Box dimensions of (×m,×n) -invariant sets
aa r X i v : . [ m a t h . D S ] S e p BOX DIMENSIONS OF ( × m , × n ) -INVARIANT SETS JONATHAN M. FRASER AND NATALIA JURGA
Abstract.
We study the box dimensions of sets invariant under the toral endomorphism( x, y ) ( m x mod 1 , n y mod 1) for integers n > m ≥
2. The basic examples of such setsare Bedford-McMullen carpets and, more generally, invariant sets are modelled by subshiftson the associated symbolic space. When this subshift is topologically mixing and sofic thesituation is well-understood by results of Kenyon and Peres. Moreover, other work of Kenyonand Peres shows that the Hausdorff dimension is generally given by a variational principle.Therefore, our work is focused on the box dimensions in the case where the underlying shiftis not topologically mixing and sofic. We establish straightforward upper and lower boundsfor the box dimensions in terms of entropy which hold for all subshifts and show that theupper bound is the correct value for coded subshifts whose entropy can be realised by wordswhich can be freely concatenated, which includes many well-known families such as β -shifts,(generalised) S -gap shifts, and transitive sofic shifts. We also provide examples of transitivecoded subshifts where the general upper bound fails and the box dimension is actually givenby the general lower bound. In the non-transitive sofic setting, we provide a formula for thebox dimensions which is often intermediate between the general lower and upper bounds. Introduction
We study compact sets invariant under the toral endomorphism T ( x, y ) = ( m x mod 1 , n y mod 1)for integers n > m ≥
2. This dynamical system is a basic and fundamental example ofan expanding non-conformal system and invariant sets have many subtle properties. Thesimplest examples of such invariant sets are the self-affine carpets introduced by Bedford andMcMullen in 1984 [1, 11]. In particular, these are modelled by a full shift. More generally,compact ( × m , × n )-invariant sets are modelled by subshifts on the associated symbolic space.Kenyon and Peres [8] studied the more general case when this subshift is topologically mixingand sofic and in [9] they resolved the Hausdorff dimension case in general by proving avariational principle. These papers provide the starting point for our investigation, whichis focused on the box dimensions in the case where the underlying shift is not topologicallymixing and sofic. We expand the theory in several directions.Let ∆ m , n = { ( a, b ) : 1 ≤ a ≤ m , ≤ b ≤ n , a, b ∈ N } . For any ( a, b ) ∈ ∆ ( m , n ) define thecontraction S ( a,b ) : [0 , → [0 , as S ( a,b ) ( x, y ) = (cid:18) m n (cid:19) (cid:18) xy (cid:19) + (cid:18) a − m b − n (cid:19) . The authors were both supported by an
EPSRC Standard Grant (EP/R015104/1). J. M. Fraser was alsosupported by a
Leverhulme Trust Research Project Grant (RPG-2019-034).
Define the coding map Π : ∆ N m , n → [0 , asΠ (( a , b )( a , b ) . . . ) := lim n →∞ S ( a ,b ) ◦ · · · ◦ S ( a n ,b n ) (0) . Consider any compact ( × m , × n )-invariant set F , meaning that T ( F ) ⊆ F . Then there existsa digit set I ⊆ ∆ m , n and a subshift Σ on the digit set I (meaning a compact σ -invariantsubset Σ ⊆ I N , i.e. σ (Σ) ⊆ Σ where σ : Σ → Σ denotes the left shift map) such that F = Π(Σ). For example, if Σ is the full shift on I then Π(Σ) is a Bedford-McMullen carpet[1, 11]. For brevity, rather than writing sequences in Σ as ( a , b )( a , b ) . . . and finite wordswhich appear in sequences of Σ as ( a , b ) . . . ( a n , b n ) we will for the most part denote bothinfinite sequences and finite words by variables such as i , j , and k .Given a subshift Σ, let Σ ∗ denote the language of Σ, meaning the collection of finite wordswhich appear in sequences i ∈ Σ. For n ∈ N let Σ n denote words in Σ ∗ which have length n . We say Σ is topologically transitive if for all i , j ∈ Σ ∗ there exists k ∈ Σ ∗ such that ikj ∈ Σ ∗ . We say Σ is topologically mixing if there exists N ∈ N such that for all i , j ∈ Σ ∗ there exists k ∈ Σ N such that ikj ∈ Σ ∗ . Recall that the topological entropy of Σ is definedas h (Σ) := lim n →∞ n log n , where the limit exists by submultiplicativity arguments.The ( × m , × n )-invariant sets are typically fractal and a key question of interest is in com-puting their dimensions, especially Hausdorff and box dimensions, see [1, 5, 8, 9, 11]. For morebackground on Hausdorff and box dimensions, see [6]. We write dim H , dim B , and dim B for the Hausdorff, lower and upper box dimensions, respectively. The lower and upper boxdimensions 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 . It is useful to keep in mind that, for all bounded sets E in Euclidean space,dim H E ≤ dim B E ≤ dim B E. Moreover, if the upper and lower box dimensions coincide we simply refer to the box dimen-sion, written dim B . In the case where Σ is a full shift (over a restricted alphabet I ⊆ ∆ m , n ),the box and Hausdorff dimensions were computed independently by Bedford [1] and McMul-len [11]. If Σ is a topologically mixing sofic subshift, then the box and Hausdorff dimensionswere given by Kenyon and Peres [8]. We say that a subshift is sofic if it can be presented bya finite directed labelled graph G (see Section 3 for a more precise definition). If Σ is a topo-logically transitive subshift of finite type, then the box dimension was computed by Deliu etal [5]. The only progress beyond the sofic setting is provided by Kenyon and Peres [9] wherethey show that for any compact ( × m , × n )-invariant set the Hausdorff dimension is given bya variational principle, that is, as the supremum of the Hausdorff dimensions of ( × m , × n )-invariant measures supported on the set. It is also shown that there exists a maximising(ergodic) measure, which achieves the Hausdorff dimension of the set. Moreover, it is shownin [9] that the Hausdorff dimension of an ergodic ( × m , × n )-invariant measure is given by aLedrappier-Young formula. In some sense, this settles the question of Hausdorff dimension.The box dimensions of ( × m , × n )-invariant sets remains an interesting open programme. Werecall the box dimension result of Kenyon and Peres which is the current state of the art. Let π : Σ → π Σ denote the projection mapping π (( a , b )( a , b ) . . . ) = a a . . . . In particular, π Σ is itself a subshift.
OX DIMENSIONS OF ( × m , × n )-INVARIANT SETS 3 Theorem 1.1 (Proposition 3.5, [8]) . Suppose Σ is a topologically mixing sofic subshift. Then dim B Π(Σ) = h ( π Σ)log m + h (Σ) − h ( π Σ)log n . (1)It is straightforward to construct an example where (1) does not hold for a general soficsubshift Σ. For example fix m = 2, n = 4 and I = { (1 , , (1 , , (1 , , (1 , , (2 , } anddenote Σ = { (1 , , (2 , } N , Σ = { (1 , , (1 , , (1 , } N . Consider the subshift of finite typeΣ = Σ ∪ Σ . Then,dim B Π(Σ) = max { dim B Π(Σ ) , dim B Π(Σ ) } = 1 <
1+ log 3 − log 2log 4 = h ( π Σ)log m + h (Σ) − h ( π Σ)log n where in the second equality we apply (1) to dim B Π(Σ ) and dim B Π(Σ ). This exampleheavily relies on a lack of transitivity.We fully resolve the sofic case by finding a formula that holds for any sofic subshift (whichis not just the maximum over irreducible parts as above) and which simplifies to (1) inthe transitive case, thus generalising Theorem 1.1 from topologically mixing to topologicallytransitive.We say a graph G is irreducible if given any pair of vertices v, w ∈ G there is a path in G from v to w . Given a finite directed labelled graph G which presents Σ, let { G i } ki =1 denotethe irreducible components of G , meaning the maximal irreducible subgraphs of G . Eachsubgraph G i therefore presents a subshift Σ G i ⊆ Σ. Given 1 ≤ i ≤ k we let { i } + denote theset of all indices 1 ≤ j ≤ k such that there is a path in G from a vertex in G i to a vertex in G j , noting that { i } + is necessarily non-empty since we always have i ∈ { i } + . Theorem 1.2.
Let Σ be a sofic subshift which is presented by a graph G . Let { G , . . . , G k } be the irreducible components of G . Then dim B Π(Σ) = max ≤ i ≤ k (cid:26) h (Σ G i )log n + max j ∈{ i } + h ( π Σ G j ) (cid:18) m − n (cid:19)(cid:27) . (2)As in [8, Proposition 3.5], each entropy h (Σ G i ) and h ( π Σ G i ) can be expressed in termsof the spectral radius of the adjacency matrix of an appropriate right-resolving presentation(of Σ G i and π Σ G i respectively). When Σ is topologically transitive and sofic, Σ can bepresented by an irreducible labelled graph, therefore (2) simplifies to (1). Additionally, wecan also recover (1) for some sofic subshifts which are not topologically transitive, undersome assumptions on the “position” of the entropy maximising irreducible components, seeCorollary 3.1. Moreover, the “position” of the entropy maximising irreducible componentscan also determine whether or not the Hausdorff and box dimensions are equal, see Corollary3.2.Next, we turn to more general subshifts. By bounding dim B Π(Σ) (and dim H Π(Σ)) belowby the box dimension of its projection and by a crude estimate involving entropy and thelarger Lyapunov exponent, we show (see Proposition 2.1) that any invariant set satisfies atrivial lower bound of dim B Π(Σ) ≥ max n h ( π Σ)log m , h (Σ)log n o . On the other hand, we also show (seeProposition 2.1) that the right hand side of (1) is a trivial upper bound on dim B Π(Σ) ingeneral. While Theorem 1.2 demonstrates that the box dimension can drop from this trivialupper bound if Σ is not topologically transitive, it is interesting to ask whether transitivityis sufficient for (1) to hold for general subshifts. We answer this in the negative:
JONATHAN M. FRASER AND NATALIA JURGA
Theorem 1.3.
There exists a topologically transitive subshift Σ with < h ( π Σ) < h (Σ) and dim B Π(Σ) = max (cid:26) h (Σ)log n , h ( π Σ)log m (cid:27) . In particular, in the above example the trivial lower bound is in fact the exact value ofthe box dimension. Moreover this box dimension is clearly strictly smaller than the trivialupper bound and we can modify our example such that either of the trivial lower boundsequals the box dimension. The subshift Σ that we construct towards the proof of Theorem1.3 falls into the class of coded subshifts . Coded subshifts, which were first introduced in [3]and include the well-known subclasses of S -gap shifts, β -shifts and Dyck shifts, are subshiftswhich can be presented by an irreducible (but not necessarily finite), directed labelled graph(see Section 4). In particular, they clearly extend the class of transitive sofic subshifts andprovide a natural and interesting class to investigate which, unlike subshifts of finite type andsofic subshifts in general, cannot be handled by techniques that depend on finiteness of thepresentation.A useful equivalent characterisation of coded subshifts is that a subshift Σ is coded if thereexists a countable collection of finite words C , which we call generators, such that Σ is theclosure of the set of sequences obtained by freely concatenating the generators. In particular, π Σ is also a coded subshift which is generated by π C . We say that a coded subshift Σ has unique decomposition with respect to C if no finite word can be written as a concatenation ofgenerators in C in distinct ways.We will show that if the entropy of a coded subshift Σ and π Σ can be realised by countingwords which can be obtained by concatenating their (respective) generators, then the boxdimension dim B Π(Σ) equals the trivial upper bound given in Proposition 2.1. In particular let G n denote all words of length n in Σ ∗ which can be written by concatenating generators from C . Analogously, π G n are all words of length n in ( π Σ) ∗ which can be written by concatenatinggenerators from π C . We denote h := lim sup n →∞ n log G n and h π := lim sup n →∞ n log π G n . Theorem 1.4.
Let Σ be a coded subshift and suppose h = h (Σ) and h π = h ( π Σ) . Then dim B Π(Σ) = h ( π Σ)log m + h (Σ) − h ( π Σ)log n . (3)Note that the example constructed in Theorem 1.3 satisfies h < h (Σ). A drawback ofTheorem 1.4 is that in general it may not be straightforward to verify the equalities h = h (Σ)and h π = h ( π Σ). However, under the assumption of unique decomposition of Σ and π Σ weprovide a more practical way of checking that the conclusion of Theorem 1.4 holds. This isbased on the fact that under the assumption of unique decomposition of Σ and π Σ (withrespect to C and π C ), h and h π can be understood as the Gurevic entropies of countablegraphs associated with the coded subshifts Σ and π Σ (see Section 4). This allows us to employclassical tools from the theory of countable Markov shifts which yields checkable criteria forTheorem 1.4 to hold, see Theorem 1.5 below, whose statement requires the introduction ofsome further notation.
OX DIMENSIONS OF ( × m , × n )-INVARIANT SETS 5 Let L n denote all words of length n in Σ ∗ which appear at the beginning or end of somegenerator in C , analogously π L n are all words of length n which appear at the beginning orend of some generator in π C . We denote ℓ := lim sup n →∞ n log L n and ℓ π := lim sup n →∞ n log π L n . Let C n denote words in C of length n ∈ N , analogously π C n denotes words in π C of length n . Finally, define functions f, f π : [0 , ∞ ) → (0 , ∞ ] by f ( x ) = ∞ X n =1 C n e − nx and f π ( x ) = ∞ X n =1 π C n e − nx . (4) Theorem 1.5.
Suppose Σ is a coded subshift such that Σ and π Σ have unique decompositionwith respect to C and π C respectively. Additionally, assume f ( ℓ ) > and f π ( ℓ π ) > . Then dim B Π(Σ) = h ( π Σ)log m + h (Σ) − h ( π Σ)log n . The usefulness of Theorem 1.5 lies in the fact that C n , π C n , ℓ and ℓ π are often easy tocompute, which we demonstrate by applying it to generalised S -gap shifts in § π Σ to be uniquely decomposing withrespect to an arbitrary generating set C π rather than π C . In particular if π C n is replaced by C π (words of length n in C π ) in the definition of f π , then Theorem 1.5 remains true underthe assumption that π Σ satisfies unique decomposition with respect to C π .2. Preliminaries
We write a . b to mean there exists a constant C > a ≤ Cb . The implicitconstant C may depend on parameters which are fixed in the hypotheses, such as m, n andΣ, but crucially do not depend on variables in the proofs, such as the covering scale δ . If wewish to emphasise that the C depends on something else, not fixed in the hypothesis such as ε , then we write a . ε b . Similarly, we write a & b to mean b . a and a ≈ b to mean a . b and a & b both hold (analogously a & ε b and a ≈ ε b ). For i ∈ Σ k , we write [ i ] for the cylinderconsisting of elements of Σ with prefix i . We also refer to Π([ i ]) as cylinders, although theseare subsets of the fractal, rather than the symbolic space. Given i ∈ Σ or i ∈ Σ ∗ of lengthat least n + 1 ≥ i | n denote the truncation of i to its first n digits. We also write A to denote the cardinality of a (usually finite) set A .Let δ >
0. Throughout the paper we will let k ( δ ) denote the unique positive integersatisfying n − k ( δ ) ≤ δ < n − k ( δ ) and l ( δ ) denote the unique positive integer satisfying m − l ( δ ) ≤ δ < m − l ( δ ) , noting that k ( δ ) < l ( δ ) for sufficiently small δ . Observe that by definition l ( δ ) ≈ − log δ log m and k ( δ ) ≈ − log δ log n for sufficiently small δ .Here we prove the trivial lower and upper bounds that we alluded to in the introduction.The general strategy of relating covers to allowed words in Σ and π Σ will underpin all of oursubsequent proofs, therefore we take care to include all of the details here.
Proposition 2.1.
For all subshifts Σ ⊂ ∆ N m , n , max (cid:26) h ( π Σ)log m , h (Σ)log n (cid:27) ≤ dim H Π(Σ) ≤ dim B Π(Σ) ≤ dim B Π(Σ) ≤ h ( π Σ)log m + h (Σ) − h ( π Σ)log n . JONATHAN M. FRASER AND NATALIA JURGA
Proof.
Fix ε > δ > k ( δ ) and consider covers of the level k ( δ )cylinders, Π([ i ]), independently. For i ∈ Σ k , write M ( i , l ) = π ( j ∈ Σ l : j | k ( δ ) = i )for the number of children of i at level l > k ( δ ) which lie in distinct columns. Then N δ (Π(Σ)) ≈ X i ∈ Σ k ( δ ) N δ (Π([ i ])) ≈ X i ∈ Σ k ( δ ) M ( i , l ( δ )) ≤ X i ∈ Σ k ( δ ) π Σ l ( δ ) − k ( δ ) (using shift invariance)= k ( δ ) π Σ l ( δ ) − k ( δ ) . ε exp(( h (Σ) + ε ) k ( δ )) exp(( h ( π Σ) + ε )( l ( δ ) − k ( δ ))) . In particular since l ( δ ) ≈ − log δ log m and k ( δ ) ≈ − log δ log n we havelog N δ (Π(Σ)) − log δ . ε ( h (Σ) + ε ) − log δ log n − log δ + ( h ( π Σ) + ε )( − log δ log m − − log δ log n ) − log δ , therefore letting δ → ε > B Π(Σ) ≥ dim H Π(Σ) ≥ dim H π Π(Σ) = h ( π Σ)log m ,where the second inequality follows since the projection π : [0 , → [0 ,
1] to the first co-ordinate is Lipschitz, and the final equality follows from Furstenberg’s result expressing theHausdorff dimension of a subshift in terms of entropy [7]. To see the second lower bound, let µ be a measure of maximal entropy for Σ projected onto Π(Σ). Let ε > δ >
0. A ball of radius δ > . k ( δ ) cylinderseach with mass . ε exp( − k ( δ ) h (Σ)(1 − ε )) . Therefore since k ( δ ) ≈ − log δ log n we deduce that dim B Π(Σ) ≥ dim H Π(Σ) ≥ h (Σ)log n by the massdistribution principle, upon letting ε → (cid:3) Sofic ( × m , × n ) -invariant sets Fix n > m ≥ I ⊆ ∆ m , n . We say that a subshift Σ of the full shift on I is sofic ifthere exists a labelled directed graph G with a finite set of vertices V and edges E , whereeach edge e ∈ E has a label ℓ ( e ) ∈ I , such that for each i ∈ Σ, there exists an infinite path e e . . . ( e i ∈ E ) such that i = ℓ ( e ) ℓ ( e ) . . . . In this case we say that G presents Σ.Given a presentation G of a sofic subshift Σ, there is a unique set of maximal irreduciblesubgraphs { G , . . . , G k } of G , where by maximal we mean that no neighbouring vertices can OX DIMENSIONS OF ( × m , × n )-INVARIANT SETS 7 be added to the subgraph while maintaining irreducibility. We call these the irreduciblecomponents of G . For each 1 ≤ i ≤ k , define the subshift Σ G i ⊆ Σ byΣ G i = { ℓ ( e ) ℓ ( e ) . . . : e ∈ E i } where E i denotes the set of edges in G i . Note that π Σ is a subshift which is presented by thelabelled, directed graph πG , which is constructed from G by projecting each label to its firstcoordinate. Its subgraphs πG i are irreducible components of πG .Construct a labelled directed graph H whose set of vertices is { , . . . , k } and where thereis an edge labelled a from i to j if there is an edge labelled a in G from some vertex in G i tosome vertex in G j . Note that H contains no cycles by definition of irreducible components.Then for each 1 ≤ i ≤ k we can define { i } + , { i } − ⊂ { , . . . , k } by { i } + := { ≤ j ≤ k : there is a path in H from i to j }{ i } − := { ≤ j ≤ k : there is a path in H from j to i } noting that the definition of { i } + is equivalent to that provided in the introduction. We saythat an irreducible component G i is a source if { i } + = { , . . . , k } and we say that G i is a sink if { i } − = { , . . . , k } .Before proving Theorem 1.2 we provide a couple of corollaries which follow from it. First,by exploiting the fact that h (Σ) = max { h (Σ G i ) } ki =1 and h ( π Σ) = max { h ( π Σ G i ) } ki =1 , we canrecover a simpler formula for the box dimension in the case that a source or sink has certainentropy maximising properties. Corollary 3.1.
Let G be a presentation of Σ with irreducible components { G , . . . , G k } .Suppose that either:(a) for some ≤ i ≤ k , G i is a source and h (Σ G i ) = h (Σ) or(b) for some ≤ i ≤ k , G i is a sink and h ( π Σ G i ) = h ( π Σ) .Then dim B Π(Σ) = h ( π Σ)log m + h (Σ) − h ( π Σ)log n . Secondly, by [9] we can describe which conditions guarantee (or preclude) equality of theHausdorff and box dimensions.
Corollary 3.2.
The equality dim H Π(Σ) = dim B Π(Σ) holds if and only if dim B Π(Σ) = max ≤ p ≤ k (cid:26) h ( π Σ G p )log m + h (Σ G p ) − h ( π Σ G p )log n (cid:27) (5) and the measure of maximal entropy on Σ G p (for some p which maximises the expression onthe right hand side of (5) ) projects to the measure of maximal entropy on π Σ G p .In particular, if the maximum in (2) is not obtained for a pair i = j , that is, max ≤ i ≤ k (cid:26) h (Σ G i )log n + max j ∈{ i } + h ( π Σ G j ) (cid:18) m − n (cid:19)(cid:27) > max ≤ p ≤ k (cid:26) h ( π Σ G p )log m + h (Σ G p ) − h ( π Σ G p )log n (cid:27) , (6) then dim H Π(Σ) < dim B Π(Σ) . We will prove Corollaries 3.1 and 3.2 following the proof of Theorem 1.2 in Section 3.2.
JONATHAN M. FRASER AND NATALIA JURGA
Example.
Before providing the proofs of the results of this section, we illustrate The-orem 1.2 with an example. Put n = 5 and m = 3. Let Σ be the subshift of finite typepresented by the graph G in Figure 1. (2 , , ,
1) (2 ,
3) (1 ,
3) (2 , , , , , , Figure 1.
The graph GG has three irreducible components G , G , G . Σ G is the full shift on { (1 , , (2 , , (3 , } and h (Σ G ) = h ( π Σ G ) = log 3. Σ G is the full shift on { (3 , , (3 , , (3 , , (3 , } and h (Σ G ) = log 4, h ( π Σ G ) = 0. Σ G is the full shift on { (1 , , (2 , } and h (Σ G ) = h ( π Σ G ) =log 2.By Theorem 1.2,dim B Π(Σ) = max (cid:26) log 3log 5 + log 3 (cid:18) − (cid:19) , log 4log 5 + log 2 (cid:18) − (cid:19) , log 2log 5 + log 2 (cid:18) − (cid:19)(cid:27) = max (cid:26) , log 2 (cid:18) (cid:19)(cid:27) = log 2 (cid:18) (cid:19) . Note that dim B Π(Σ) < − log 3log 5 = h ( π Σ)log 3 + h (Σ) − h ( π Σ)log 5 . Also note that dim B Π(Σ) > max (cid:26) log 2log 3 + log 3 − log 2log 5 , log 4log 5 , log 2log 3 (cid:27) = max ≤ p ≤ k (cid:26) h ( π Σ G p )log 3 + h (Σ G p ) − h ( π Σ G p )log 5 (cid:27) . Proofs.
We begin by proving Theorem 1.2. Fix a presentation G of Σ and let 1 ≤ i ≤ k and j ∈ { i } + be parameters which achieve the maximum in (2). Roughly speaking, we showthat the box dimension is exhausted by covering all regions Π([ i ]) where i ∈ Σ l ( δ ) labels apath in G which stays in the irreducible component G i for roughly k ( δ ) time steps beforetravelling to the irreducible component G j and staying inside it until time l ( δ ).Given a vertex v in G , let Σ v + n denote all strings in Σ n which label a path beginning at v and Σ v − n denote all strings in Σ n which label a path ending at v . For the lower bound we willrequire the following standard result which relates the entropy of an irreducible sofic subshiftto paths in G . OX DIMENSIONS OF ( × m , × n )-INVARIANT SETS 9 Lemma 3.3.
Let Σ be an irreducible sofic subshift with irreducible presentation G . Then lim n →∞ n log v ± n = h (Σ) . (7) Proof.
Since v ± n ≤ n ≤ P v ∈ V v ± n we have that lim n →∞ n log (max v ∈ V v ± n ) existsand equals the entropy h (Σ). By irreducibility of G , there exists M ∈ N such that for n > M , v ± n ≥ max w ∈ V w ± n − M . Hencelim inf n →∞ n log v ± n ≥ lim n →∞ n log (cid:18) max v ∈ V v ± n (cid:19) = h (Σ) , completing the proof of (7). (cid:3) Proof of lower bound in Theorem 1.2.
Fix ε > δ >
0. Let 1 ≤ i ≤ k and j ∈ { i } + be theindices that maximise the expression in (2). Let v be a vertex in G i and w be a vertex in G j .Since j ∈ { i } + , there exists a path of some length N in G from v to w which is labelled by k ∈ Σ N . We may assume δ is small enough to ensure k ( δ ) > N . Given i ∈ Σ v − G i ,k ( δ ) − N , [ j ∈ Σ w + Gj,l ( δ ) − k ( δ ) Π([ ikj ]) ⊆ Π([ i ]) . Therefore, N δ (Π[ i ]) & N δ [ j ∈ Σ w + Gj,l ( δ ) − k ( δ ) Π([ ikj ]) ≈ π Σ w + G j ,l ( δ ) − k ( δ ) . Hence N δ (Π(Σ)) & X i ∈ Σ v − Gi,k ( δ ) − N N δ ([ i ]) & v − G i ,k ( δ ) − N · π Σ w + G j ,l ( δ ) − k ( δ ) . Therefore by (7), N δ (Π(Σ)) & ε exp (( h (Σ G i ) − ε )( k ( δ ) − N )) exp (cid:0) ( h ( π Σ G j ) − ε )( l ( δ ) − k ( δ )) (cid:1) and by letting δ → ε > (cid:3) For the upper bound we will require the standard result that the entropy of a sofic subshiftequals the maximum entropy of its irreducible subshifts.
Lemma 3.4.
Let Σ be a sofic subshift presented by a graph G which has irreducible components G , . . . , G k . Then h (Σ) = max ≤ i ≤ k h (Σ G i ) .Proof. It is only necessary to prove the upper bound which follows by bounding n aboveby X ≤ m ,...,m l ≤ k X n m + ··· + n ml = n G m ,n m · G m ,n m · · · G ml ,n ml , where Σ G i ,n denotes all distinct strings of length n that appear in the subgraph G i , andbounding G mi ,n mi in terms of h (Σ G mi ). (cid:3) Let Σ G i + n denote all words in Σ n which label paths in G that start at any vertex in G i , andΣ G i − n denote all words in Σ n which label paths in G that end at any vertex in G i . Proof of upper bound in Theorem 1.2.
Fix ε > δ >
0. Fix any 1 ≤ i ≤ k and i ∈ Σ G i − n .Writing M ( i , l ) = π ( j ∈ Σ l : j | k ( δ ) = i ) and noting that by shift invariance we have M ( i , l ( δ )) ≤ π Σ G i + l ( δ ) − k ( δ ) it follows that N δ (Π(Σ)) . k X i =1 X i ∈ Σ Gi − k ( δ ) N δ (Π([ i ])) ≈ k X i =1 X i ∈ Σ Gi − k ( δ ) M ( i , l ( δ )) . G i − k ( δ ) π Σ G i + l ( δ ) − k ( δ ) . Note that any path that ends at a vertex in G i is contained in the minimal subgraph E of G which contains the irreducible components { G j } j ∈{ i } − and all edges between these com-ponents. Similarly, any path in G that begins at a vertex in G i is contained in the minimalsubgraph F of G where F contains the irreducible components { G j } j ∈{ i } + , and all edgesbetween these components. By Lemma 3.4,lim sup n →∞ n log G i − n ≤ h (Σ E ) = max j ∈{ i } − h (Σ G j )and lim sup n →∞ n log π Σ G i + n ≤ h ( π Σ F ) = max j ∈{ i } + h ( π Σ G j ) . Therefore, N δ (Π(Σ)) . ε exp (cid:18) k ( δ )( max j ∈{ i } − h (Σ G j ) + ε ) (cid:19) exp (cid:18) ( l ( δ ) − k ( δ ))( max j ∈{ i } + h ( π Σ G j ) + ε ) (cid:19) , and by letting δ → ε > (cid:3) Proof of Corollary 3.1 .
First, to see (a), let G i be the source. By assumption h (Σ G i ) = h (Σ).By Lemma 3.4 there exists 1 ≤ j ≤ k such that h ( π Σ G j ) = h ( π Σ). Moreover j ∈ { i } + bydefinition of a source. Hence by (2),dim B Π(Σ) ≥ h (Σ G i )log n + h ( π Σ G j ) (cid:18) m − n (cid:19) = h ( π Σ)log m + h (Σ) − h ( π Σ)log n . On the other hand, the upper bound follows from Proposition 2.1, completing the proof of(a).Similarly, to see (b), let G j be the sink. By assumption h ( π Σ G j ) = h ( π Σ). Also, by Lemma3.4 there exists 1 ≤ i ≤ k such that h (Σ G i ) = h (Σ). Moreover i ∈ { j } − by definition of asink. Hence by (2),dim B Π(Σ) ≥ h (Σ G i )log n + h ( π Σ G j ) (cid:18) m − n (cid:19) = h ( π Σ)log m + h (Σ) − h ( π Σ)log n . The upper bound follows from Proposition 2.1, completing the proof of (b). (cid:3)
Proof of Corollary 3.2 .
First we recall that by [9], any ergodic invariant measure µ on Σsatisfies the Ledrappier-Young formula:dim H µ = h ( πµ )log m + h ( µ ) − h ( πµ )log n , (8)where dim H µ denotes the Hausdorff dimension of µ , h ( µ ) denotes the measure-theoreticentropy of µ with respect to the left shift map on Σ and h ( πµ ) denotes the measure-theoreticentropy of the pushforward measure πµ with respect to the left shift on π Σ. OX DIMENSIONS OF ( × m , × n )-INVARIANT SETS 11 First, suppose the equality (5) holds and let 1 ≤ p ≤ k be an index that maximises theright hand side of (5). Let µ be the ergodic invariant measure which maximises entropy onΣ G p , which we will assume projects to the measure which maximises entropy on π Σ G p . Thenby (8), dim H Π(Σ) ≥ dim H µ = h ( πµ )log m + h ( µ ) − h ( πµ )log n = dim B Π(Σ)by (5).For the converse, we assume that dim B Π(Σ) = dim H Π(Σ). By [9] there exists an ergodicinvariant measure µ of maximal Hausdorff dimension. Since µ is ergodic, its support mustbe contained in Σ G i for some irreducible component G i of G . Therefore, using (8) we obtaindim H Π(Σ) = dim H µ = h ( πµ )log m + h ( µ ) − h ( πµ )log n ≤ max ≤ p ≤ k (cid:26) h ( π Σ G p )log m + h (Σ G p ) − h ( π Σ G p )log n (cid:27) ≤ dim B Π(Σ) . Now, if (5) does not hold, then the second inequality above is strict and thus we get acontradiction. On the other hand, if (5) holds but the measure of maximal entropy µ p onΣ G p does not project to the measure of maximal entropy on π Σ G p for any 1 ≤ p ≤ k thatmaximises the right hand side of (5), then the first inequality above is strict yielding acontradiction and completing the proof. (cid:3) Coded subshifts
Fix n > m ≥ I ⊆ ∆ m , n . Let C = { c i } ∞ i =1 be a countable family of words on thealphabet I . We call C the generators. Let C n := C ∩ I n . Define B := { sc i c i . . . : c i j ∈ C , s is a suffix of a word in C} . Note that B is σ -invariant but may not be compact. We define Σ = B and say that Σ isa coded subshift. Note that π Σ is also a coded subshift which is generated by π C . Recallthat we say that the coded subshift Σ satisfies unique decomposition with respect to C if nofinite word in Σ ∗ can be written by concatenating generators in C in distinct ways. Notethat if Σ satisfies unique decomposition with respect to C , this does not necessarily meanthat π Σ satisfies unique decomposition with respect to π C , although it may satisfy uniquedecomposition with respect to a different generating set (for instance if Σ satisfies uniquedecomposition with respect to C and { (1 , , (1 , , (1 , , } ⊂ C then since { , } ⊂ π C , π Σ does not satisfy unique decomposition with respect to π C ).Construct a directed labelled graph by fixing a vertex v and, for each i ∈ N , adding a pathwhich begins and ends at v which is labelled by the generator c i , such that the paths do notintersect each other apart from at the start and end points. We call these generating loops.We say that G presents the coded subshift Σ. Similarly, construct the graph πG from G by The statement of [9, Theorem 1.1] does not make explicit that a measure of maximal Hausdorff dimensioncan be taken to be ergodic, however this is clear from its proof. Note that this notion of the presentation of a coded subshift differs from the notion of the presentation ofa sofic subshift. If Σ is sofic then all infinite sequences in Σ label an infinite path in its presentation, whereasif Σ is coded then this is need not be the case (i.e. if Σ \ B = ∅ ). projecting each label to its first coordinate and removing any generating loop which bears thesame sequence of labels as another generating loop (so that each generating loop is labelleduniquely by a generator in π C ). Then πG presents the coded subshift π Σ.Let G n denote all words in Σ n which label a path in G that begins and ends at the vertex v , and G = S ∞ n =1 G n . In particular, G consists of concatenations of generators. Analogously, π G are all words in π Σ which label a path in πG that begins and ends at the vertex v . Wedenote h := lim sup n →∞ n log G n and h π := lim sup n →∞ n log π G n . Note that the lim sup is necessary in the definitions above, for instance consider a codedsubshift generated by a set of generators which all have even length. Also, note that thesedefinitions are equivalent to those recorded in the introduction.We begin by proving Theorem 1.4, namely that if h = h (Σ) and h π = h ( π Σ) then dim B Π(Σ)equals its trivial upper bound.
Proof of Theorem 1.4.
By Proposition 2.1 it suffices to prove the lower bound. Fix ε > h = h (Σ) and h π = h ( π Σ) we can choose m ε , n ε ∈ N such that G m ε ≥ e m ε ( h (Σ) − ε ) and π G n ε ≥ e n ε ( h ( π Σ) − ε ) . In particular, for all k ∈ N , G km ε ≥ e km ε ( h (Σ) − ε ) and π G kn ε ≥ e kn ε ( h ( π Σ) − ε ) since G kn ≥ ( G n ) k and π G kn ≥ ( π G n ) k . Let δ > k ( δ ) ≥ m ε and l ( δ ) − k ( δ ) ≥ n ε . Hence we can find k ( δ ) − m ε < k ′ ( δ ) ≤ k ( δ ) which is a multiple of m ε ,that is, G k ′ ( δ ) ≥ e k ′ ( δ )( h (Σ) − ε ) . Similarly we can find l ( δ ) − n ε < l ′ ( δ ) ≤ l ( δ ) such that l ′ ( δ ) − k ′ ( δ ) is a multiple of n ε , that is, π G l ′ ( δ ) − k ′ ( δ ) ≥ e ( l ′ ( δ ) − k ′ ( δ ))( h ( π Σ) − ε ) . Denoting M ( i , l ) = π ( j ∈ Σ l : j | k ′ ( δ ) = i ), we have N δ (Π(Σ)) & X i ∈G k ′ ( δ ) N δ (Π([ i ])) & X i ∈G k ′ ( δ ) M ( i , l ′ ( δ )) ≥ G k ′ ( δ ) π G l ′ ( δ ) − k ′ ( δ ) ≥ e k ′ ( δ )( h (Σ) − ε ) e ( l ′ ( δ ) − k ′ ( δ ))( h ( π Σ) − ε ) & ε e k ( δ )( h (Σ) − ε ) e ( l ( δ ) − k ( δ ))( h ( π Σ) − ε ) . The lower bound follows since ε was chosen arbitrarily. (cid:3) Conversely, examples can be constructed where either h < h (Σ) or h π < h ( π Σ) and theconclusion of Theorem 1.4 does not hold, that is, the dimension dim B Π(Σ) drops from thetrivial upper bound. In particular, in § h < h (Σ)and dim B Π(Σ) equals the trivial lower bound max n h (Σ)log n , h ( π Σ)log m o thereby settling Theorem1.3. OX DIMENSIONS OF ( × m , × n )-INVARIANT SETS 13 The drawback of Theorem 1.5 is that generally it is not straightforward to verify theequalities h = h (Σ) and h π = h ( π Σ). However, under the assumption of unique decompositionof Σ and π Σ we can provide more checkable conditions that guarantee the box dimensiondim B Π(Σ) to equal its trivial upper bound (Theorem 1.5).4.1.
Coded subshifts with unique decomposition.
Throughout this short section wewill assume that Σ is a coded subshift with unique decomposition with respect to C and thatthe coded subshift π Σ satisfies unique decomposition with respect to π C . Let G and πG bethe presentations of Σ and π Σ as detailed in the previous section. Let p G ( v, n ) denote thenumber of paths of length n in G which begin and end at v and p πG ( v, n ) denote the numberof paths of length n in πG which begin and end at v and write h G := lim sup n →∞ n log p G ( v, n ) and h πG := lim sup n →∞ n log p πG ( v, n ) . In particular, h G is the Gurevic entropy of G and h πG is the Gurevic entropy of πG , notingthat the limsups are actually independent of the choice of vertex. Since Σ and π Σ satisfyunique decomposition with respect to C and π C respectively, we have h = h G and h π = h πG .This will enable us to apply techniques from the theory of countable Markov shifts.Recall from the introduction the functions f, f π : [0 , ∞ ) → (0 , ∞ ] which we defined by f ( x ) = ∞ X n =1 C n e − nx and f π ( x ) = ∞ X n =1 π C n e − nx . (9)We can apply the classical work of Vere-Jones [13] to deduce behaviour of f and f π at h and h π . Lemma 4.1.
Let Σ and π Σ be coded subshifts with unique decomposition with respect togenerating sets C and π C respectively. Then f ( h ) ≤ f π ( h π ) ≤ . (10) Proof.
Let q G ( v, n ) denote the number of generating loops of length n in G . Let q πG ( v, n )denote the number of generating loops of length n in πG . In particular, q G ( v, n ) = C n and q πG ( v, n ) = π C n , so f ( x ) = P ∞ n =1 q G ( v, n ) e − nx and f π ( x ) = P ∞ n =1 q πG ( v, n ) e − nx . By usingthe recurrence relation p ( v, n ) = P nk =1 p G ( v, n − k ) q G ( v, k ) and an application of a renewaltheorem, Vere-Jones [13, Lemma 2] showed that P ∞ n =1 q G ( v, n ) e − nh G ≤
1, and analogously P ∞ n =1 q πG ( v, n ) e − nh πG ≤
1. This implies the result since h = h G and h π = h πG by uniquedecomposition. (cid:3) Next recall the definitions from the introduction ℓ := lim sup n →∞ n log L n and ℓ π := lim sup n →∞ n log π L n where L n and π L n denote words of length n which appear at the beginning or end of generatorsin C and π C respectively. In [4] it was shown that ℓ < h implies existence of a measure ofmaximal entropy for the coded subshift Σ. The behaviour of f at a quantity related to ℓ wasused in [12] to characterise coded subshifts in terms of the properties of their measures ofmaximal entropy. To prove Theorem 1.5 we will show that f ( ℓ ) > ℓ < h by using (10) and the factthat f is strictly decreasing, and then by naturally decomposing words in Σ into concaten-ations of generators and subwords of generators we will deduce that this implies h = h (Σ)(respectively h π = h ( π Σ)).
Lemma 4.2.
Suppose Σ is a coded subshift which satisfies unique decomposition with respectto a generating set C and f ( ℓ ) > . Then lim sup n →∞ n log G n = h (Σ) . Proof.
Assume that f ( ℓ ) >
1. Since h G = h by unique decomposition it follows that f ( h ) = f ( h G ) ≤ f is strictly decreasing we have ℓ < h G = h ≤ h (Σ)(the second inequality follows trivially from the definition of G ). We will show that ℓ < h (Σ)implies that h = h (Σ), using arguments similar to those contained in [4, § ε > ℓ − h (Σ) + 2 ε < n ≤ X i + j + k = n L i G j L k . ε X i + j + k = n e ( i + k )( ℓ + ε ) G j = n X j =0 ( n − j ) e ( n − j )( ℓ + ε ) G j . (11)Hence n X j =0 e ( n − j )( ℓ − h (Σ)+2 ε ) ( n − j ) G j j & ε n X j =0 e ( n − j )( ℓ + ε ) ( n − j ) G j j j n & ε . In particular, there exists c ε > n X j =0 e ( n − j )( ℓ − h (Σ)+2 ε ) ( n − j ) G j j ≥ c ε . (12)Since ℓ − h (Σ) + 2 ε < N ∈ N sufficiently large that n − N X j =0 e ( n − j )( ℓ − h (Σ)+2 ε ) ( n − j ) G j j ≤ n − N X j =0 e ( n − j )( ℓ − h (Σ)+2 ε ) ( n − j ) ≤ X m ≥ N e m ( ℓ − h (Σ)+2 ε ) m ≤ c ε . Hence by (12) n X j = n − N +1 e ( n − j )( ℓ − h (Σ)+2 ε ) ( n − j ) G j j ≥ c ε . (13) OX DIMENSIONS OF ( × m , × n )-INVARIANT SETS 15 If we let C be a uniform upper bound on e m ( ℓ − h (Σ)+2 ε ) m (for m ≥ n X j = n − N +1 G j j ≥ c ε C hence for all n ≥ N + 1 we have G j ≥ c ε CN j for some n − N + 1 ≤ j ≤ n . This impliesthat h = lim sup n →∞ n log G n = h (Σ). (cid:3) Clearly by combining Lemma 4.2 with Theorem 1.4 we establish Theorem 1.5: that if Σand π Σ satisfy unique decomposition with respect to C and π C and we have that f ( ℓ ) > f π ( ℓ π ) > B Π(Σ) = h ( π Σ)log m + h (Σ) − h ( π Σ)log n . (14)Hence to establish (14) for uniquely decomposing coded subshifts Σ and π Σ, it is sufficientto calculate f ( ℓ ) and f π ( ℓ π ), which solely depend on C n , π C n , L n and π L n which areoften easy to compute. We demonstrate this with some examples in the next section.4.2. Examples.
In this section, we illustrate Theorems 1.3, 1.4 and 1.5 with some examples.First, in § β -shifts. In § S -gap shifts. Finally in § h < h (Σ) and h π = h ( π Σ) anddim B Π(Σ) = max (cid:26) h (Σ)log n , h ( π Σ)log m (cid:27) < h ( π Σ)log m + h (Σ) − h ( π Σ)log n thereby proving Theorem 1.3.4.2.1. β -shifts. Fix n > m and I ⊂ ∆ m , n . We begin by describing a subshift on the set ofdigits I which is conjugate to the β -shift on the usual digit set { , . . . , ⌊ β ⌋} , for more detailssee [2] or [4] and references therein.Fix a bijection O : { , . . . , |I| − } → I which will determine an ordering on the elementsin I . We extend O to finite and infinite words with digits in { , . . . , |I| − } by O ( i i . . . ) = O ( i ) O ( i ) . . . . Fix |I| < β < |I| + 1 and let ( b n ) n ∈ N be the greedy β -expansion of 1, meaningthe lexicographically maximal solution to ∞ X n =1 b n β − n = 1 . We defineΣ = n O (( x n ) n ∈ N ) : ( x n ) n ∈ N ∈ { , . . . , |I| − } N s.t. σ k (( x n ) n ∈ N ) (cid:22) ( b n ) n ∈ N ∀ k ∈ N o where (cid:22) stands for the lexicographic order. In particular, Σ is conjugated by O to the β -shifton the set of digits { , . . . , ⌊ β ⌋} . Therefore it is known [2] that Σ is a coded subshift wherethe set of generators is given by C = [ n ≥ b n > {O ( b . . . b n − , . . . , O ( b . . . b n − ( b n − } . Note that any word in i ∈ Σ ∗ can be written i = c . . . c k w where c i ∈ C and w is a wordthat appears at the beginning of a generator in C . Hence n ≤ n X k =1 G n − k (15)since for each k ∈ N , O ( b . . . b k ) is the unique word of length k that appears at the beginningof a generator in C . Similarly, we have π Σ n ≤ n X k =1 π G n − k . (16)Using (15) and (16) it is easy to adapt the set of inequalities (11) and the estimates that followit to deduce that h = h (Σ) and h π = h ( π Σ). In particular dim B Π(Σ) = h ( π Σ)log m + h (Σ) − h ( π Σ)log n by Theorem 1.4.4.2.2. Generalised S -gap shifts. We begin by considering the following natural generalisationof the S -gap shifts [4, 10]. Fix any n > m , I ⊂ ∆ m , n and ( i , j ) ∈ I such that π ( I \{ ( i , j ) } ) ∩{ i } = ∅ . Fix a countable set S ⊂ N . Put C = { w ( i , j ) : w ∈ ( I \ { ( i , j ) } ) s , s ∈ S } . We consider the coded subshift Σ generated by C . Under the assumptions on I , both Σ and π Σ satisfy unique decomposition with respect to C and π C respectively. The classical S -gapshifts correspond to the case that I = 2, however since analysis of the box dimension ofΠ(Σ) is trivial for subshifts on 2 symbols we are primarily interested in the case that I ≥ C n = ( ( I − n − n ∈ S n / ∈ S. Also clearly ℓ = log( I − f ( ℓ ) = ( I − n − ( I − n = X n ∈ S I − ∞ > . Similarly we can calculate that π C n = ( ( π I − n − n ∈ S n / ∈ S. and ℓ π = log( π I − f π ( ℓ π ) = ∞ . In particular, Theorem 1.5 is applicable and wededuce that dim B Π(Σ) = h ( π Σ)log m + h (Σ) − h ( π Σ)log n .4.2.3. Example whose box dimension equals the trivial lower bound.
Fix m ≥ n ≥ max { m + 1 , } . Let I = { (1 , , (1 , , (1 , , (1 , , (1 , , (2 , } and Ω = { (1 , , (1 , , (1 , } . Put C = { (1 , } ∪ { (2 , } ∪ { w (1 , m : w ∈ Ω ∗ , m ≥ | w | } where (1 , m denotes the concatenation of m instances of the digit (1 , C . Note that π Σ = { , } N . It is easy to see that h π = h ( π Σ) =
OX DIMENSIONS OF ( × m , × n )-INVARIANT SETS 17 log 2, and we will show that h ≤ log 2 < log 3 = h (Σ), see Lemma 4.4. The graph G (seeFigure 2) presents Σ. We will be interested in words which label a path that begins and endsat the vertex v . v (1 , , w (1 , m ( w ∈ Ω ∗ , m ≥ | w | ) Figure 2.
The graph G Definition 4.3.
For each n ∈ N let I n denote all strings in Σ n which can be presented by apath on G ending at v . Let I = S ∞ n =1 I n . Lemma 4.4.
We have lim sup n →∞ n log I n ≤ log 2 . Proof.
Suppose a word in I n has c digits from Ω and a digits from { (1 , , (2 , } .By definition of the code words C , we must have a + c + 2 c ≤ n therefore c ≤ log ( n − a ) . Now, assuming c >
0, for each 1 ≤ k ≤ c there are (cid:0) c − k − (cid:1) ways to divide the c digits into k groups.Following each of the k blocks of digits from Ω there must be a string of (1 , n − c − c − a extra(1 , k blocks of (1 , (cid:0) n − c − c − a + kk (cid:1) different ways in which wecan distribute the excess (1 , a digits from { (1 , , (2 , } directly preceding any of the k blocks of (1 , (cid:0) a + kk (cid:1) possibilities for distributing the a digits from { (1 , , (2 , } .Note that since k ≤ c ≤ log ( n − a ) we have (cid:18) a + kk (cid:19) ≤ (cid:18) a + log ( n − a )log ( n − a ) (cid:19) ≤ ( e + en ) log n where we have used that (cid:0) Nk (cid:1) ≤ ( eNk ) k . Similarly (cid:18) n − c − c − a + kk (cid:19) ≤ (cid:18) n − c − c − a + log ( n − a )log ( n − a ) (cid:19) ≤ ( e + en ) log n . Also, since c ≤ log ( n − a ), (cid:18) c − k − (cid:19) ≤ log ( n − a ) ≤ log n where we have first bounded (cid:0) c − k − (cid:1) by the central binomial term and used that (cid:0) NN (cid:1) ≤ N . Therefore I n ≤ n X a =0 log ( n − a ) X c =0 c X k =1 (cid:18) c − k − (cid:19) c (cid:18) n − c − c − a + kk (cid:19)(cid:18) a + kk (cid:19) a ≤ ( e + en ) n log n n X a =0 log ( n − a ) X c =0 c X k =1 c a ≤ ( e + en ) n log n log n n X a =0 log ( n − a ) X c =0 c X k =1 a from which the result follows. (cid:3) Using the above estimate for I n , it is now easy to compute the entropy of Σ. Lemma 4.5. h (Σ) = log 3 .Proof. The lower bound h (Σ) ≥ log 3 follows from the fact that Ω N ⊂ Σ. So it is sufficientto prove the upper bound. Fix any ε >
0. Suppose i ∈ Σ n . Then i falls into one of thefollowing mutually exclusive categories:(i) i ∈ I n .(ii) i = jk for j ∈ I and k = w (1 , m where w ∈ Ω ∗ , m ≥ i = w (1 , m for w ∈ Ω ∗ and m ≥ . ε (1+ ε ) n . The number of strings incategory (iii) is given by n X m =0 n − m . ε (1+ ε ) n . Finally, the number of strings in category (ii) is given by n − X j =1 n − j − X m =0 I j n − m − j ≤ n − X j =1 n − j − X m =0 j n − m − j . ε (1+ ε ) n . Hence n log n . ε (1 + ε ) log 3 which concludes the proof of the upper bound since ε > (cid:3) We will now prove Theorem 1.3 by showing thatdim B Π(Σ) = max (cid:26) log 3log n , log 2log m (cid:27) = max (cid:26) h (Σ)log n , h ( π Σ)log m (cid:27) . Note that dim B Π(Σ) can attain either h (Σ)log n or h ( π Σ)log m . For instance if n = 5, m = 2 thendim B Π(Σ) = 1 = h ( π Σ)log m . Whereas if n = 6, m = 5 then dim B Π(Σ) = log 3log 6 = h (Σ)log n . Proof of Theorem 1.3.
The lower bound corresponds to the trivial lower bound from Propos-ition 2.1. So we just need to prove the upper bound. Fix ε > δ >
0. Let k = k ( δ ) and l = l ( δ ) and i ∈ Σ l . Then i falls into one of the following mutually exclusive categories.(1) i = jk where j ∈ Σ k and π ( k ) = 1 l − k . OX DIMENSIONS OF ( × m , × n )-INVARIANT SETS 19 (2) i = jk (2 , l where: for some 1 ≤ m ≤ l − k , π ( l ) ∈ { , } m − ; π ( k ) = 1 l − k − m ; j ∈ Σ k has the form j = uw for u ∈ I and w ∈ Ω ∗ with length 1 ≤ | w | ≤ log ( l − k − m ) . (3) i = jk (2 , l where: for some 1 ≤ m ≤ l − k , π ( l ) ∈ { , } m − ; π ( k ) = 1 l − k − m ; j = uw (1 , z where u ∈ I , 1 ≤ z ≤ k , w ∈ Ω ∗ with length 0 ≤ | w | ≤ log ( l − k + z − m ).For each j = 1 , , A j := [ i ∈ Σ l in category ( j ) Π([ i ]) . Then N δ (Π(Σ)) ≤ X j =1 N δ ( A j ) . (17)Firstly, N δ ( A ) = k . ε (1+ ε ) k by Lemma 4.5. Secondly, N δ ( A ) = l − k X m =1 log ( l − k − m ) X | w | =1 I k −| w | | w | m − . ε l − k X m =1 log ( l − k − m ) X | w | =1 (1+ ε )( k −| w | ) | w | m − . ε (1+ ε ) k (cid:18) ε (cid:19) log ( l − k ) (1+ ε )( l − k ) . ε (1+2 ε ) l . Finally, N δ ( A ) = l − k X m =1 k X z =1 log ( l − k + z − m ) X | w | =0 I k −| w |− z | w | m − . ε l − k X m =1 k X z =1 log ( l − k + z − m ) X | w | =0 (1+ ε )( k −| w |− z ) | w | m − . ε l − k X m =1 k X z =1 (1+ ε )( k − z ) (cid:18) ε (cid:19) log ( l − k + z − m ) m − . ε (1+ ε ) k (cid:18) ε (cid:19) log l (1+ ε )( l − k ) . ε (1+2 ε ) l . By (17) we deduce that dim B Π(Σ) ≤ max (cid:26) log 3log n , log 2log m (cid:27) , as required. (cid:3) References [1] T. Bedford,
Crinkly curves, Markov partitions and dimension , University of Warwick, 1984.[2] F. Blanchard, β -expansions and symbolic dynamics , Theoret. Comput. Sci. (1989), no. 2, 131–141.[3] F. Blanchard and G. Hansel, Syst`emes cod´es , Theoret. Comput. Sci. (1986), no. 1, 17–49.[4] V. Climenhaga and D. J. Thompson, Intrinsic ergodicity beyond specification: β -shifts, S -gap shifts, andtheir factors , Israel J. Math. (2012), no. 2, 785–817. [5] A. Deliu, J. S. Geronimo, R. Shonkwiler, and D. Hardin, Dimensions associated with recurrent self-similarsets , Math. Proc. Cambridge Philos. Soc. (1991), no. 2, 327–336.[6] K. Falconer,
Fractal geometry , Third, John Wiley & Sons, Ltd., Chichester, 2014. Mathematical founda-tions and applications.[7] H. Furstenberg,
Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation ,Math. Systems Theory (1967), 1–49.[8] R. Kenyon and Y. Peres, Hausdorff dimensions of sofic affine-invariant sets , Israel J. Math. (1996),157–178.[9] , Measures of full dimension on affine-invariant sets , Ergodic Theory Dynam. Systems (1996),no. 2, 307–323.[10] D. Lind and B. Marcus, An introduction to symbolic dynamics and coding , Cambridge University Press,Cambridge, 1995.[11] C. McMullen,
The Hausdorff dimension of general Sierpi´nski carpets , Nagoya Math. J. (1984), 1–9.[12] R. Pavlov, On entropy and intrinsic ergodicity of coded subshifts. , to appear in Proc. Amer. Math. Soc.,available at https://arxiv.org/abs/1803.05966 .[13] D. Vere-Jones,
Geometric ergodicity in denumerable Markov chains , Quart. J. Math. Oxford Ser. (2) (1962), 7–28. Mathematical Institute, University of St Andrews, Scotland, KY16 9SS
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