DYADIC DIOPHANTINE APPROXIMATION AND KATOK S HORSESHOE APPROXIMATION

DYADIC DIOPHANTINE APPROXIMATION AND KATOK S HORSESHOE APPROXIMATION TOMAS PERSSON AND JÖRG SCHMELING Preprints in Mathematical Sciences 2006:3 Centre for Mathematical Sciences Mathematics CENTRUM SCIENTIARUM MATHEMATICARUM

DYADIC DIOPHANTINE APPROXIMATION AND KATOK S HORSESHOE APPROXIMATION TOMAS PERSSON AND JÖRG SCHMELING Abstract We consider approximations of real numbers by rational numbers with denominator 2 n We will exploit results on hitting times for the underlying dynamical system on the full shift In the second part we transfer the results to the β-shifts This will give us an estimate on the approximation speed of arbitrary β-shifts by finite type β-shifts This is a particular case of Katok s horseshoe approximation of non-uniformly hyperbolic systems 1 Introduction In this paper we deal with two questions which are connected to a general principle in dynamical systems theory This principle is the asymptotics of hitting times By this we mean the following Given a dynamical system f : M M on some compact metric space M we specify a point y M We are interested in the set of points which meet a shrinking base of neighborhoods C n (y) of this point with a given speed More precisely let ξ be a finite sufficiently regular partition of M We consider cylinder sets C n (y) n 1 k=0 f k ξ containing x The first hitting time of x M is defined as τ n (x, y) := inf k N : f k (x) C n (x) The quantity we are interested in is the lower asymptotic of the first hitting time when n tends to infinity Using results obtained earlier we are going to apply this setup to two different questions The first one is purely number-theoretical The approximation of real number by rational numbers ordered with respect to their denominator is a well-studied and important problem in number theory We are going to ask what happens if one does not allow all rationals to approximate a real number but rather rationals with denominators being powers of a given number In this paper we chose this number equal to 2 We remark that there is nothing special in this case The proofs and statements will be the same for any other natural number There is a major difference to the standard Diophantine approximation While the underlying dynamical system (for a fixed irrational number) in the standard case is an irrational rotation the underlying case in the dyadic approximation is the multiplication by 2 These dynamical systems exhibit a completely different Key words and phrases Diophantine approximation, Non-uniformly hyperbolic systems, horseshoes 1

2 DYADIC AND HORSESHOE APPROXIMATION behaviour Irrational rotations are uniquely ergodic and of zero entropy while the multiplication by two has a very rich space of invariant measures Moreover it has positive topological entropy Therefore one can expect a more complex and even irregular behaviour of the dyadic approximation than of the standard approximation These differences are discussed in the corresponding sections The second question is more of a dynamical origin although it has a clear interpretation in the expansions of a real number with respect to a non-integer basis (β-expansions) Given a real number β > 1 one can associate the dynamical system x βx (mod 1) If one considers the partition of [0, 1] corresponding to the different branches of this piecewise linear map one obtains a coding space, the β-shift S β For some β s these shift are especially simple, namely subshifts of finite type For most of the numbers these shifts are more complicated In any case Katok s horseshoe theorem [7] states that these shifts can be approximated from inside by subshifts of finite type Here we want to make some remarks on this First the theory of subshifts of finite type is well understood In particular one has control over the statistics of most orbits This control depends on the size of the transition matrix defining this subshift So one can expect to have a good control over the statistics of general subshifts if one has a control over the speed of approximations by finite type subshifts This is the question we address in the second part of the paper 2 Preliminaries For simpler notation and formulae we will only use logarithms of 2 throughout this paper 21 Dimension Let Y be a subset of R n Let N(ɛ) denote the minimal number of ɛ balls needed to cover Y Definition 1 For a subset Y of R n, the lower box dimension of Y, denoted by dim LB Y is given by lim inf ɛ 0 log N(ɛ) log 1/ɛ The upper box dimension dim UB is defined similarly, replacing the lim inf by lim sup If dim UB Y and dim LB Y both exist and are equal, we define the box dimension of Y to be this value, and write dim B Y = dim UB Y = dim LB Y For a subset U of R n, we let diam(u) denote the diameter of the set U Definition 2 Let s [0, ] The s-dimensional Hausdorff measure H s (Y ) of a subset Y of R n is defined by the following limit of covering

sums: H s (Y ) = lim ɛ 0 TOMAS PERSSON AND JÖRG SCHMELING 3 ( inf (diam U i ) s : Y i=1 i=1 U i, sup diam U i ɛ i ) It is easy to see that there exists a unique s 0 = s 0 (Y ) such that H s for s < s 0 (Y ) = (1) 0 for s > s 0 Definition 3 The unique number s 0 given by Equation (1) is defined to be the Hausdorff dimension of Y and is denoted by dim H Y Standard arguments give that for a subset Y of R n, dim H Y dim LB Y dim UB Y There are examples which show that these inequalities may be strict The box dimension can also be defined in terms of covering sums The only change being that the covering sets all have equal diameter We note that in order to estimate the box dimension, it suffices that the diameters of the covering sets tend to 0 along a geometric sequence Lastly, we define the Hausdorff dimension of a measure: Definition 4 Let µ be a Borel probability measure on X Then the Hausdorff dimension of the measure µ is defined by dim H µ = inf Y dim H Y : µ(y ) = 1 We remark that the dimension of a measure is clearly always less than or equal to the dimension of its (Borel) support There is a well known method of computing lower bounds on the dimension of a measure or a set Let U(x, r) denote the ball of radius r centred at x Lemma 1 If for some finite measure µ log µ(u(x, r)) lim inf s on a set of µ-positive measures r 0 log r then the dimension of the measure is at least s A survey of the methods and results in dimension theory can be found in [3, 10] 22 Entropy The notion of topological entropy for non-compact sets was introduced by Bowen in [2] Later it was considered by Pesin and Pitskel in [11] The main idea is to replace the diameter as a geometric measure by a measurement based on the complexity Roughly speaking we are going to measure our system with some finite precision Then we measure n times This gives us classes of trajectories We define the diameter of such a class as e n Now ask how many classes of trajectories we can distinguish? The scaling of the number of different

4 DYADIC AND HORSESHOE APPROXIMATION classes versus its diameter defines a dimension-like quantity based on complexity arguments This quantity is the entropy In order to make the situation simpler we assume that R is a finite generating partition For every set Z M, and every real number δ, we set N(Z, δ, R) = lim k inf Γ k U Γ k exp ( δm(u)), (2) where the infimum is taken over all finite or countable collections Γ k j k j 1 n=0 f n R that cover Z and m(u) = j if U j 1 n=0 f n R When δ goes from to +, the quantity in (2) jumps from + to 0 at a unique critical value The number h(f Z) = infδ : N(Z, δ, R) = 0 = supδ : N(Z, δ, R) = + is called the topological entropy of Z If the set Z is compact and invariant then the above definitions of entropy coincide with the classically defined topological entropies (see [9]) The entropy with respect to the partition R is ( j 1 ) 1 h(f, R) := lim n n H µ f n R, where n=0 H µ (P) := ξ P µ(ξ) log µ(ξ) The entropy of the measure µ is defined as h µ := sup h(f, R) R This supremum is attained for generating partitions R, ie h µ = h(f, R) and moreover 1 H ( j 1 n µ n=0 f n R ) decreases to h(f, R), see [14] Hence ( j 1 ) 1 h µ = inf n n H µ f n R for any generating partition R We can also define the entropy of a measure µ as h µ := inf h(f Z) : µ(z) = 1 For ergodic measures it coincides with the standard definition, but it is not an affine function on the space of measures! n=0

TOMAS PERSSON AND JÖRG SCHMELING 5 The very important Theorem of Shannon-McMillan-Breiman asserts that we can compute the entropy locally for an invariant ergodic measure h µ = lim log µ(c n(x)) µ-ae n n where C n (x) is the cylinder of length n containing x, ie x C n n 1 k=0 f k R Remark 1 The Shannon-McMillan-Breiman Theorem has a very nice and useful interpretation Let us consider the generic points for an ergodic measure µ, ie all points x for which 1 n 1 lim φ(f k x) = φ dµ n n k=0 for all continuous functions φ Since M is compact by separability and Birkhoff s Theorem this set has full measure So its entropy is at least h µ (In fact its entropy equals h µ by the variational principle for noncompact sets (see [9])) If n is sufficiently large then M µ(c n (x)) e hµn and since the total mass of the cylinders covering the generic points is 1 we get that the number of generic (for µ) n-cylinders is at least (approximable) e hµn If Z is compact and f-invariant, then h(f Z) = h(f 1 Z) This is not true for non-invariant sets! 23 Linear expanding maps We first consider the case f(x) = 2x (mod 1) From now on we will drop the notation (mod 1) since it will cause no confusion The multiplication by 2 exhibits a natural Markov partition S 1 = R/Z = I 0 I 1 where I 0 = [0, 1/2) and I 1 = [1/2, 1) We can code a point x S 1 by its orbit where x x = x 0 x 1 x n x n = 0 if 2 n x I 0 1 if 2 n x I 1 This coding corresponds to the binary coding of a real number in the unit interval It is one-to-one except for a countable set where it is twoto-one Since the measures we are going to consider are non-atomic we can neglect this ambiguity and work in the coding space Σ 2 = 0, 1 N

6 DYADIC AND HORSESHOE APPROXIMATION equipped with the product topology We denote the coding ρ: Σ 2 S 1 by ρ(x) = x We define the metric 1 d (x, y) := inf 2 : x k 0 = y 0 x k = y k Pushing this metric to the circle we obtain a metric equivalent to the standard metric (if we neglect the points with non-unique coding) Hence we can work in the coding space endowed with the metric d Moreover the shift operator σ(x 0 x 1 x n ) = x 1 x n commutes with the map f(x) = 2x, ie f ρ = ρ σ A crucial role will be played by cylinder sets C k (x) = y Σ 2 : y 0 = x 0 y k = x k Then such a cylinder has diameter 1 2 k+1 We can define a metric d f on S 1 as the unique metric with diam I n (y) = diam C n (y) 24 Measures on S 1 Let φ: S 1 R be Hölder continuous We may assume that P (φ) = 0 The corresponding Gibbs state is the unique measure µ φ with C 1 µ φ (C n (y)) exp[ n i=0 φ(σi y)] C for some C > 0 and any n Gibbs states are invariant and ergodic Let φ depend only on the first coordinate: φ(x) = φ(x 0 ) and set φ(i) =: φ i e φ i =: p i Then P (φ) = 0 implies p 1 + p 2 = 1 The corresponding Gibbs state is called a Bernoulli measure Remark 2 Gibbs states wrt Hölder continuous potentials have approximately similar computation rules as Bernoulli measures In the following section we will restrict our arguments to Bernoulli measures and Hölder functions which depend only on the first coordinate In particular, we assume that the expansion rate of f is constant on the intervals I 1 and I 2 ie the map is piecewise linear All results are valid for Gibbs states with arbitrary Hölder potential and for maps which are C 1+α on I i Also the results can be generalised to higher-dimensional conformal systems

TOMAS PERSSON AND JÖRG SCHMELING 7 25 Hitting times Let f(x) = 2x (mod 1) and let µ φ be a Gibbs state with respect to a Hölder potential φ Put l N = 1 The associated cylinder of length l N in base 2 is C n (y) with n ν log N We N ν want to study the first hit time when iterates of x firsts hit C n (y) The first hit time of the cylinder C n (y) is Since τ n (x, y) := min k : f k (x) C n (y) f n (x) C n (y) σ n (x) C ν log N (y) = C n (x) we need τ n (x, y) N 2 n log τ n (x, y) ν or 1 n ν to have that f n (x) hits C n (y) infinitely often We want to study τ n (x, y) The main problem is that there is no nice asymptotics in general for hitting times The case of Gibbs states is much better Theorem 1 (Chazottes) For µ φ µ ae (x, y) holds log τ n (x, y) lim n n = h µ (y) = lim log µ φ(c n (y)) n n 3 Standard Diophantine Approximation and Diophantine classes Definition 5 An irrational number α is of Diophantine class ν = ν(α) R + if qα < 1 q µ (3) has infinitely many solutions in integers q for µ < ν and at most finitely many for µ > ν If α is not of a Diophantine class ν R + then α is said to be a Liouville number If α is of Diophantine class ν, we write α Dioph(ν) In a slight abuse of terminology, we say that a Liouville number α (0, 1) has infinite Diophantine class and write α Dioph( ) 4 Approximation by dyadic numbers The original motivation of this question was to find the exact speed of approximation of arbitrary β-shifts by subshifts of finite type We will deal with a simpler case first By using the correspondence between the shift space and the circle with multiplication by 2 we address the question in the symbolic space In case of no confusion we will omit the correspondence and identify numbers on the circle with their dyadic expansion Since only dyadic

8 DYADIC AND HORSESHOE APPROXIMATION numbers have two different expansions (01 = 10 ) we can ignore this ambiguity We will use the metric x n y n d(x, y) := 2 n n=1 This metric is equivalent to the metric on the circle We consider a point x = x 1 x 2 Σ 2 and ask the question how fast does x(n) := x 1 x n 0 x 1 x 2 or x(n) + := x 1 x n 1 x 1 x 2 This means we have to study the occurence of blocks of 0 s or 1 s in the symbolic sequence x The faster such blocks occur the better the approximation will be Since the numbers x 1 x n 0 and x 1 x n 1 correspond to dyadic numbers with denominator 2 n we deal with approximations of numbers by dyadic numbers The situation here is different from the usual Diophantine approximation 41 Badly approximable numbers The numbers j (mod 1); j = 2 n 0,, 2 n 1 are equally spaced on S 1 with distance 1 Therefore for 2 n all x R and all n N min j x j 2 1 which gives an upper n 2 n+1 bound on the dyadic approximation The worst approximable numbers are There x 1 = 1 3 = 01010101 x 2 = 2 3 = 101010101 r n (x) : = mind(x i (n), x i ), d(x i (n) +, x i ) 1 = 2 = 1 n+2k 3 2 n k=1 In these two numbers neither a block of 2 consecutive zeros nor of two consecutive ones occurs On the other hand if a number x contains infinitely many blocks of N > 1 consecutive zeros or ones then min j x j 2 n = min d(x i (n), x i ), d(x i (n) +, x i ) 1 2 < 1 n+n 3 2 n infinitely often (with equality if x = x 1 x n 1 10 N 1 ) These are examples of numbers where the speed of convergence is not faster than 2 n From a result with Fan [5] follows Theorem 2 Proof Let dim H x : c > 0 st rn (x) > c2 n n N = 1 BAN := x : c > 0 st r n (x) > c2 n n N be the set of badly approximable numbers We start by remarking that those numbers correspond to numbers which have bounded strings of

TOMAS PERSSON AND JÖRG SCHMELING 9 0 s and 1 s Let x be such that there are at most N consecutive zeros or ones in its dyadic expansion Then x n+k r n (x) = min 2, 1 x n+k 1 n+k 2 n+k 2 n+n+1 k=1 k=1 where we used that the remainder is minimised by having a block of N consecutive zeros (or ones) starting at place n + 1, ie x = x 1 x n 1 10 N 1 or x = x 1 x n 1 01 N 0 This gives c = 2 N+1 Clearly, if N is unbounded, ie we have arbitrarily long blocks of zeros or ones, we cannot find such a number c > 0 Now the set x : c > 0 st rn (x) > c2 n n N = S N n N where S N is the subshift of finite type where blocks of N +1 consecutive zeros and ones are forbidden If N the dimension of S N clearly tends to 1 Remark 3 One can normalise the approximation by considering the distances 2 n x on the circle By Dirichlet s Theorem we have for standard continued fraction approximations that for any irrational x (0, 1) and hence, min 1 p q px < 1 q lim inf q N qx = 0 The above numbers show that this is not the case for dyadic approximations Actually the numbers in Theorem 2 are those where lim inf n N 2n x = c > 0 This fact can be explained that irrational rotations are uniquely ergodic and the invariant measure is Lebesgue measure independent of the irrational number x This means that the orbit (qx (mod 1)) q is uniformly distributed in S 1 This is no longer true for the multiplication by 2 There we have uncountably many different ergodic measures with completely different asymptotics 42 The Liouville case In this section we consider those numbers which have an arbitrary speed of approximations In standard Diophantine approximation those numbers are called Liouville numbers They form a residual subset of the real line no matter what the speed of approximation is We are going to show that a similar statement holds for dyadic approximations

10 DYADIC AND HORSESHOE APPROXIMATION Theorem 3 Let Φ(n) 0 be an arbitrary function Then the set L Φ = x R : r n (x) < Φ(n) 12 infinitely often n is residual in R Proof The proof is similar to the one for standard Diophantine approximations and is included for completeness We have that x L Φ if and only if there is a subsequence n k of the naturals such that x nk +1 x log Φ(nk ) = 0 log Φ(n k) Clearly for each n N the sets O n = [x 1 x n 0 log Φ(n) ] (x 1,,x n) 0,1 n or x nk +1 x log Φ(nk ) = 1 log Φ(n k) (x 1,,x n) 0,1 n [x 1 x n 1 log Φ(n) ] are open Moreover, the sets G N = n N O n are dense in R for all N N Hence, the set R = N N G N = L Φ is a dense G δ set and hence, residual Remark 4 As can be seen from the proof the set of points having arbitrary good approximations from the left (respectively from the right), ie with arbitrary long sequences of zeros (respectively ones) forms also a residual set Let Σ A be a subshift of finite type This is a closed subset of Σ 2 [0, 1] and hence a Baire space A slight modification of the above proof (choose the cylinders [x 1 x n 0 log Φ(n) ] and [x 1 x n 1 log Φ(n) ] in Σ A ) gives: Theorem 4 If Σ A is a subshift of finite type that is not contained in x : c > 0 st r n (x) > c2 n n N, ie arbitrary long blocks of zeros or ones are allowed for points in Σ A, then L Φ Σ A is residual in Σ A and hence uncountable 43 The almost Liouville case In this section we want to investigate those numbers which have a dyadic approximation of the order of a power of the denominator This corresponds to the case of Diophantine numbers in the standard continued fraction approximation A number x R is called Diophantine if there is a number β > 1 such that qx < 1 q β has infinitely many solutions in integers q Jarník s theorem [4] states that for β > 1 dim H x R : qx < 1 has many solutions q N = 2 q β β + 1

TOMAS PERSSON AND JÖRG SCHMELING 11 or equivalent for γ > 2 dim H x R : x p q < 1 q has many solutions p γ q Q = 2 γ Therefore for any β > 1 or γ > 2 the set x R : qx < 1 q β has only finitely many solutions q N or equivalently x R : x p q < 1 q has only finitely many solutions p γ q Q has full Lebesgue measure Let Σ A be a mixing subshift of finite type We denote L A α : = x Σ A : r n (x) < 2 αn infinitely often = x [0, 1] : x p 2 n < 1 2 αn In contrast we have for dyadic approximations Theorem 5 Let α > 1 and Σ A be a subshift of finite type not contained in BAN Then dim H x ΣA : r n (x) < 2 αn often, = dim H Σ A α Remark 5 We emphasise that in contrast to the standard Diophantine approximation we loose the 2 in the nominator Ie we get only half of the dimension Proof We note first that since (Σ 2 \ BAN ) Σ A arbitrary long sequences of zeros or ones are allowed in Σ A Let T > 0 be such that A T > 0 A number x is in the set L A α if and only if there is a sequence n k (x) such that r nk (x) < 2 αn k, ie xnk +1 = 0,, x nk +(α 1)n k = 0 (or ones respectively) Let us fix a sequence (n k ) k such that k 1 (2T + (α 1)n i ) < log n k i=0 We consider the following map f from L A α(n k ) := x Σ A : r nk (x) < 2 αn k k N into Σ 2 For x L A α(n k ) we will collapse the sequence of zeros (or ones) occurring at the places n k More precisely, the point is sent by f to x = x 1 x nk T 1x nk T x nk 0 (α 1)n k x nk +(α 1)n k +1 x nk +(α 1)n k +T 1x nk +(α 1)n k +T f(x) = x 1 x nk T x nk +(α 1)n k +T +1

12 DYADIC AND HORSESHOE APPROXIMATION This point does not have to belong to Σ A since the transition from x 1 x nk T 1 to x nk +(α 1)n k +T might be forbidden in Σ A However, since if y = y 1 y nk T 1 Σ A the sequence x = y 1 y nk z 1 z T 0 (α 1)n k w 1 w T y nk +1 L A α(n k ) for suitable chosen z 1 z T and w 1 w T Hence, Σ A f(l A α(n k )) We are going to estimate the Hölder exponent of f For simplicity we will use the metric d which has equivalent diameters of cylinder sets Let x, y L A α(n k ) Then their distance d (x, y) is determined by the first digit where these sequences differ Let this digit be x m y m (x i = y i for 1 i < m) and k be such that αn k m < n k+1 (m cannot be in the intervals (n k, αn k ) since there both sequences are 0) Then for f(x) and f(y) we erase at most k 1 i=0 (2T +(α 1)n i) < log n k symbols before n k T < m and at most 2T +(α 1)n k symbols between n k T and m Thus for some m > m (log n k + 2T + (α 1)n k ) we have f(x) i = f(y) i ; 1 i m and f(x) m f(y) m This yields for sufficiently large k d (f(x), f(y)) = 2 m 2 m+(log n k+2t +(α 1)n k ) 2 m+(α 1+ɛ)n k 2 α 1+ɛ m+ α m ( 1 ɛ 2 m) α Since a Hölder continuous map with Hölder exponent κ can increase the dimension at most by a factor 1/κ we get that dim H Σ A α 1 ɛ dim H L A α(n k ) letting ɛ 0 gives us the right lower bound For the upper bound we remark that L A α = (n k ) k L A α(n k ) and L A α(n k ) can be covered by cylinders of length αn k for each k N The number of those cylinders is at most the number of cylinders of length n k intersecting Σ A But for each ɛ > 0 and k sufficiently large this number is bounded by 2 (dim H Σ A +ɛ) n k Hence, we get for those covering sums diam C dim H Σ A +ɛ α i = diam C dim H Σ A +ɛ α i n N C i L A α C i C j = n N C i L A α C i =n 2 (dim H Σ A +ɛ) n ( ) 1 dim H Σ A +ɛ α 2 αn n N 2 (1 α)ɛn < Remark 6 Those numbers have the property that infinitely often x n = 0 x αn = 0 or x n = 1 x αn = 1

TOMAS PERSSON AND JÖRG SCHMELING 13 Therefore no invariant measure besides the delta measure on the sequence 0, respectively 1 can sit on this set To see this we remark that by Birkhoff s ergodic theorem the frequencies # 1 n N : x n = i F (x, i) := lim N N for i = 0, 1 have to exist for an invariant measure µ and for µ-ae x We claim that the frequency of 0 s (or 1 s) must be 1 Assume that µ is a invariant measure on set of points with x n = 0 x αn = 0 infinitely often Then for µ-ae x we have that F (x, 0) exists If n k is an infinite sequence such that x nk x αnk = 0 then where F (x, i) N := F αnk (x, 0) = n kf nk (x, 0) + (α 1)n k αn k, # 1 n N:xn=i Hence we have N αf αnk (x, 0) F αnk (x, 0) = α 1 Since F N (x, 0) converges to F (x, 0) we must have F (x, 0) = 1 Now δ 0 is the only invariant measure having frequency one of the digit 0 A similar argument holds for long blocks of ones 44 The Diophantine case By Chazotte s theorem 1 we know that for equilibrium measures µ φ the hitting times have the following asymptotics (in logarithmic scale): τ k (x, 0 ) 2 hµ φ (0 )k and τ k (x, 1 ) 2 hµ φ (1 )k So we expect, for a typical with respect to µ φ point x, to have a block of k consecutive zeros (or ones) at place approximately 2 hµ (0 )k φ or equivalently at n = 2 hµ φ (0 )k we expect to have a block of h 1 µ φ log n consecutive zeros This means that we should expect (with 0 < β = 1/ minh µφ (0 ), h µφ (1 ) < 1) r n (x) < 1 infinitely often 2 n n β Clearly according to the previous theorem these sets have full dimension For β > 0 we are interested when this approximation speed is the best possible Ie we are interested in the set 1 D β : = x Σ 2 : r n (x) < = x : lim sup log 2n r n (x) n log n finitely often for any ɛ > 0 and 2 n n β+ɛ infinitely often for any ɛ < 0 = β This corresponds to the Diophantine classes for standard Diophantine approximation (see sections 3 and 43)

14 DYADIC AND HORSESHOE APPROXIMATION For dyadic approximations we have the following Theorem 6 For any 0 < β < 1 dim H x : lim sup log 2n r n (x) n log n = β = 1 Proof This proof follows the methods developed in [5] Theorem 41 Let us consider the set BAN = x Σ 2 : c > 0 st r n (x) > c2 n n N Then by Theorem 2 this set has full dimension Also this set consists of sequences x having both bounded sequences of zeros and ones Now let n k = 2 k/β We are going to modify the sequences in BAN Let x BAN For k N we insert a block of k consecutive zeros between x nk and x nk +1, ie x 1 x nk x nk +1 x 1 x nk 0 k x nk +1 By the proof of Theorem 41 in [5] (k n k ) this procedure does not decrease the dimension Therefore the obtained set BAN modified has dimension 1 Clearly the fastest approximation is obtained exactly at the places n k Hence we have for x modified BAN modified that lim sup n log 2n r n(x) = β log n Remark 7 This means that we have uncountably many disjoint sets of full dimension with different approximation speed 45 The regular Diophantine case The dynamical system underlying the standard Diophantine approximation is the rotation For an irrational number x the rotation by angle x: x nx (mod 1) has a unique invariant measure, the Lebesgue measure on S 1 By Birkhoff s ergodic theorem 1 lim n n # 0 k < n : kx < 1 = 2 q q Hence, we can regularise the approximation by setting 1 log lim n # 0 k < n : kx < 1 n q τ q (x) := log q and τ(x) = lim τ q (x) q This number gives the average approximation rate over the orbit of x from the discussion above follows that τ(x) = 1 for all irrational x This equals the Diophantine approximation rate if the Diophantine class β(x) = 1 It is well known that the numbers for which β(x) = τ(x) = 1 has full Lebesgue measure In general we have 1 = τ(x) β(x)

TOMAS PERSSON AND JÖRG SCHMELING 15 The situation in dyadic Diophantine approximation is different Here we define τm reg (x) := log lim inf 1 N # 0 k < N : 2 k x < 1 N 2 m m and τ reg (x) = inf τ m reg (x) m In symbolic language this is τm reg (x) := log lim inf N max F N (x, 0 m ), F N (x, 1 m ) m We are interested in the size of the set of points D reg β := x Σ 2 : τ reg (x) = log β for β 2 Let x D reg β Then for each m N there is a sequence (n(m) k ) k such that log lim k max F (m) n (x, 0 m ), F (m) k n (x, 1 m ) k = log β m m and inf m β m = β By using a diagonal argument we can choose a universal sequence (n k ) k along which all the limits exist Consider the set V σ (x) := n 1 k 1 µ inv : (n k ) k such that lim δ σ k n i x = µ k By using subsequences of the universal sequence (n k ) k from above we derive that there is a measure µ V σ (x) with Therefore, i=0 µ([0 m ]) = β m m or µ([1 m ]) = β m m max 1 m log µ([0m ]), 1 m log µ([1m ]) log β We are going to use the following result of Bowen [2] Theorem 7 (Bowen) h top x Σ 2 : µ V σ (x) with h µ t t This immediately implies Proposition 1 Let β 2 Then dim H D reg sup β h µ inv : max 1 m log µ([0m ]), 1 m log µ([1m ]) log β The next step is to evaluate the supremum over those measures

16 DYADIC AND HORSESHOE APPROXIMATION Theorem 8 Let β 2 Then sup h µ inv : max 1 m log µ([0m ]), 1 m log µ([1m ]) log β = 1 β log 1 ( β 1 1 ) ( log 1 1 ) β β This means the dimension is carried by a Bernoulli measure Proof Clearly, sup h µ inv : max 1 m log µ([0m ]), 1 m log µ([1m ]) log β sup h µ inv : max log µ([0]), log µ([1]) log β = sup h µ inv : min µ([0]), µ([1]) β 1 But µ([0]) = 1 µ([1]) Without loss of generality we may assume that µ([0]) β 1 12 Then h µ H µ ([0], [1]) 1 β log 1 ( β 1 1 ) ( log 1 1 ) β β The lower bound is obtained by considering the ( 1 β, 1 1 β ) -Bernoulli measure µ β which sits on the set in question since µ β ([0 m ]) = β m and µ β ([1 m ]) = (1 β 1 ) m Remark 8 In the definition of τ reg we cannot substitute inf with lim inf Consider the set BAN There lim inf n τm reg (x) = giving that 1 log 1 ( ) ( ) 1 1 β β β log 1 1 β = 0 but the dimension of BAN is 1 Remark 9 We have proved that a maximal ergodic measure is sitting on the sets with regular approximation Therefore for ae point with respect to this measure we have that actually we can replace lim inf and inf in the definitions of τm reg and τ reg with limits Remark 10 The Lebesgue measure on the circle is obtained by pulling back the ( 1 2, 1 2) -Bernoulli measure Therefore Leb(D reg 2 ) = dim H x S 1 : τ reg (x) = log 2 = 1 Remark 11 If we have an ergodic measure µ then τ reg (x) = max h µ (0 ), h µ (1 ) µ-ae x where h µ (y) = lim inf 1 n n log µ(c n(y)) In [6] it was proved (as a slight modification of Chazotte s Theorem) that for an equilibrium state µ with respect to a Hölder continuous potential lim m 1 m log min n > 1 : σ n (x) C m (y) = h µ (y) µ-ae x

TOMAS PERSSON AND JÖRG SCHMELING 17 Therefore for β 2 dim H x S 1 : r(n) 2 n 2 log n β = 1 2 n n, n > n 0(x) 1/β = 1 β log 1 ( β 1 1 ) ( log β 1 1 β Moreover we have a maximising ergodic measure sitting on these sets In particular we have Leb x S 1 : r(n) 1 2 n n, n > n 0(x) = 1 5 Approximation by finite β-expansions Let 1 < β < 2 and let T β : [0, 1] [0, 1) be the β-expansion defined by T β : x βx mod 1 Denote by i(x, β) = i n (x, β) n=0 0, 1 N the sequence defined by i n (x, β) = [βt n β (x)] The set S β = closurei(x, β) : x [0, 1) together with the left shift σ : i n n=0 i n+1 n=0 is called the β-shift S β can be characterised by S β = j n n=0 0, 1 N : σ n j n i(1, β) n 0, where denotes the lexicographical ordering Let Ξ be the map β i(1, β) For more details on the β-expansion, see for example [1, 8, 12] 51 Extension of theorems to the β-expansion If we in section 4 exchange the full two-shift Σ 2 with the β-shift we can state similar theorems as in section 4 Since for β < 2 the maximal number of consecutive 1 s in S β is bounded we will get r n (x) := d(x i (n), x i ) The following theorems are analogous to those in section 4, hence we state them without proof Theorem 9 dim H x : c > 0 st r n (x) > cβ n, n N = 1 Theorem 10 Let Φ(n) 0 be an arbitrary function Then the set L Φ = x R : r n (x) < Φ(n) 1 infinitely often βn is residual in R )

18 DYADIC AND HORSESHOE APPROXIMATION Theorem 11 Let α > 1 and Σ A S β be a subshift of finite type not contained in BAN Then dim H x ΣA : r n (x) < β αn infinitely often = dim H Σ A α Theorem 12 For any 0 < α < 1 dim H x : lim sup log βn r n (x) n log n Theorem 13 Let α 2 Then sup h µ inv : 1 m log µ([0m ]) log α = α = 1 = 1 α log 1 α ( 1 1 α ) ( log 1 1 ) α 52 Diophantine approximation with β-expansions A cylinder C n ([i 0 i n 1 ], β) is the set C n ([i 0 i n 1 ], β) = x [0, 1] : i k (x, β) = i k, k = 0,, n 1 The collection of all cylinders of length n is denoted C n A cylinder in the parameter space is the set C p n([i 0 i n 1 ]) = β (1, ) : i k (1, β) = i k, k = 0,, n 1 The collection of all cylinders of length n in the parameter space is denoted C p n Let α > 0 and fix β 0 < β 1 (1, 2) We are going to calculate the dimension of the set D α (β 0, β 1 ) = β (β 0, β 1 ) : d(σ n Ξ(β), 0 N ) < β αn infinitely often = β (β 0, β 1 ) : i n (1, β) i (1+α)n (1, β) = 0 0 infinitely often For this purpose we will need the following lemmata Lemma 2 dim H (D α (β 0, β 1 )) 1 1+α Proof Let K β1 be the set of x [0, 1] such that i(x, β 1 ) = Ξ(β) for some β (1, β 1 ) Let ρ β1 : K β1 (1, 2) be the map x β We estimate the dimension of ρ 1 β 1 (D α ) from above The set B α = [i 0 i n 1 0 0 ] S β1 N=1 n N i αn satisfies ρ β1 (B α ) D α (β 0, β 1 ) There are constants c 0, c 1 > 1 such that the number of cylinders of length n in S β1 is bounded by c 0 β n 1 and the length of a cylinder of

TOMAS PERSSON AND JÖRG SCHMELING 19 length n is bounded by c 1 β1 k We use this to bound the Hausdorff dimension of B α from above by 1 ; For any ε > 0 1+α lim inf [i 0 i 1 i n 1 0 0] 1+ε 1+α N n N i lim N n N c 0 β n 1 (c 1 β (1+α)n 1 ) 1+ε 1+α = 0 This implies that dim H (B α ) 1 1+α The estimate of the Hölder exponent of the map ρ β1 in [12] implies that dim H (ρ β1 (B α )) 1 log β 1 1+α log β 0 and so dim H (D α (β 0, β 1 )) 1 log β 1 1+α log β 0 Given any ε > 0 we can decompose the interval (β 0, β 1 ) into finitely many intervals (β 0, β 1 ) = k (β 0,k, β 1,k ) with log β 1,k log β 0,k < 1 + ε Then dim H (D α (β 0, β 1 )) = max k dim H (D α (β 0,k, β 1,k )) 1+ε We conclude 1+α that dim H (D α (β 0, β 1 )) 1 1+α Lemma 3 Among any n consecutive cylinders in C p n there is at least one C p n with C p n > β n, where β = sup C p n Proof Fix n and take i = i(1, β) for some β (1, ) Then the sequence i 0 i n 1 00 is admissible If i 0 i n 1 100 is admissible then we are done, so assume that this sequence is not admissible Then there is a k 1 < n such that i k1 i n 1 0 = i 0 i n k1 The sequence (i 0 i n k1 )(i 0 i n k1 ) is admissible and so i 0 i n k1 1100 is admissible as well and the cylinders Cn([i p 0 i n 1 ) and Cn([i p 0 i n k1 110 0]) are neighbors The sequence i 0 i n k1 1100 is admissible and if the sequence i 0 i n k1 1100 0100 is admissible then we are done If the sequence i 0 i n k 1 100 0100 is not admissible then we let i (2) = i 0 i n k1 100 and proceed with i (2) as was done above with i; There is a k 2 < n such that i (2) k 2 i (2) n 10 = i (2) 0 i (2) n k 2 and the two cylinders Cn([i p (2) 0 i (2) n 1) and Cn([i p (2) 0 i (2) n k 2 110 0]) are both nonempty and neighbors This process is continued until we get a i (j) with both i (j) 0 and i (j) 0 i (j) n 1100 admissible Note that n < k 1 < k 2 <, so after at most n steps we end up with a cylinder Cn([i p (j) 0 i (j) n 1]) with Cn([i p (j) 0 i (j) n 1]) β n Corollary 1 For any β (1, ) and ε > 0 there is a number n = n(β, ε) and a set V p B(β, ε) such that V p can be written as a union of n cylinders of length n and B(β, ε) > (β + ε) n 1 Proof Let β (1, ) and ε > 0 be fixed Put β = β + ε and let n = maxk N : β k > 2ε Take V p to be a union of n consecutive

20 DYADIC AND HORSESHOE APPROXIMATION cylinders of length n V p = n C p n,i, i=1 where β Cn,n p and α i < α i+1 if α i C p n,i and α i C p n,i+1 Lemma 3 implies that V p > β n > 2ε = B(β, ε) and so V p B(β, ε) Lemma 4 Let β 0 < β 1 < β 2 be such that Ξ(β 2 ) terminates with zeros Then dim H (ρ 1 β 2 (D α (β 0, β 1 ))) 1 log β 1 1+α log β 2 Proof Let i = Ξ(β 1 ) Take n so large that β 0 U n, where U n = Cn([i p 0 i n 1 ]) and (i 0 i n 1 ) Ξ(U n ) Let β n (β 0, β 1 ) \ U n be such that Ξ(β n ) terminates with zeros Then the subshift S βn is of finite type and Ξ(U n ) σ n 1 S βn K β1, where K β1 is the set of sequences x S β1 such that x = Ξ(β) for some β Since (i 0 i n 1 ) Ξ(U n ) we have Ξ(U n ) σ n 1 S βn Let β 2 > β 1 such that j = Ξ(β 2 ) terminates with zeros Then S β2 is of finite type Take m so large that σ m (Ξ(U n )) S β1 and let k(l) l=0 be a sparse sequence such that k(0) > m Let Kn α S β2 be the set of sequences x Ξ(U n ) σ n 1 S βn with the property that x k(l) x (1+α)k(l) = 0 0 for every l N We construct a measure µ on S β2 with support in Kn α in the following way For any l N define S l (x) = C (1+α)k(l) ([x 0 x 1 x k(l) 1 0 0 ], β 2 ) αk(l) zeros Define the measure of S l (x) by µ(s 0 (x)) = (#S0 : S 0 K α n ) 1 if S 0 (x) K α n 0 otherwise For l > 0 we define µ(s l ) recursively µ(s l (x))) = µ(s l 1 (x)) #S l (y) S l 1 (x) Since S βn is a subshift of finite type there is a constant c 2 > 0 such that c 1 2 #S l(y) S l 1 (x) c βn k(l) (1+α)k(l 1) 2 and since S β2 is of finite type there is a constant c 3 such that c 1 3 β (1+α)k(l) 2 S l c 3 β (1+α)k(l) 2

This implies that log µ(s l ) log S l log TOMAS PERSSON AND JÖRG SCHMELING 21 µ(s l 1 ) c 2 β k(l) (1+α)k(l 1) n log c 1 3 β (1+α)k(l) 2 = log µ(s l 1) log c 2 k(l) log β n + (1 + α)k(l 1) log β n log c 3 (1 + α)k(l) log β 2 1 log β n, 1 + α log β 2 as k(l) Let β n β 1 Theorem 14 dim H (D α (β 0, β 1 )) = 1 1+α Proof Fix β 2 > β 1 such that S β2 is of finite type Let U p k be a δ-cover of D α (β 0, β 1 ) By Corollary 1 we can for each k find a set V p k U p k such that and V p k is a union of n(k) cylinders of length n(k) V p k n(k) = i=1 C p n(k),i To each V p k there is a corresponding set V k S β2, defined in a natural way as the union of the corresponding cylinders in S β2 There is a constant c 1 such that c 1 n(k)β n(k) 2 V k c 1 1 n(k)β n(k) 2 and V 2,k covers ρ 1 β 2 (D α (β 0, β 1 )) Since β 2 > β 1 there is a constant c 2 such that n(k) 1 > c 2 n(k)β n(k) 2 > c 2 c 1 1 V k, where β k = sup U p k This U p k > β k implies that U p k s (c 2 c 1 (c 2 c 1 1 ) s inf 1 ) s V k s Uk s : U k is a (c 1 c 1 2 δ)-cover of ρ 1 β 2 (D α (β 0, β 1 )) But U p k is an arbitrary δ-cover of D α(β 0, β 1 ) so this proves that inf U p k s : U p k is a δ-cover of D α(β 0, β 1 ) (c 2 c 1 1 ) s inf Uk s : U k is a (c 1 c 1 2 δ)-cover of We conclude that dim H (D α (β 0, β 1 )) dim H (ρ 1 β 2 (D α (β 0, β 1 ))) 1 1 + α Let β 2 β 1 ρ 1 β 2 (D α (β 0, β 1 )) log β 1 log β 2

22 DYADIC AND HORSESHOE APPROXIMATION 53 Convergence of dimensions in β-shifts The technique from Lemma 4 can be used to state the following theorem of convergence of dimensions for a general set Theorem 15 Let β 0 > 1 If E S β0 is such that for any β < β 0 and any cylinder C n of length n there exists an m = m(c n ) such that dim H (E S β ) = dim H (C n σ m (E S β )) and K β0 is the set of x S β0 such that x = Ξ(β) for some β < β 0, then dim H (E K β0 ) = lim β β0 dim H (E S β ), where the dimension of the sets are the dimensions as subsets of S β0 Proof We will first prove that dim H (E K β0 ) lim β β0 dim H (E S β ) It is clear that K β0 β<β 0 S β Hence, if β n n=1 is an increasing sequence with β n < β 0 and β n β 0, then dim H (E K β0 ) dim H = sup n 1 n=1 (E S βn ) dim H (E S βn ) = lim n dim H (E S βn ) It remains to show that dim H (E K β0 ) lim β β0 dim H (E S β ) Fix β < β 0 We proceed as in the proof of Lemma 4 and take n so large that β U n := C p n(β 0 ) = C p n([i 0 i n 1 ]) and (i 0 i n 1 ) U n Then Ξ(U n ) σ m(ξ(un)) (S β ) is a non-empty subset of K β0, if we assume that m(ξ(u n )) > n The required estimate follow by dim H (E K β0 ) dim H (E (Ξ(U n ) σ m (S β ))) = dim H (Ξ(U n ) σ m (E S β )) = dim H (E S β ) References [1] F Blanchard, β-expansions and symbolic dynamics, Theor Comp Sci 65 (1989), 131 141 [2] R Bowen, Topological entropy for noncompact sets, Trans Amer Math Soc 184 (1973), 125 136 [3] K Falconer, Fractal geometry Mathematical foundations and applications, John Wiley & Sons Ltd, Chichester, 1990, ISBN: 0-471-92287-0 [4] V Jarník, Zur metrischen Theorie der diophantischen Approximationen, Prace Mat-Fiz, (1928 1929), 91 106 [5] A Fan, J Schmeling, On fast Birkhoff averaging, Math Proc Cambridge Philos Soc 135 (2003), no 3, 443 467 [6] A Fan, J Schmeling, and S Troubetzkoy, Dynamical defined Dvoretzky coverings, preprint [7] A Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms, Inst Hautes tudes Sci Publ Math No 51 (1980), 137 173 [8] W Parry, On the β-expansion of real numbers, Acta Math Acad Sci Hung 11 (1960), 401 416

TOMAS PERSSON AND JÖRG SCHMELING 23 [9] W Parry, and M Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 1990, 187 188 [10] Ya Pesin, Dimension theory in dynamical systems, Contemporary views and applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, 1997, ISBN 0-226-66221-7; 0-226-66222-5 [11] Ya Pesin, B S Pitskel, Topological pressure and the variational principle for noncompact sets, (English translation), Functional Anal Appl 18 (1984), no 4, 307 318 [12] J Schmeling, Symbolic dynamics for β-shifts and self-normal numbers, Ergod Th & Dynam Sys, 17 (1997), 675 694 [13] J Schmeling, Entropy preservation under Markov coding, J Stat Ph 104 (2001), no 3/4, 799 815 [14] P Walters, An introduction to ergodic theory, Graduate Texts in Mathematics 79, Springer-Verlag, New York-Berlin, 1982, ISBN: 0-387-90599-5 Mathematics Centre for Mathematical Sciences, Lund Institute of Technology, Lund University Box 118 SE-221 00 Lund, Sweden E-mail address: tomasp@mathslthse URL: http://wwwmathslthse/matematiklth/personal/tomasp/ Mathematics Centre for Mathematical Sciences, Lund Institute of Technology, Lund University Box 118 SE-221 00 Lund, Sweden E-mail address: joerg@mathslthse URL: http://wwwmathslthse/matematiklth/personal/joerg/

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