1 Iterated function systems Many of the well known examples of fractals, including the middle third Cantor sets, the Siepiński gasket and certain Julia sets, can be defined as attractors of iterated function systems In the standard case these are iterations of a finite family of contracting functions More precisely let T 1,, T k : d d be Lipschitz maps with Lipschitz constants 0 < r 1,, r k < 1 That is T i (x) T i (y) r i x y Theorem 1 (Hutchinson 1984) Let {T 1,, T k } be a family of contracting functions as above There is a unique non-empty compact set, Λ such that k T i(λ) = λ For each probability vector (p 1,, p k ) there is a unique probability measure, µ supported on Λ such that for any Borel set A d, µ(a) = k p i µ(ti 1 (A)) A sketch of the proof It is possible to fine a complete metric on the space of compact subsets of d, K( d ) For A K( d ) and δ > 0 let A δ = {x d : dist(x, A) δ} For A, B K( d ) let d(a, B) = inf{δ : A δ B and B δ A} d is referred to as the Hausdorff metric and forms a complete metric on K( d ) The map A k T i(k) is a contraction with respect to this metric and thus by the contraction mapping theorem there is a unique non-empty compact set such that A = k T i(a) For the second part we note that the map µ k p i T i (µ) is a contraction with respect to the weak-*-topology and once again apply the Contraction Mapping Theorem 1
2 2 Shift Spaces A convenient method of coding iterated function systems is via shift spaces Let Σ k = {1,, k} We define σ : Σ k Σ k by σ(x 1, x 2, x 3, ) = (x 2, x 3, ) If {T 1,, T k } is an iterated function systems on d with attractor Λ then we can define a natural projection Π : Σ Λ by Π(i) = T i1 T in (0) Let ν be (p 1,, p k )-Bernoulli measure on Σ k and note that it is ergodic and invariant with respect to the shift map, σ The projection of ν by Π gives the stationary measure associated to the IFS {T 1,, T k } and the probability vector (p 1,, p n ) If we restrict our attentions to an IFS of the form T i (x) = r x + a i where 0 < r i < 1 and a i d then the associated stationary measures are called self-similar measures We will use the following result Lemma 1 For ν almost all i Σ k 1 (1) log µ([i n 1,, i n ]) = p i log p i 1 (2) log diam(π([i n 1,, i n ])) = Λ π i Proof The first part follows from applying the Birkhoff Ergodic Theorem to the function g 1 : Σ k given by g 2 (i) = log p i1 For the second part note that T i1 T in (Λ) = r i1 r in Λ The result then follows by applying the Birkhoff Ergodic Theorem to the function g 2 : Σ k given by g 2 (i) = i1 The first part of this result is a special case of the Shannon-Macmillan- Brieman Theorem (we will see a more general case later) The second part is also a general case of a special scheme When the functions are not similarities we can often use the function log T i 1 (Π(σ(i))) 3 Dimension of self-similar measures In this situation we use Lemma 1 to compute the dimension of self similar measures Let {T 1,, T k } : d d be an iterated function system defined by T i (x) = r i x + a i where 0 < r i < 1 We will denote the attractor by Λ and for simplicity assume that diam(λ) = Λ = 1 Additionally we will assume that T i (Λ) T j (Λ) = for all distinct i, j
(we will refer to this as the strong separation condition) Note that the results stated would still be true if we made the weaker assumption that there exists an open set V such that T i (V ) V for all i and T i (V ) T j (V ) = for all distinct i, j (this is known as the open set condition) We will now compute the dimension of the self-similar measure defined by (p 1,, p k )-Bernoulli measure Recall that the Hausdorff dimension of a probability measure, µ on d is defined by dim µ = inf{dim A : µ(a) = 1 and A is Borel } For the definition of Hausdorff dimension see David Preiss s lectures In practise the mass distribution principle (strongly related to Frostman s Lemma in David Preiss s notes) is often the best way to calculate the Hausdorff dimension of a measure This means that if for µ almost all x, = s r 0 then dim µ = s Theorem 2 Let µ be an self-similar measure satisfying the above condition We have that 3 dim µ = i = 1 k p i log p i r i log p i Proof We choose a sequence i in Σ k such that both parts of Lemma 1 hold Let x = Πi Note that the µ measure of points x which can be found in this way is 1 Fix r > 0 and choose n such that r i1 r in r < r i1 r in 1 We will denote i 1,, i n for diam(π([i 1,, i n ]) We have where C = max k Thus inf r 0 log ν([i 1,, i n ] log ν([i 1,, i n ]) log i 1,, i n 1 log ν([i 1,, i n ]) log i 1,, i n in log ν([i 1,, i n ]) log i 1,, i n C inf log ν([i 1,, i n ]) log i 1,, i n
4 By Lemma 1 it follows immediately that inf log ν([i 1,, i n ]) log i 1,, i n p i log p i r i log p i For the other direction we let ɛ = min{dist(t i (Λ), T j (Λ) : i j and 1 i, j k} and note that it is strictly positive We now fix r and choose n such that r i1 r in ɛ r > r i1 r in+1 ɛ Thus since the nth level cylinders are separated by r i1 r in ɛ the ball B(x, r) can intersect at most 1 nth level cylinder which has to be [i 1,, i n ] Hence log(µ(b(x, r))) where K = min i = 1 k i Thus sup r 0 log ν([i 1,, i n ]) log ν([i 1,, i n ]) log i 1,, i n+1 log ν([i 1,, i n ]) log i 1,, i n + K sup Again by Lemma 1 it follows immediately that sup log ν([i 1,, i n ]) log i 1,, i n Putting all of this together we have that r 0 = log ν([i 1,, i n ]) log i 1,, i n p i log p i r i log p i p i log p i r i log p i This holds for any similarly chosen x thus for mu-almost all x k = p i log p i r 0 k r i log p i and hence dim µ = i = 1 k p i log p i r i log p i This is the simplest example of the results where the dimension of a measure is given by the entropy divided by the Lyapunov exponent Here the entropy of the measure is given as p i log p i and the Lyapunov exponent can be thought of as p i i
4 Multifractal analysis The term multifractal analysis first came into use in the mid 1980 s It was first introduced in physical settings However since the mid 1990 s there have been a string of papers on multifractal analysis in settings relating to iterated function systems and hyperbolic dynamical systems This includes Mauldin-Cawley, several papers by Olsen and coauthors, papers by Pesin-Weiss and papers by Barreira and Saussol Multifractal analysis looks at sets when some locally defined quantity is the same for all points A good example was first studied by Besicovitch in the 1930 s (50 years before the term multifractal was first used!) Consider a point x [0, 1] such that x = a 0 + a 1 2 + a 22 2 + where each a i {0, 1} In other words a 0 + a 1 2 + gives the binary expansion of x Let A n (x) = #{k : 1 k n and a k = 0} A and P (x) = n(x) when the it exists P (x) can be said to n be the frequency of 0 in the binary expansion of x This is a local quantity (arbitrary close points may have completely different frequencies) Lebesgue almost all points will have frequency 1 but large sets 2 of points will have different its between 0 and 1 and for some the it will not exist Theorem 3 (Besicovitch) dim{x [0, 1] : P n (x) = p} = p log p + (1 p) log(1 p) log 2 This is an example of a multifractal result In the setting of iterated function systems a quantity which often has multifractal phenomenon is the local dimension For a measure, µ on d the local dimension at a point x is defined by loc µ (x) = r 0 log(µ(b(x, r))) when the it exists We can then examine the dimension of sets of point where this quantity is the same The rest of these notes concentrate on this problem Before remove on other quantities that can be looked at are Birkhoff averages, local Lyapunov expoonents and local entropies Also we can look at the packing dimension rather than the Hausdorff dimension of the set or indeed the topological entropy However it usually does not make sense to look at the box dimension as 5
6 the multifractal sets are often dense and have the same box dimension as the whole set The analogue of box dimension in multifractal analysis comes from what is often called the L q spectrum and will not be covered in these notes 5 Multifractal analysis for self-similar measures Let T 1,, T m : be of the form T i (x) = r i x + a i We assume the system satisfies the open set condition and let µ be the self-similar measure associated to the probability vector, (p 1,, p m ) Π : Σ k will be the natural projection and ν is (p 1,, p m )-Bernoulli measure on Σ m Note that µ = ν Π 1 We wish to consider the level sets log(µ(b(x, r))) X α = {x : = α} r 0 To do this we will first consider a related subset on Σ n log(µ([i 1,, i n ])) Σ α = {i Σ m : log Π([i 1,, i n ]) = α} The reason for doing this is that the dimension of Π(X α ) is much easier to calculate than X α In fact the function f s : α dim Π(X α ) is often referred to as the symbolic multifractal spectrum for the measure (X α ) However once dim Π(Σ α ) is calculated the question is whether it is equal to dim(x α ) (this question is not necessarily trivial) To compute the symbolic multifractal spectrum of µ we introduce the function β This is defined implicitly by the formula m p q i rβ(q) i = 1 Theorem 4 [Cawley-Mauldin] 1 For α 2 For α / ( ) min m log p i we have that dim(π(σ α )) = inf{β(q) + αq} q [ ] min m log p i we have that Σ α = It is straightforward to see that part 2 of the Theorem is true so we will just concentrate on part 1 In the case that p i = ri s where s is the dimension of the attractor part 1 gives no information since min m log p i i = max m log p i i In fact in this case log(µ([i 1,, i n ])) log Π([i 1,, i n ]) = s
for all i Σ Proof of the upper bound To show that inf q+αq is an upper bound note that for ɛ > 0 and N the set Σ α (N, ɛ) = {x Σ m : α ɛ log(µ([i 1,, i n ])) α+ɛ for all n N} log Π([i 1,, i n ]) It is easily seen that for any ɛ we have Σ α n Σ α (N, ɛ) We now fix q 0 For simplicity we will assume that Λ = 1 Note that for i Σ α (N, ɛ) we have for n N, Letting r = min m r i we have H β(q)+q(α+ɛ) r n (Π(Σ α(n, ɛ))) i 1,, i n q(α+ɛ) µ(i 1,, i n ) α+ɛ = i Σ α(n,ɛ) i Σ α(n,ɛ) i 1,, i n (β(q)+q(α+ɛ) µ([i 1,, i n ]) q i 1,, i n β(q) (p i1 p in ) q (r i1 r in ) B(q) i Σ m p q i r(β(q) i = 1 It then follows that H β(q)+q(α+ɛ) (Π(Σ α (N, ɛ))) 1 and thus dim(π(σ α (N, ɛ)) β(q) + q(α + ɛ) This is true for all ɛ > 0 and q 0 thus dim(π(σ α )) β(q) + q for q 0 By using the same methods for q < 0 we get that dim(π(σ α (N, ɛ)) β(q) + q(α ɛ) for all ɛ > 0 and q < 0 which gives dim(π(σ α )) inf{β(q) + q} q Proof of the Lower Bound For the lower bound we need to look at the derivatives of β(q) (1) β (q) = P m (2) β 0 pq i rβ(q) i log p i P m pq i rβ(q) i i Note that β (q) attains all values in ( Hence for α min m log p i 7 ( min m log p i, max m log p i ) i ) we can find q such that β (q) = α and so αq + β(q) will be at a unique minimum From now on we will use this value of q Let ν q be the Bernoulli measure defined by the probability vector (p q 1r β(q) 1,, p q mr β(q) m ) and let µ q = ν q Π 1
8 We now apply the Birkhoff Ergodic Theorem with respect to the functions i log p i1 and i i and the measure ν q This gives that for ν q almost all i Σ, log(µ([i 1,, i n ])) = log [i 1,, i n ] pq i rβ(q) i log p i = β (q) = α pq i rβ(q) i i Thus ν q (Σ α ) = 1 and so µ q (Π(Σ α )) = 1 Since µ q is simply a self-similar measure we can calculate its dimension Finally dim µ q = pq i rβ(q) i pq i rβ(q) i log(p q i rβ(q) i i = q pq i rβ(q) i log(p i ) pq i rβ(q) i = β(q) + αq log(r i ) + β(q) pq i rβ(q) i pq i rβ(q) i dim(π(σ α )) dim µ q = β(q) + αq i i Symbolic local dimension and standard local dimension It remains to show that looking at the symbolic local dimension is the same thing as looking at the standard local dimension In the case when the images of the similarities applied to the attractor are pairwise disjoint this is the case (in fact it is also the case for the weaker open set condition) Lemma 2 Let x Λ and i Σ k such that Π(i) = x We have that if the iterated function system satisfies that for all 1 i j k T i (Λ) T j (Λ) = then x X α if and only if i Σ α Proof This can be proved using the methods from the prove of Theorem 2 Finally we have that Theorem 5 (Cawley-Mauldin) Let µ be a self-similar measure satisfying the strong separation condition with associated contraction ratios {r 1,, r k } and probabilities {p 1,, p k } Let We then have X α = {x : r 0 log(µ(b(x, r))) = α}
1 For α 2 For α / ( ) min m log p i we have that dim(x α ) = inf{β(q) + αq} q [ ] min m log p i we have that X α = The proof follows immediately from Theorem 4 and Lemma 2 9