The role of transfer operators and shifts in the study of fractals: encoding-models, analysis and geometry, commutative and non-commutative

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1 The role of transfer operators and shifts in the study of fractals: encoding-models, analysis and geometry, commutative and non-commutative Palle E. T. Jorgensen, University of Iowa and Dorin E. Dutkay, University of Central Florida

2 Abstract Dynamics on X = B N is determined by the one-sided shift in X, and by a given transition-operator R. Our results apply to any positive operator R in C(B) such that R1 = 1. From this we obtain induced measures Σ on X, and we study spectral theory in the associated L 2 (X,Σ). For the second class of dynamics, we introduce a fixed endomorphism r in the base space B, and specialize to the induced solenoid Sol(r). The solenoid Sol(r) is then naturally embedded in X = B N, and r induces an automorphism in Sol(r). The induced systems will then live in L 2 (Sol(r),Σ).

3 Applications: Wavelet analysis, both in the classical setting of R n, and Cantor-wavelets in the setting of fractals induced by affine iterated function systems (IFS). Such hyperbolic systems as the Smale-Williams attractor, with the endomorphism r there prescribed to preserve a foliation by meridional disks. Julia set-attractors in complex dynamics. Keywords Fractal, solenoid, shift-spaces, transition operators, Ruelle-operators, attractors, Cantor, infinite-product measures, wavelets, wavelet representations.

4 1. Introduction To every affine function system S with fixed scaling matrix, and a fixed set of translation points in R n, we may associate to S a solenoid. By this we mean a measure space whose L 2 space includes L 2 (R n ) in such a way that R n embeds densely in the solenoid. (In the more familiar case of n = 1, we speak of a dense curve in an infinite-dimensional torus. The paper is organized as follows: Starting with a compact Hausdorff space B, and a positive operator R in C(B), we pass to a family of induced probability measures Σ (depending on R) on the infinite product Ω = B N.

5 An additional structure: a prescribed endomorphism r in the base space B, and we study the corresponding solenoid Sol(r), contained in Ω, and its harmonic analysis, including applications to generalized wavelets. We introduce wavelet-filters, in the form of certain functions m on B.

6 Definition 2.1 We denote by M(B) the set of positive Borel measures on B, and by M 1 (B) those that have µ(b) = 1. Let V be some set. B V = V B = all functions from V to B. For example V = N or Z. For x B V we denote by π v (x) := x v, v V. If V = N, then we denote by π 1 1 (x) = {(x 1,x 2,...) B N : x 1 = x}. Let r : B B be some onto mapping, and µ a Borel probability measure on B, µ(b) = 1.

7 To begin with we do not introduce µ and r, but if r is fixed and 1 #r 1 (x) <, for all x B, (2.1) then we introduce two objects (1) R = R W, the Ruelle operator; (2) Sol(r), the solenoid. For (1), fix W : B [0, ) such that W(y) = 1, for all x B, r(y)=x and set (R W ϕ)(x) = r(y)=x W(y)ϕ(y). (2.2)

8 For (2), Sol(r) = { x B N : r(x i+1 ) = x i,i = 1,2,... } (2.3) σ(x) i = x i+1, (x B N ), r(x) = (r(x 1 ),x 1,x 2,...). (2.4) More generally, consider R : C(B) C(B) or R : M(B) M(B), (2.5) where M(B) is the set of all measurable functions on B.

9 Definition 2.2 We say that R is positive iff ϕ(x) 0 for all x B implies (Rϕ)(x) 0, for all x B. (2.6) We will always assume R1 = 1 where 1 indicates the constant function 1 on B. This is satisfied if R = R W in (2.2), but there are many other positive operators R with these properties. Given Set and p(x, dy) such that p(x, ) M(B),x B. (Rϕ)(x) = ϕ(y) p(x,dy),r : C(B) C(B) B (R ν)(a) = B p(x,a) dν(x), A B(B), R : M(B) M(B).

10 Definition 2.3 A subset S of B N is said to be shift-invariant iff σ(s) S, where σ is as in (2.4), σ(x) i = x i+1. Remark 2.4 Every solenoid Sol(r) is shift-invariant.

11 Example 2.5 The solenoids introduced in connection with generalized wavelet constructions: Let r : B B as above and let µ be a strongly invariant measure, i.e., 1 f dµ = #r 1 f(y)dµ(x) (x) for all f C(B). r(y)=x

12 Example 2.5 (cont.) A quadrature mirror filter (QMF) for r is a function m 0 in L (B,µ) with the property that 1 N r(w)=z m 0 (w) 2 = 1, (z B) (2.7) As shown by Dutkay and Jorgensen (2005, 2007), every quadrature mirror filter (QMF) gives rise to a wavelet theory. Various extra conditions on the filter m 0 will produce wavelets in L 2 (R)

13 Theorem 2.6 Let m 0 be a QMF for r. Then there exists a Hilbert space H, a representation π of L (B) on H, a unitary operator U on H and a vector ϕ in H such that (i) (Covariance) Uπ(f)U = π(f r), (f L (B)) (2.8) (ii) (Scaling equation) Uϕ = π(m 0 )ϕ (2.9)

14 Theorem 2.6 (cont.) (iii) (Orthogonality) π(f)ϕ, ϕ = f dµ, (f L (B)) (2.10) (iv) (Density) span { U n π(f)ϕ : f L (B),n 0 } = H (2.11) The system (H,U,π,ϕ) in Theorem 2.6 is called the wavelet representation associated to the QMF m 0.

15 Remark 2.7 #r 1 (x) =, the modification of the operators R, extending those from Example 2.5, is as follows: Consider (i) W : B [0, ) Borel (ii) p : B B(B) [0, ) such that for all x B, p(x, ) M(r 1 (x)), so is a positive measure such that W(y)p(x,dy) = 1, (x B). r 1 (x) Then set (Rϕ)(x) = ϕ(y)w(y)p(x, dy). r 1 (x)

16 Example 2.8 G = (V,E) infinite graph, V are the vertices, E are the edges. i(e) = initial vertex, t(e) = terminal vertex. (2.12) S (G) = solenoid of G = { ẽ E N : t(e j ) = i(e j+1 ) for all j N }. (2.13) For example V = Z 2 and the edges are given by x y iff x y = 1. For details and applications, see Jorgensen and Pearse (2010).

17 Definition 2.9 Now back to C, the cylinder sets C C, subsets of B V, indexed by finite sytems v 1,...,v n, O 1,...,O n, v i V, O i B open subsets, i = 1,...,n, n N. C vi,o i := { x B V : x vi O i for all i = 1,...,n }. (2.14) Notation: C generates the topology and the σ-algebra of subsets in B V in the usual way, and B V is compact by Tychonoff s theorem.

18 Lemma 2.10 ϕ 1 ϕ n = (ϕ 1 π 1 )(ϕ 2 π 2 )...(ϕ n π n ). (2.15) Let A C be the algebra of all cylinder functions. Then A C is dense in C(B N ).

19 Theorem 2.11 Let R be a positive operator as in (2.5), with R1 = 1. Then for each x B there exists a unique Borel probability measure P x on B N such that B N ϕ 1 ϕ n dp x = (M ϕ1 RM ϕ2...rm ϕn 1)(x), (ϕ i C(B),n N). Proof. (2.16) We only need to check that the right-hand side of (2.16) for ϕ 1...ϕ n equals the right-hand side of (2.16) for ϕ 1...ϕ n 1; but this is immediate from (2.16) and the fact that R1 = 1. The existence and uniqueness of P x the follows form the inductive method of Kolmogorov.

20 Corollary 2.12 Let B and R : C(B) C(B) be as in Theorem 2.11, and let µ M 1 (B) be given. Let Σ = Σ (µ) be the measure on Ω = B N given by f dσ := f dp x dµ(x). (2.17) B π 1 1 (x) Then (i) V 1 : L 2 (B,µ) L 2 (Ω,Σ) given by V 1 ϕ := ϕ π 1 is isometric. (ii) For its adjoint operator V1, we have V1 : L 2 (Ω,Σ) L 2 (B,µ) with (V 1 f)(x) = π 1 1 (x) f dp x. (2.18)

21 Proof. The assertion (i) is immediate from Theorem To prove (ii) we must show that the following formula holds: ( ) f dp x ψ(x)dµ(x) = f ψ π 1 dσ (2.19) B π 1 1 (x) Ω for all f L 2 (Ω,Σ) and all ψ C(B). Recall that V 1 is determined by V 1 f, ψ L2 (µ) = f, V 1ψ L2 (Ω,Σ). (2.20)

22 Lemma 2.13 Let B and R be specified as above. Given µ M 1 (B), let Σ = Σ (µ) denote the corresponding measure on Ω,i.e., f dσ = f dp (R) Ω B π 1 1 (x) x dµ(x) (2.21) We shall consider V 1 : L 2 (B,µ) L 2 (Ω,Σ) and its adjoint operator V1 : L 2 (Ω,Σ) L 2 (B,µ), where V 1 ϕ = ϕ π, for all ϕ L 2 (B,µ). Note that the adjoint operator V1 makes reference to the choice of R at the very outset. The following two hold: (i) R naturally extends to L 2 (B,µ); and (ii) RV 1 f = V 1 (f σ), (f L 2 (Ω,Σ)) (2.22)

23 Proposition 2.14 Let Σ M(B N ), then (P x ) x B has the form (2.16) in Theorem 2.11 if and only if there is a positive operator R such that R1 = 1 and E x (f σ) = (R(E f))(x) (2.23) holds for all x B and for all f L 2 (B N,Σ), where in (2.23) we use the notation E x (...) =... dp x = E (Σ) (... π 1 = x) (2.24) π 1 1 (x) for the field of conditional expectations, and E f denotes the map x E x f.

24 2.2 Subalgebras in L (Ω Σ) and a conditional expectation V n ϕ = ϕ π n, (ϕ C(B),n N). Since V 1 : L 2 (B,µ) L 2 (Ω,Σ) is isometric, it follows that Q 1 := V 1 V 1 (2.25) is a projection in each of the Hilbert spaces L 2 (Ω,Σ (µ) ).

25 Theorem 2.15 With B,R,µ,Σ = Σ (µ) and V n specified as above, we have the following formulas: (i) V 1 V n+1 = R n on L 2 (B,µ), n = 0,1,2...; (ii) Q 1 := V 1 V 1 is a conditional expectation onto A 1 := {ϕ π 1 : ϕ L (B,µ)} Q 1 ((ϕ π 1 )f) = (ϕ π 1 )Q 1 (f) for all ϕ L (B,µ), f L (Ω,Σ). (2.26) (iii) Q 1 (ϕ π n+1 ) = (R n ϕ) π 1 for all ϕ C(B), n = 0,1,2,...

26 Proposition 2.16 Let B, R : C(B) C(B) and Σ = Σ (µ) M 1 (Ω) be as above, i.e., µ = Σ π1 1. Let R be the adjoint of the operator R when considered as a bounded operator in L 2 (B,µ). For V2 we have V 2 ((ϕ 1 π 1 )(ϕ 2 π 2 )...(ϕ n π n )) = R (ϕ 1 )ϕ 2 R(ϕ 3...R(ϕ n )...). (2.27)

27 The next result is an extension of Lemma 2.13(ii). Note that (2.22) is the assertion that V1 intertwines the two operations, R and f f σ. The next result shows that, by contrast, V2 acts as a multiplier. Corollary 2.17 Let B,R,Σ and µ = Σ π 1 1 be as in Proposition 2.16, and set ρ := R 1 L 2 (B,µ); then (V 2 (f σ))(x) = ρ(x)e x (f) = ρ(x)(v 1 f)(x), (x B,f L 2 (Ω,Σ)).

28 Proof. This is immediate from Proposition 2.16, see (2.27). Recall that the span of the tensors is dense in L 2 (Ω,Σ) and that if then f = (ϕ 1 π 1 )(ϕ 2 π 2 )...(ϕ n π n ), f σ = (ϕ 1 π 2 )(ϕ 2 π 3 )...(ϕ n π n+1 ).

29 We calculate the multiplier ρ for the special case of the wavelet representation from Example 2.5 Corollary 2.18 Let B and R be as in Theorem 2.15, and let µ M 1 (B) be given. The induced measure on Ω = B N is denoted Σ (µ) and specified as in (2.17). We then have the following equivalence: (i) h L 2 (B,µ) and Rh = h, i.e., h is harmonic; and (ii) There exists f L 2 (Ω,Σ (µ) ) such that f = f σ and h(x) = f dp (R) π 1 1 (x) x. (2.28)

30 Proof. The implication (ii) (i) follows from Lemma For (i) (ii), let h be given and assume it satisfies (i). An application of (iii) from Theorem 2.15 now yields Q 1 (h π n+1 ) = R n (h) π 1 = h π 1 = V 1 h. Using (2.25), we get V 1 (h V1 (h π n+1 )) = 0 for all n = 0,1,2,...; and therefore h = V1 (h π n+1 ), (n N). (2.29) Recalling V1 (h π n+1 )(x) = h π n+1 dp (R) π 1 1 (x) x (2.30)

31 Proof (cont.) and using Theorem 2.11, we conclude that {h π n+1 } n N is a bounded L 2 (Ω,Σ (µ) )-martingale. By Doob s theorem, there is a f L 2 (Ω,Σ (µ) ) such that lim f h π n+1 L2 (Σ n (µ) ) = 0. Since π n+1 σ = π n, it follows that f σ = f. Taking the limit in (2.29) and using that the operator norm of V1 is one, we get that h = V1 f and therefore the desired formula (2.28) holds.

32 Theorem 2.19 Let B be compact Hausdorff and R : C(B) C(B) positive, R1 = 1. Let µ M 1 (B) such that µ(b) = 1, µ R = µ. Set X n (ϕ) = ϕ π n, (ϕ C(B),n N) and f dσ = B f dp (R) π 1 1 (x) x dµ(x) (2.31)

33 Theorem 2.19 (cont.) Then E(...) =... dσ satisfies E(X n (ϕ)x n+k (ψ)) = B ϕ(x)(r k ψ)(x)dµ, (n,k N,ϕ,ψ C(B)); (2.32) i.e., R k is the transfer operator governing distances k. Asymptotic properties as k goes to infinity govern long-range order.

34 Corollary 2.20 Let (r,w) be as in Definition 2.1, and let R W be the Ruelle operator in (2.2), P (W) x - the random walk measure with transition probability specified as follows Prob(x y) = { W(y), if r(y) = x 0, otherwise (2.33) Then P x from Theorem 2.11 is equal to P (W) x.

35 Proof. We apply Theorem 2.11 to R = R W in (2.2) and we compute by induction (M ϕ1 R W...R W M ϕn+1 )(x) = ϕ 1 (x) y 1 y n W(y 1 )W(y 2 )...W(y n)ϕ 2 (y 1 )...ϕ n+1 (y n), where r(y i+1 ) = y i, 1 i < n, r (n) (y n ) = x. Further, = Prob(x y 1 )Prob(y 1 y 2,y 1 ) =...Prob(y n 1 y n y n 1 )ϕ(y 1 )...ϕ(y n) dp (W-transition R W-measure) x ϕ 1 ϕ n (2.34)

36 Remark 2.21 The assertion in (2.34) applies to any random walk measure, for example, the one in Example 2.8. Let G = (V,E) be as in Example 2.8, with E un-directed edges. Let c : E [0, ) be such that c (xy) = c (yx) for all (xy) E, c (xy) 0 if (xy) E. (2.35) A function as in (2.35) is called conductance. Set p = p c, where p xy = c xy z,z x c xz = c xy c(x), (2.36) where c(x) = c xz, and z x means (zx) E. z,z x

37 Remark 2.21 (cont.) Then there is a unique P (c) x such that ϕ 1 ϕ ndp (c) x = y 1 y n p xy1 p y1 y 2...p yn 1 y n ϕ 1 (y 1 )...ϕ n(y n), where the sums are over all y 1,y 2,...,y n such that (y i y i+1 ) E. Note that P (W) x is supported on the solenoid, and P (c) x is supported on S (G) (see (2.13)).

38 Remark 2.22 The last application is useful in the setting of harmonic functions on graphs G = (V,E) with prescribed conductance function c as in (2.35). Set ( ϕ)(x) = c xy (ϕ(x) ϕ(y)) (2.37) y V,y x the graph Laplacian with conductance c. A function ϕ on V satisfies ϕ 0, iff ϕ(x) = p (c) xyϕ(y), (2.38) where p (c) xy = cxy c(x) as in (2.36). y V,y x

39 3.1 Preliminaries about r : B B 3.2 Compact groups (1) B fixed compact Hausdorff space (2) µ fixed Borel measure on B(B) New measure space (Ω,Cyl,Σ) (1)-(2)+(3) (3) B r B Reduced measure space (Solenoid(r),Cyl,Σ (r) )

40 3.1 Preliminaries about r : B B 3.2 Compact groups 3.1 Preliminaries about r : B B Example 3.1 (Rϕ)(x) = 1 #r 1 (x) r(y)=x Classical wavelet theory on the real line. Let Let m(y) 2 ϕ(y), N = 2, B = T = {z C : z = 1} R/Z ( 1 2, 1 2 ] via z = e2πiθ, θ R/Z; µ = dθ; L 2 (B,µ) = L 2 (( 1 2, 1 2 ],dθ), r : B B, r(z) = z 2, or equivalently r(θmodz) = 2θmodZ. (3.1) m 0 (θ) = n Z h n e 2πinθ, or equivalently m 0 (z) = n Zh n z n, (3.2) where we assume h n = 2, n Z h n 2 <. n Z

41 3.1 Preliminaries about r : B B 3.2 Compact groups Lemma 3.2 With m 0 as in (3.2), the condition (2.7) is equivalent to h k h k 2n = 1 2 δ n,0. (3.3) k Z

42 3.1 Preliminaries about r : B B 3.2 Compact groups Proposition 3.3 See Bratteli and Jorgensen (2005), Dutkay and Jorgensen (2005) Suppose that m 0 is as above, and that there is a solution ϕ L 2 (R) satisfying 1 ( x ϕ = 2 2) n Zh n ϕ(x n), (3.4) and The translates ϕ( n) are orthogonal in L 2 (R), n Z. (3.5)

43 3.1 Preliminaries about r : B B 3.2 Compact groups Proposition 3.3 (cont.) Set W : L 2 (T) L 2 (R), (Wξ)(x) = n Z ξ(n)ϕ(x n) =: π(ξ)ϕ, (3.6) where ξ L 2 (T), and ξ(n) = T e nξdµ; (S 0 ξ)(z) = m 0 (z)ξ(z 2 ), (z T); (3.7) and (Uf)(x) = 1 2 f ( x 2), f L 2 (R). (3.8)

44 3.1 Preliminaries about r : B B 3.2 Compact groups Proposition 3.3 (cont.) (i) Then S 0 is isometric, and (L 2 (R),ϕ,π,U) is a wavelet representation. (ii) The dilation W : L 2 (T) L 2 (R) then takes the following form: W is isometric and it intertwines S 0 and the unitary operator U, i.e., we have (WS 0 ξ)(x) = (UWξ)(x) = 1 ( x (Wξ), (ξ L 2 2) 2 (T),x R). (3.9)

45 3.1 Preliminaries about r : B B 3.2 Compact groups Remark 3.4 With m 0 as specified in Proposition 3.3, we conclude that the wavelet representation can be realized on L 2 (R). We will see in Corollary 4.1 that it can be also realized on the solenoid. The two representations have to be isomorphic. The identifications can be done via the usual embedding of R into the solenoid x (e 2πix,e 2πix/2,e 2πix/22,...). The measure Σ in this case is supported on the image of R under this embedding. For details, see Bratteli and Jorgensen (2002) and Dutkay (2006).

46 3.1 Preliminaries about r : B B 3.2 Compact groups Proposition 3.5 Let B and R : C(B) C(B) be as stated in Theorem For every µ M 1 (B) we denote the induced measure on Ω = B N by Σ (µ). If some r : B B satisfies R((ϕ r)ψ) = ϕr(ψ), (ϕ,ψ C(B)) (3.10) then every one of the induced measures Σ (µ) has its support contained in the solenoid Sol(r).

47 Corollary Preliminaries about r : B B 3.2 Compact groups Let B and R : C(B) C(B) be as in Corollary Let µ M 1 (B) and consider Σ = Σ (µ). Let r : B B be an endomorphism. (i) For the operators V 1 : L 2 (B,µ) L 2 (Sol(r),Σ) and U (R) : f f r acting in L 2 (Sol(r),Σ) we have the following covariance relation: V 1 is isometric and (ii) Assume that µ = µ R and (V 1 U (R) V 1 )(ϕ) = ϕ r, (ϕ C(B)). R((ϕ r)ψ) = ϕr(ψ), (ϕ,ψ C(B)) (3.11) holds. For functions F on Ω, say F L (Sol(r)), let M F be the multiplication operator defined by F. Then U (R) is unitary and the following covariance relation holds: (U (R) ) M F U (R) = M F σ.

48 3.1 Preliminaries about r : B B 3.2 Compact groups 3.2 Compact groups Proposition 3.7 Assume B is a compact group with normalized Haar measure µ. Let r : B B be a homomorphism r(xy) = r(x)r(y) for all x,y B, and assume for N N, N > 1 Set then #r 1 (x) = N, (x B). (Rϕ)(x) = 1 ϕ(y), (ϕ C(B)); (3.12) N r(y)=x (i) Sol(r) is a compact subgroup of B N. (ii) The induced measure Σ = Σ (µ,r) is the Haar measure on the group Sol(r).

49 3.1 Preliminaries about r : B B 3.2 Compact groups In wavelet theory, one often takes B = R n /Z n, and a fixed n n matrix A over Z such that the eigenvalues λ satisfy λ > 1. For r : B B, then take r(xmodz n ) = AxmodZ n, (x R n ) and it is immediate that r satisfies the multiplicative property in Proposition 3.7. There are important examples when r : B B does not satisfy this property.

50 3.1 Preliminaries about r : B B 3.2 Compact groups Example 3.8 (Non-group case: the Smale-Williams attractor) Take B = T D, where T = R/Z and D = {z C : z 1} the disk. For (t,z) T D, set r(t,z) = (2tmodZ, 1 4 z e2πit ). Then Sol(r) is the Smale-Williams attractor, see Kuznetsov and Pikovsky (2007), a hyperbolic strange attractor.

51 Corollary 4.1 Let B and r be as specified in section 3, let µ be strongly invariant and let the function m be quadrature mirror filter, as in Example 2.5. Assume in addition that m is non-singular, i.e. µ({x : m(x) = 0}) = 0.

52 Corollary 4.1 (cont.) Then, with R, we get a wavelet representation as in Theorem 2.6 to L 2 (Sol(r),Σ (µ) ) with (i) H = L 2 (Sol(r),Σ (µ) ); (ii) Uf = (m π 1 )(f r), for all f L 2 (Sol(r),Σ (µ) ); (iii) π(g)f = (g π 1 )f, for all g L (B), f L 2 (Sol(r),Σ (µ) ); (iv) ϕ = 1.

53 Thank you for your attention.

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