 r ≤ N − k ( δ ′ N ) − w − X u =1 X u ∈I u X w ∈ Ω w M ( uw (1 , k ( δ ′ N ) − u − w , l ( δ ′ N )) . ε X w =13 r ≤ N − k ( δ ′ N ) − w − X u =1 (4 e ε ) u w l ( δ ′ N ) − w − u . ε X w =13 r ≤ N − (cid:18) · e ε · (cid:19) w (cid:18) e ε (cid:19) k ( δ ′ N ) l ( δ ′ N ) . ε N − (4 e ε ) k ( δ ′ N ) − N − l ( δ ′ N ) − k ( δ ′ N ) − N − ≈ N − (4 e ε ) N − − N − ( log 12log 2 − N − − N − . Since ε > − log δ ′ N = 13 N − log 12,lim inf N →∞ log N δ ′ N (Π( X e )) − log δ ′ N ≤ √ log 10 + (1 − √ ) log 4log 12 + log 2 − √ log 12 ! . (19)Since the upper bounds in (17) and (19) are strictly greater than the upper bounds in (9),(11) and (13) the proof is complete. (cid:3) Proof of Theorem 1.2.
Π(Σ) is invariant under the smooth expanding map T ( x, y ) = ( mx mod 1 , ny mod 1). Note that to four decimal placeslog 10log 12 + log 2 (cid:18) − (cid:19) ≈ . √ log 10 + (1 − √ ) log 4log 12 + log 2 − √ log 12 ! ≈ . . By Lemmas 4.1 and 4.2dim B Π(Σ) > lim sup N →∞ log N δ N (Π(Σ)) − log δ N > lim inf N →∞ log N δ ′ N (Π(Σ)) − log δ ′ N > dim B Π(Σ) . In particular, the box dimension of Π(Σ) does not exist. (cid:3)
Remark 4.3.
Lemmas 4.1 and 4.2 can also be used to demonstrate the existence of infinitelygenerated self-affine sets whose box dimensions are distinct. Consider the countable familyof affine contractions { S (1 , } ∪ { S (1 , } ∪ ∞ [ N =1 [ w ∈ Ω N { S w (1 , N } which generates the infinitely generated self-affine set E = Π( ˜Σ) where˜Σ := { u u . . . : u i ∈ C for all i ∈ N } . Since E ⊂ F , dim B E ≤ dim B F . On the other hand, for all N ∈ N , w ∈ Ω k ( δ N ) and u ∈ { (1 , , (2 , } l ( δ N ) − k ( δ N ) , [ w (1 , N u ] ∩ ˜Σ = ∅ . Therefore by bounding N δ N ( E ) in the same way as in Lemma 4.1 we deduce that dim B E < dim B E . 5. Further questions
Here we suggest some possible directions for future work.
Question 5.1.
Does there exist an expanding repeller whose box dimension does not exist?Namely, does there exist a smooth expanding map f : M → M of a Riemannian manifold M and compact set Λ = f (Λ) such that Λ = { x ∈ U : f n ( x ) ∈ U, ∀ n ∈ N } for some openneighbourhood U of Λ? Question 5.2.
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