FAST DYNAMO ON THE REAL LINE

Transcription

1 FAST DYAMO O THE REAL LIE O. KOZLOVSKI & P. VYTOVA Abstract. In this paper we show that a piecewise expanding ap on the interval, extended to the real line by a non-expanding ap satisfying soe ild hypthesis does induce a fast dynao action on the functions on the real line in the sense that there exist a L function whose nor grows exponentially under induced action. This is the first step towards a solution of the kineatic fast dynao proble.. Introduction In this work we establish the fast dynao theore for the induced action on vector fields on the unstable anifold of the Poincaré ap of the provisional flow. The unstable anifold is one diensional and the settings are the following. Vector fields on a one-diensional real anifold ay be identified with functions R R; and an induced action on vector fields on R is given by a transfer operator f vy = dfxvx. x f y Theore Fast dynao on R. There exist a easure-preserving piecewise-c transforation f : R R and an essentially bounded, absolutely integrable vector field v such that li li δ 0 n n ln expδ f n v L > 0, The ap f ay be realised as an induced action on the unstable anifold by the Poincaré ap of the provisional fluid flow. After introducing necessary tools, we deduce Theore in the Section 3 fro two technical results: the noise Lea 3. and the invariant cone Theore 5. In the Section 4 we fix technical notation; and the last two Sections 5 and 6 are dedicated to the proof of Theore 5.. Basic constructions In this Section we introduce obects central for our investigations: sall rando perturbations of a dynaical syste and a nor in the space of vector fields. We also specify the type of cones in the space vector fields we are interested in.

2 FAST DYAMO O THE REAL LIE.. Sall rando perturbations. We construct a rando dynaical syste using skew-products. Let X be a real anifold and let f : X X be a transforation. We consider its extension f : X R n X fx, def = fx +. Let Σ l R n be a shift-invariant subset of two-sided bounded sequences of vectors in R n. We introduce a skew product over the Bernoulli shift σ f : Σ X Σ X The induced transforation on fibers we denote by σ f, z def = σ, fz,. f : X X, f z def = fz,. 3 Its iterations are given by f k z def = ff k z, k. 4 Reark. The following identities follow fro the definition of the ap f. f k = f k f k... f ; 5 f k f n k = f k = f f... f k ; 6 = f n f k = f k f n = f n k σ n f n k, if n < k; σ k, if n > k. 7 Definition. We call the ap f a rando perturbation of the ap f associated to the sequence Σ... or in the space of vector fields. Piecewise constant vector fields are proved to be very useful to us. We define a nor in the space of essentially bounded and absolutely integrable vector fields Φ, using partitions. Definition. A nor in the space of essentially bounded and absolutely integrable functions, associated to a partition Ω = Ω of R is given by Z = f Ω = ax fx dx, / sup f. 8 Ω Ω The first ter we refer to as the weighted L -nor and write f Ω,L : = fx dx, Ω Z Ω it depends, of course, on the partition chosen.

3 O. KOZLOVSKI & P. VYTOVA The subspace of Φ, consisting of piecewise constant vector fields associated to the partition Ω we denote by Φ Ω. Observe that for any step function φ = Z c χ Ω Φ Ω we have that φ Ω = ax c, / sup c. 9 Z.3. Cones in vector fields on R. We reserve a notation for a cone of radius r with the ain axis χ [,] in the space Φ Ω of piecewise constant functions, associated to a partition Ω: { Cone r, Ω def = η = dχ [,] + ϕ ϕ = c χ Ω ; Z r } c = 0; ϕ Ω dr. 0 = l We extend the cone Cone r, Ω to include general functions fro the ain space: } Ĉone r, ε, Ω def = {f = η + g, η Cone r, Ω, g Ω ε η Ω. We say that the cone Ĉone r, ε, Ω is saller than the cone Ĉone r, ε, Ω and write Ĉone r, ε, Ω Ĉone r, ε, Ω, if r > r and ε > ε ; we do not assue here that Ĉone r, ε, Ω Ĉone r, ε, Ω. 3. Fast dynao theore in diension one In this Section we give a proof of our ain result, Theore. The arguent has three steps. The first step is the noise Lea 3., which suggests to replace the operator expδ f n in our considerations with the operator expδ f n t for soe sequence t. The second step is to choose a large and to construct explicitly an invariant cone for the operator exp δ f t exp δ with t l R, t δ. The third step is to deduce the fast dynao theore fro the existence of an invariant cone. An instant proof of the noise Lea 3. is given in this Subsection. The invariant cone Theore 5 for a specific ap is proved in the Section 6 after the preparatory Section 5. We begin with a siple observation that the exponent of the Laplacian operator, is the convolution with the Gaussian kernel, in particular expδ v = w δ v, where w δ x = πδ exp x δ. : v d v in the case of the real line 3

4 FAST DYAMO O THE REAL LIE The latter operator is also known as the Weiertstrass transfor W δ v def = w δ v; this notation we use throughout. The following stateent is generally known, but we give a proof for copleteness. Lea 3. oise Lea. For any ap f : R R and for any function v we have W δ f W δ f vx = w δ t w δ t... w δ t W δ f R 0t vxdt dt... dt, where 0t = 0, t, t,..., t R. Proof. Observe that f x t = ft x, because f t x = fx+t. By straightforward calculation, W δ f W δ f vx = W δ f W δ f W δ f vx = f W δ f = W δ R w δ tf vx tdt = = W δ f W δ f w δ t f t vxdt =... = R = W δ w δ t... w δ t f f t... f t vxdt... dt = R = w δ t... w δ t W δ f R 0t vxdt... dt. Let s s, be two real nubers such that log s s = κ ; and let δ = α be a sall real nuber with 5 α <. Consider the ap 6 s x + s, if < x < s ; lx = s x + s, if s < x < ; 3 x, otherwise; and define its extension l: R R by lx, y = lx + y. We then associate a sall perturbation l to any sequence l R and δ. The existence of an invariant cone for the operator W δ l W δ end of Section 6 in the following is established in the Theore 5. Invariant cone. For any sequence l R with δ there exists an, a partition Ω, and four nubers r r ; ε ε 4

5 O. KOZLOVSKI & P. VYTOVA such that W δ l W δ : Ĉone r, ε, Ω Ĉone r, ε, Ω Ĉone r, ε, Ω. 06 f Ĉone r, ε, Ω: W δ l W δ f Ω f Ω. 07 We choose δ = α, the partition Ω = [, + ], and fix four diension paraeters of two cones r = δs 4 δ, r = δ 64, ε = δ 3 and ε = δ 4 such that the Theore holds true. Lea 3.. In the notations introduced above, for any f Ĉone r, ε, Ω t... w δ t W δ l t W fdt δ... dt Ĉone e r, e ε, Ω. [ δ,δ] w δ Proof. By Theore 5 we know that for any t [ δ, δ] and any f Ĉone r, ε, Ω W δ l t W δ f = dχ [,] + ψ t + g t Ĉone r, ε, Ω, where ψ t Φ Ω, ψ t Ω dr and g t Ω dε. Observe that Ω is independent on t. Therefore, [ δ,δ] w δ t... w δ t χ [,] dt... dt = = χ [,] δ for large enough. Since ψ t Φ Ω for any t [ δ, δ], [ δ,δ] w δ and we calculate Ω -nor. [ δ,δ] w δ δ 4 w δ tdt = χ[,] e χ [,], 5 t... w δ t ψ t dt... dt Φ Ω, Ω t... w δ t ψ t dt... dt Z Ω [ δ,δ] w δ t... w δ t ψ t x dx dt... dt Ω Siilarly, sup ψ t Ω dr. t [ δ,δ] w δ Ω t... w δ t g t dt... dt dε. 5

6 FAST DYAMO O THE REAL LIE Observe that [ δ,δ] w δ t... w δ t ψ t xdt... dt dx = = [ δ,δ] w δ t... w δ t dt... dt ψ t xdx = 0. Suing up, for any f Ĉone ε, r, Ω [ δ,δ] w δ t... w δ t W δ l t W fdt δ... dt Ĉone e r, e ε, Ω. Lea 3.3. In the notations introduced above, W δ l W δ l W δ : Ĉone r, ε, Ω Ĉone r, ε, Ω Ĉone r, ε, Ω ; W δ l W δ l W δ Proof. By Lea 3. for any f Cone r, Ω W δ l W δ l W δ = R w δ = f = Ĉoner,ε,Ω 5 t... w δ t W δ l t W δ fdt dt... dt = R \[ δ,δ] + [ δ,δ] By Lea 3. we know that for any f Ĉone r, ε, Ω [ δ,δ] = We estiate the first ter R \[ δ,δ] Z = Ω 3 = w δ t W δ l t W δ fdt. 6 w δ t W δ l t W δ fdt Ĉone e r, e ε, Ω. w δ t W δ l t W δ R \[ δ,δ] = fdt Ω w δ t Ω 3 W δ l t W δ fx dx dt sup W δ l t W δ f Ω w δ t dt. t R \[ δ,δ] = 6

7 O. KOZLOVSKI & P. VYTOVA We shall find an upper bound for the integral: R \[ δ,δ] + δ = w δ t dt + w δ tdt + δ [ δ,δ] w δ We ay also observe that for any f Ĉone r, ε, Ω, sup W δ l δ t W δ f Ω t δs δ δs l t f Ω inf Ω l t f L We put the last two together and we see that δ δs t... w δ t dt... dt s s sup W δ l t W f δ Ω w δ t dt t R \[ δ,δ] = e + e. f Ω s δ s 3 δ f Ω. We need to verify δ δs s s f Ω s δ s 3 δ e f Ω. It is equivalent to s δ e ε s 3 δ, where ε = δ 4. s +α α log s s 3 e < α 4 ; and holds true for κ = log s s sufficiently sall. For the second inequality we recall Corollary of Theore 5 again Then [ δ,δ] = f Cone r, ε, Ω : W δ l W δ f Ω f Ω. w δ t W δ l t W δ inf W l t W t [ δ,δ] δ δ f Ω fdt Ω [ δ,δ] = w δ t e f Ω. 7 7

8 FAST DYAMO O THE REAL LIE Taking into account we get the result. R \[ δ,δ] = w δ t W δ l t W δ fdt ε f Ω, Ω Theore.[Fast dynao on R] There exist a easure-preserving piecewise-c transforation f : R R and an essentially bounded, absolutely integrable vector field v such that li li δ 0 n n ln expδ f n v L > 0, The ap f ay be realised as an induced action of the Poincaré ap of the provisional fluid flow on the unstable anifold. Proof. Let us choose f = l, where l: R R is given by 3. It follows by straightforward calculation that W δ W δ = W δ for any nuber δ > 0. The Theore follows fro Lea 3.3 with v = W δ χ [,]. The reaining part of the paper is dedicated to the proof of existence of an invariant cone for the operator W δ l t W δ large is fixed. for arbitrary t δ. Therefore, we assue that a 4. otation In this Section we fix notation we use throught the proof of Invariant Cone Theore 5. The following letters are reserved for constants: α, β, γ, γ, κ, s, s. The adissible range of values will be specified later. Given a subset I R n we denote by I its Lebesgue easure. We say that two sets I and I are δ-close and write I I < δ if I belongs to the δ-neighbourhood of I or I belongs to the δ-neighbourhood of I. Otherwise, we write I I > δ. The indicator function of a set I we denote by χ I. Let δ i be the Dirac delta function:, if i = δ i = 0, otherwise. The supreu nor of a sequence of real nubers l R we denote by = sup k. Whenever supreu or infiu are taken along the whole range k of values, we oit the range. Let δ = α be a sall real nuber with 5 6 < α <. 8

9 O. KOZLOVSKI & P. VYTOVA We write x y when x is exponentially sall copared to y, naely, there exist a sall nuber 0 < ε < such that x < ε y. Definition 3. We say that a collection of intervals Ω = {Ω } Z akes a partition of the class G, δ, s, s, if Ω = R, Ω i Ω = if i, and the following conditions hold true. The interval [, ] contains at least and at ost intervals of the partition, and {±} are the end points of soe intervals of the partition. The length of intervals Ω is bounded away fro zero and fro infinity Ω s +. s s 3 Any interval I R of the length I = δ contains not ore than intervals of the partition. δ = + δ log s = α log s + 4 Any interval of the partition Ω R \ [ δ, + δ] has length Ω =. We write G, δ, s, s to indicate dependence on, δ, s, and s ; we will abuse notations and oit, δ, s, or s, when it leads to no confusion and the dependence is of no iportance. We nuber intervals of a partition Ω in the natural order, starting fro Ω 0 0. We set Ω l to be the ost left interval of Ω inside [, ], and Ω r to be the ost right interval of Ω inside [, ]. Here we deal with essentially bounded absolutely integrable functions on the real line. We refer to the space Φ def = L R L R as the ain space. Any function refers to a function fro the ain space always. Given a partition Ω = {Ω } Z of the class G, we denote the associated space of step functions by Φ Ω and address the basis {χ Ω } Z as the canonical basis of Φ Ω. Definition 4. We associate a weighted transfer operator f, acting on the ain space, to a ap f on the real line by f φx: = y f x sgn dfyφy. 8 Transfer operator is a bounded linear operator. In this case, it is chosen to be one diensional analogue of induced action on vector fields by area-preserving transforations. Transfer operators with negative coefficients have been considered, for instance, in [5]. 9

10 FAST DYAMO O THE REAL LIE 5. Transfer operator as a dynao operator In this section we first show that any generalised toy dynao operator has an invariant cone. Afterwards, we prove that there exists a ap l: R R such that for any sall perturbation l with δ we can find a linear operator A: Φ Φ and two partitions Ω and Ω of R such that A: Φ Ω Φ Ω and l AW δ γ l + A for soe γ > 0. Moreover, the atrix of A ΦΩ satisfies conditions D D4. In other words, for any sequence δ, the operator l ay be approxiated by a generalised toy dynao. 5.. The odel atrix. The plan is to choose suitable subspaces of Φ and approxiate the operator l by a siple atrix. In this Subsection we describe the atrix we would like to obtain and show that for any atrix A, satisfying these conditions, there exists a pair of cones C, C Φ such that C C and AC C but C C. Let A be a linear operator acting on the ain space. Assue that there exists two partitions Ω, Ω of the class G such that A: Φ Ω Φ Ω. Here and below we denote by l and r the indices of the first and the last intervals of the partition Ω inside [, ], respectively; and let l and r be the indices of the first and the last intervals of the partition Ω inside [, ], respectively. In other words, the sets Ω i Ω with l i r, and l r ake a partition of the unit square. We define several sets of indices in order to describe the properties of the operator A iportant to us. Let a i be coefficients of the atrix of A in the canonical bases of the subspaces Φ Ω and Φ Ω. Accelerator: Ar: = { } {l,..., r } #{i {l,..., r } a i = } δ. 9 Inflow diffusion: D in : = { i, {l,..., r } {l,..., r } a i }. 0 Outflow diffusion: D out : = {l δ,..., r + δ } {l δ,..., r + δ } {l,..., r } {l,..., r }. Indifferent subspace: Sp: = Z \ {l δ,..., r + δ } {l δ,..., r + δ }. 0

11 O. KOZLOVSKI & P. VYTOVA We are interested in linear operators A such that the following conditions hold true for the atrix coefficients in the canonical bases. D ax a i + s ; s D #D in δs ; D3 for any pair i, Sp we have a i = 0 whenever i > δ ; D4 #Ar. Definition 5. We say that a linear operator A: L R L R L R L R is a generalised toy dynao if there exist two partitions Ω and Ω of the class G such that AΦ Ω Φ Ω above. and the conditions D D4 hold true in the settings introduced Reark. All theores and the ain result hold true for an operator A that satisfies conditions D D4 with right parts of the inequalities ultiplied by polynoials in. When we have several partitions, e.g. Ω, Ω, and Ω 3 of the class G we refer to the nors associated to the partitions by,, and 3, respectively. We will need the following fact. Reark 3. For any s s, satisfying log s log s, and δ = α there exists a nuber 0 < γ = α < /4 such that for large enough. δ s 3 s < γ. 3 Lea 5.. Let A: Φ Ω Φ Ω be a generalised toy dynao and let φ = Z c χ Ω be a step function. Then r i= l r = l Proof. By straightforward calculation, r i= l r = l c a i = c a i 3/+γ φ. i, D in c a i sup a i #D in sup c s s δs / φ 3/+γ φ.

12 FAST DYAMO O THE REAL LIE Definition 6. Let Ω, Ω be two partitions of the class G. We define the kernel of A χ [,] : Φ Ω Φ Ω Ker A χ [,] : = to be the set { } φ Φ Ω Aφxdx = 0 = = { φ = c Ω : Z r i= l Z } a i c Ω i = 0. 4 Proposition 5.. Let γ < s <. Then for any two partitions of the class G and a generalised toy dynao A: Φ Ω Φ Ω Φ Ω = χ [,] Ker A χ [,]. In other words, for any φ Φ Ω there exist ψ Ker A χ [,] and d R such that φ = dχ [,] + ψ. 5 Proof. Let χ [,] = Z u χ Ω, where u = for l r and u = 0 otherwise. Let φ = Z c χ Ω Φ Ω be a step function. We want to find a function ψ Φ Ω such that ψ Ker A. By definition of the kernel 6, using 5 we write Aψxdx = Aϕ dχ [,] xdx = a i c du χ Ω i xdx = i, Z = r i= l a i c du Ω i = 0. We want to solve the last equality for d. It is sufficient to show that for any generalised toy dynao A we have that Z r i= l u a i Ω i 0. Z By straightforward calculation, r i= l u a i Ω i = Z r = l r i= l a i Ω i = r l + r = l r i= l a i Ω i.

13 O. KOZLOVSKI & P. VYTOVA Using conditions D D4, we estiate the last ter as follows r = l r i= l a i Ω i = D in a i Ω i #D in sup a i sup Ω i s δ s s s + s. We see that r i= l u a i Ω i 0 under condition that Z s δ s s s + s < r l. 6 Recall that, since Ω is of the class G, we have r l >. We also know fro 3 that there exists γ < /4 such that δ s 3 s < γ. Therefore 6 holds true under condition that γ < s <. Lea 5.. Let η = dχ [,] + ψ Φ Ω be a step function such that ψ dr for soe r. Then η Cone r r, Ω. Proof. We would like to write ψ = βχ [,] + ψ, where ψ = c χ Ω Z Let us assue that ψ = Z c χ Ω, then and r = l c = 0. r = l c χ Ω = β r = l χ Ω + r = l c χ Ω. iplies c = c β and consequently r for β : β = r l r = l c Therefore we deduce that η = d + βχ [,] + ψ C = l c β = 0. Thus we have an upper bound c = ψ dr. Z β d β, Ω 3 r C r, Ω.

14 FAST DYAMO O THE REAL LIE Definition 7. Given Ω and Ω, two partitions of the class G, we define a linear operator E : Φ Ω Φ Ω by the atrix, if l i r and l r, E i = 7 δ i, otherwise. Reark 4. The operator E is a generalised toy dynao. Lea 5.3. Consider a function ϕ Ker E χ [,]. Then Eϕ ϕ. Proof. Let ϕ Φ Ω be a step function. We ay write ϕ = c χ Ω, then Z Eϕ = r c χ [,] + + c χ Ω ; = l < l > r and the condition ϕ Ker E χ [,] iplies Eϕ = + r = l c χ Ω = c = 0. Therefore + c ϕ. < l > r < l > r Proposition 5.. Let s be sall enough so that log s 64/63. Let Ω and Ω be partitions of the class G. Consider a generalised toy dynao operator A: Φ Ω Φ Ω. Then for any φ Φ Ω A Eφ /+γ φ, where γ satisfies the inequality 3. Proof. Let φ = Z c χ Ω Φ Ω be a step function with the unit nor φ = ax c, /, sup c =, Z which iplies Z c and sup c /. By straightforward calculation, A Eφ = c a i E i χ Ω i = i Z Z = r r c a i χ Ω i + D out c a i δ i χ Ω i + Sp c a i δ i χ Ω i. i= l = l 4

15 O. KOZLOVSKI & P. VYTOVA Observe that #D out 4 δ + δ r l + δ r l = = δ δ + r l + r l. Therefore, using φ, Lea 5., definition of the set D out, and condition D3, A Eφ L,Ω r r c a i + D out c a i δ i + Sp c a i δ i i= l = l 3/+γ + sup c sup a i #D out + Z c δ sup a i /+γ + / s s δ δ + r l + r l + δ s s By straightforward calculation we see that for s sall enough so that log s 64/63. s s δ < s s α log s s s Therefore, under the sae condition, since δ, δs = γ. Finally, / s s δ < γ δ / < /+γ. / δ s s Suing up, r l + r l s s A Eφ 3 /+γ. / 4 δ < /+γ. ow, for the axiu nor, we have that A Eφ ax c a i E i χ Ω x R i x Z i Z Z c s s s s. Thus / A Eφ /+γ. Lea 5.4. Let Ω and Ω be two partitions of the class G. Let E : Φ Ω Φ Ω be a linear operator with the atrix defined by 7 in the canonical basis. Then for any function φ Φ Ω Eφ φ and Eχ [,]. 5

16 FAST DYAMO O THE REAL LIE Proof. Let φ = Z c χ Ω Φ Ω be a step function of the unit nor. Then, by straightforward calculation, Eφ = i Z c E i χ Ω i = Z r i= l r = l c χ Ω i + D out D in c δ i χ Ωi = = r c χ [,] + + c χ Ω ; = l < l > r so the weighted L -nor is Eφ,L = r = l c r l + < l + > r c +. The upper estiate for the supreu nor is easy: r Eφ = ax c χ [,] x + x R =l <l Hence Eφ φ. Obviously, + > r c χ Ω x. Eχ [,] Eφ,L = r l r l. Let us consider two cones Cone, Ω Φ Ω and Cone γ /, Ω Φ Ω in correspondence with general definition p. 3: Cone, Ω def = { φ = dχ [,] + ψ ψ = c χ Ω ; Z r } c = 0; ψ d ; 8 = l Cone γ /, Ω def = { φ = dχ [,] + ψ ψ = Z c χ Ω ; r = l c = 0; ψ d γ / }. 9 Theore. Assue that is large enough so that the inequality 3 holds true for soe 0 < γ < /4 and all sufficiently sall κ. Additionally, assue that log s 64/63. Let Ω and Ω be two partitions of the class G. Then for any generalised toy dynao A: Φ Ω Φ Ω we have A: Cone, Ω Cone γ /, Ω ; Moreover, for any η Cone, Ω we have Aη r l η η. 6

17 O. KOZLOVSKI & P. VYTOVA Proof. Let φ Cone, Ω be a step function, φ = dχ [,] + ψ, where ψ = Z c χ Ω, with ψ d and r = l c = 0. We ay write Aφ = A Eφ + Eφ = deχ [,] + A Eφ + Eψ. Obviously, φ d, thus by Proposition 5. A Eφ A E φ d /+γ +. By Lea 5.3, Eψ ψ = d. Therefore A Eφ + Eψ d /+γ + + d, so we conclude Aφ = dχ [,] + da Eχ [,] + A Eψ + Eψ, where d d and da Eχ [,] +A Eψ+Eψ d /+γ + +. Theore now follows fro Lea A dynao ap. Recall the ap we considered is Section 3 Let s s, be two real nubers such that log s s = κ. and let δ = α be a sall real nuber with 5 α <. 6 We extend an expanding ap on the interval [, ] to the line R as follows 3 s x + s, if < x < s ; lx = s x + s, if s < x < ; 30 x, otherwise. and define its extension l: R R by lx, y = lx + y. We associate a sall perturbation l to any sequence l R and δ. We associate a transfer operator to a ap l by l φx: = sgn dl yφy. 3 y l x The ap l outside the unit interval is not iportant to us and we chose a siple ap that changes direction of the vector field, to ake it non-trivial. The exact for is not relevant here. First of all we show that with any sequence we can associate a canonical partition of the class G. Then we approxiate the operator l by a generalised toy dynao Theore 3 on p. 3. 7

18 FAST DYAMO O THE REAL LIE 5.3. Canonical partition for the perturbation l. In this section we construct a partition of the class G associated to the sequence. Later we will refer to it as the canonical partition of the ap l. Recall Definition 3 of the partition G: Definition 3. We say that a collection of intervals Ω = {Ω } Z akes a partition of the class G, δ, s, s, if Ω = R, Ω i Ω = if i, and the following conditions hold true. The interval [, ] contains at least and at ost intervals of the partition, and {±} are the end points of soe intervals of the partition. The length of intervals Ω is bounded away fro zero and fro infinity Ω s +. s s 3 Any interval I R of the length I = δ contains not ore than intervals of the partition. δ = + δ log s = α log s + 4 Any interval of the partition Ω R \ [ δ, + δ] has length Ω =. We fix s and s in the definition of the ap l 3 and a sequence l R with a nor l δ = α. The ap l is piecewise linear and so any iteration is a such. Let = a k 0 < a k <... < a k k + = + 3 be all the points of discontinuity of the ap l k. Define the corresponding partition of the real line a k = k =0 ak, where a k = a k, a k + are the partition intervals. Observe that for any k we have that {±} are the endpoints of soe intervals of the partition. Let a k l k = and a k r k =. We shall odify the partition a and obtain the canonical partition for the ap l. Definition 8. We call a branch l n an of the ap l n have that l k an [, ]. Definition 9. We call a ain branch l k ak of the ap l k l k a k [, ] > s. 8 ain, if for any 0 < k < n we long, if

19 O. KOZLOVSKI & P. VYTOVA Lea 5.5. The ap l has at ost α+ ain branches that are not long, where α < α log s is chosen such that s α < α. Proof. Let a Since l a be a doain of a ain branch which is not long, that is l a [, ] < s. is an interval, a connected subset of R, we conclude l a + > s or l a + > s. Without loss of generality we ay assue that the first holds true. By definition, a we have that l k a is a point of discontinuity. Therefore, for soe k < = +k; hence we deduce that l a = l k +k. So we conclude l k + k + > s, and, consequently, k < α +. Indeed, if k > α +, then k < α, and it follows that l k + k + < s α δ < =. s s Since the ap l k has at ost k ain branches, we conclude that there are at ost α + points a such that l k a = + k. has at ost α+ ain branches that are not long. Suing up, the ap l Lea 5.6. Let k α log s and let a, b be the doain of a ain branch of the ap l k. Then l k a + < δ sk s < s δ; l k b < δ sk s < s δ. Proof. By induction in k. The case k = is obvious. Recall that a k a k+ and a k+ \ a k = {l k Therefore for x a k we have, l k, l k /s }. l k+ x = l σ k l k x = s l k x + s k +, if l k x + < /s δ l k+ x = l σ k l k x = s l k x s + k +, if l k x < /s δ In the first case we know that, by induction assuption, l k+ In the second case, l k+ x + s l k x + + k + + sk+ s δ. x s l k x + k + + sk+ s δ. 9

20 FAST DYAMO O THE REAL LIE Corollary. Let k α log s. Then for any doain a, b of a ain branch of the ap l k we have that s = s l k a, b. Proof. By assuption, l k a < lk b and fro Lea 5.6 it follows that l k a < s δ < s < s + δ < l k b. Corollary. Let n α log s. Then any ain branch of the ap l n is δ-close to one of the ends of the interval [, ]: in other words, either δ l n a, b or δ l n a, b, or both. Proof. By induction in n. The case n = is obvious. Observe that a, b cannot be an interval of continuity of the ap l n for any k < n. Therefore ln is either continuous at a, or at b, or at both end points. In any case a, b belongs to an interval of continuity of l n satisfying conditions of Corollary of Lea 5.6. By definition of l, we see that either l n a = s or l n b = s. Without loss of generality assue that l n b = s. Then we see that l n a, b n, + n δ. Siilarly, l n a = s iplies that l n a, b + n, n δ. Lea 5.7. The ap l k for any k α log s has exactly k long branches. Proof. By induction in k. The case k = is trivial. It follows fro Lea 5.6 and Corollary of Lea 5.6 that any long branch of the ap l k contains at least two long branches of the ap l k. Corollary. The ap l has at least long branches, provided α log s >. Proof. If α log s >, then α log s < α log s and therefore the ap P α log s η has at least α log s long branches for any η l R with η δ. Let η = σ α log s. Then we can decopose l = l α log s η l α log s. According to Lea 5.7 the ap l α log s has at α log s long branches. By definition of a long branch, its iage is at least s long; using Corollaries and of Lea 5.6, we deduce that for any doain a, b of a long branch we have that either +δ, + s l log s a, b or s ; δ l log s a, b. Moreover, any of two intervals, s and s, contains exactly α log s long branches of the ap l α log s. 0

21 O. KOZLOVSKI & P. VYTOVA We can find an upper bound for the length of a doain of a long branch of the ap l α log s η : it easy to show by induction in nuber of iterations that any long branch a, b has a doain of the length at least b a = s α log s δs α log s = s α log s δ s α log s Therefore any of the intervals + δ, s and s, δ contains at least α log s s α log s δ = α log s log s α α log s long branches of the ap l α log s. Therefore, the coposition has at least α log s α log s long branches, which coes as α log s >, as proised. Canonical partition construction. Let us consider the set of end points of doains of long branches D l : = { x x is an endpoint of a doain of a long branch of the ap l } {±} = = { = d < d <... < d = }, and define a partition Ω = Ω of the interval [, ] by Ω = d, d + ; =,...,. = Let us denote by U ε Ω a neighbourhood of Ω of the size ε. We shall set ε = s. If for soe Ω = d, d +, containing a long branch of the ap l, there exist points of discontinuity of the ap l in a neighbourhood U ε Ω [, ], then we extend the interval Ω to include all these points. Let Ω = d, d +, =,..., be a new collection of intervals. If there exist two intervals d, d + and d +, d +3 containing long branches of the ap l, and such that d + d + < s, then we replace the interval d, d + in Ω with the interval d, d +. ow the length of any interval of the partition Ω, containing a long branch, is not ore than s + s. Assue that there exist two intervals d, d + and d +, d +3 containing long branches of the ap l, such that d + d + > s, then we split the interval d +, d + into intervals of the size s, allowing one of the to be longer, or saller, if necessary. More precisely, let Ω = d, d + be an interval of Ω that doesn t contain a long branch. Let n := [ s d + d ] be the nuber of whole intervals of the length s that could fit inside d, d +. If d + d ns < s, we split the interval d, d + into n intervals; adding the intervals d + ks, d + k + s, 0 k < n to the partition Ω. Otherwise, we

22 FAST DYAMO O THE REAL LIE split the interval d, d + into n + intervals, adding d + ks, d + k + s, 0 k n to Ω. The intervals a 0, and, a, do not contain any long branches, and we define the partition there as described above. Finally, we define the partition on, a and a, + splitting the into equal intervals of the length. We have obtained a partition of the real line, that satisfies Conditions D and D4 of Definition 3. We have to check other conditions of Definition 3. Lea 5.8. The partition constructed satisfies Condition D3. Any interval I R of the length δ contains at ost δ < α log s + intervals of the partition. Proof. The stateent holds true for any interval I R \ [a, a ] of the length δ. Assue that I [a, a ], and I = δ. Then there are two possibilities: the interval I contains a long branch; the interval I doesn t contain a long branch. Consider the first case. Observe that for any k 0 and for any interval I 0 [, ] of the length I 0 < s k 0 such that l k I 0 [, ] for all k < k 0, the ap l k 0 is one-to-one on I 0. Easy to check by induction. Since for any long branch a we have s l a, we conclude that k 0 : = [ log δ log s ] = [α log s ] and then we see that the ap l k 0 is one-to-one on any interval I 0 of the length less than δ such that l k I 0 [, ] for all k < k 0. Thus any interval of the length δ contains at ost k 0 long branches of the ap l. Consequently, any interval I with I δ contains at ost k 0 < δ intervals of the partition with a long branch inside. Assue now that the interval I of the length I = δ contains soe intervals of the partition that do not contain a long branch inside. Let I 0 I be a axial by inclusion subinterval not containing a long branch. Then by construction of the partition, it contains at ost one interval of the partition Ω of the length less than s Since the interval I contains at ost k 0. long branches, it ay contain not ore than k 0 + intervals I 0 without a long branch inside. Therefore, the interval I contains not ore than δs + k0+ < δ intervals of the partition. In the second case, an arguent siilar to the one above shows that an interval I of the length I = δ and without a long branch inside contains not ore than δs + < δ intervals of the partition.

23 O. KOZLOVSKI & P. VYTOVA Lea 5.9. The partition constructed satisfies Condition D of Definition 3. The interval [, ] contains at least and at ost α log s + δs the partition. intervals of Proof. By Corollary of Lea 5.7, the ap l has at least long branches, provided s is chosen such that α log s >. Every long branch belongs to exactly one of intervals of the partition, and the escaping set of easure δ contains at ost δs intervals. Suing up, we conclude that the construction leads to a partition of the class G, as desired. We shall refer to the resulting partition Ω as the canonical partition of the ap l. Lea 5.0. Any interval of a canonical partition Ω has at ost two ain branches of the ap l. Proof. If an interval Ω of the partition contains ore than one ain branch, one of the ain branches is not long. Let it be a k. Then by Definition 9 l a k [, ] < s. ow we repeat the calculation of Lea 5.5. The end points of the interval a k are the points of discontinuity of the ap l. Then there exists two nubers n < and n < such that l n a k = + n and l n a k+ = n. Therefore, l a k + = l n σ n + n + s n δ; l a k+ = l n σ n n s n δ. Since by assuption l a k < s, we deduce δs n + s n s. The latter is equivalent to δs n + s n s, which iplies that either δs n s, or δs n s, or both. Hence we get an upper bound on n or n, respectively: n, < 0 : = α + 0. log s Therefore one of the end points of a k is an end point of the ain branch of the ap l n with n < 0. Observe that all ain branches of the ap l are long. Any interval of the length s 0 contains at not ore than one ain branch of the ap l 0. Therefore the distance between short ain branches of the ap l is at least s 0 s + s = sup Ω, and any interval of the partition contains not ore than one short ain branch of the ap l. Therefore, any interval of the partition contains at ost two ain branches. 3

24 FAST DYAMO O THE REAL LIE 5.4. Approxiating l by a generalised toy dynao operator. Here we prove the ain result of this Section, Theore 3, which establishes the existence of a generalised toy dynao operator, a close approxiation of l for arbitrary δ. Construction. Let a partition Ω of the class G be given. Let l be as above, and let a be a partition of the real line by its points of discontinuity and let Ω be the canonical partition of the ap l. Introduce the oint partition: a Ω = {d } Z. We assue the natural nubering: [d 0 ; d ] 0 and d < d + for any Z. Define the iage of the oint partition by { b ± : = li y d ±0 l y }. Z Then on the interval d, d + the ap l is given by l x: = b + b+ d + d x + b+ d + b + d d + d, d < x < d +. We define an approxiating ap l to be l x: = b + b+ x + b+ d + b + d, d < x < d + ; d + d d + d where x stands for the closest to x point of the partition Ω, which is saller than x; and x stands for the closest to x point of the partition Ω, which is larger than x. In particular, branches of the ap l are not longer than branches of the ap l. We define an operator T : Φ Ω Φ Ω by T φx: = y l x sgn d l yφy. 33 Lea 5.. The operator T is a linear operator between two subspaces of step functions associated to the partitions Ω and Ω see p. for definition: T : Φ Ω Φ Ω. Proof. Linearity is obvious. It is sufficient to show that for any interval Ω Ω of the first partition, T χ Ω x: = y l x sgn d l yχ Ω y Φ Ω. By definition of l, we see li l y = y d 0 [b ] and li l y = y d +0 [b+ ], therefore all points of Ω k Ω have the sae nuber of preiages with respect to l for any interval Ω k. Moreover, l Ω k does not contain any point of Ω inside, as it is piecewise onotone on a subpartition Ω a. 4

25 O. KOZLOVSKI & P. VYTOVA Definition 0. We introduce the k-escaping set E k : = {x [, ] n < k l n x [, ]}. 34 Lea 5.. In the canonical bases of Φ Ω and Φ Ω sup #{x Ω l x = y} s y Ω s i Proof. Observe that the ap is one-to-one on any interval I [, ] \ E of the length I s. Given an eleent Ω, consider a axial by inclusion interval I E Ω, such that I s. We shall show that There are two possibilities: ax y R #{ x I l x = y} 3s the ap l is continuous on I E Ω ; the ap l is not continuous on I E Ω. In the first case the ap l I is a biection and 35 holds true. ow consider the second case: the ap l is not continuous on I E Ω. We ay find the sallest k 0 such that l k 0 I [, ]. Then def = +α l k 0 I [, ], + sk 0 s k 0,. Let 0 log s. It follows by induction in k that for any k 0 k < 0 the iage l k I [, ] ay be covered by two disoint intervals in particular, where δ k = l k I [, ], + δ k + s k + δ k s k,, k s k =k 0 and δ k = k =k 0 s k. with δ, k sk k 0+ δ, and + s l k I for all k 0 k < 0. In particular, for any x, x I such that l k 0 x, +s k 0 and l k 0 x s k 0, we have for all k < 0 : l k x l k x + δ k s k + δk + s k = s k + δk δk. The ap l 0 k 0 σ k 0 is a biection on any of the intervals, +s k 0 and s k 0,. Therefore, we deduce that the ap l 0 is a biection on I. It follows that the iage l 0 I consists of not ore than 3 0 intervals each of which is not longer than s 0. Let η = σ 0 and consider the ap l 0 η. We clai that it is a biection on any interval I R of the length I s 0 3. Indeed, if l 0 η is continuous on I, 5

26 FAST DYAMO O THE REAL LIE then it is a biection. Assue that for soe k 0 0 the ap l k 0 η on I. Then is not continuous l k 0 η I [, ] ; + s 0+k δ s 0+k 0 3 δ;, and for any k 0 < k 0 l k η [, ] ; + s k+ δ + s 0+k 3 s k+ s 0+k 3 ;. By straightforward calculation we see that provided s α s 0+k 3 + s k+ δ s. Therefore for any interval I of the length I s 0 3 and for any two points x, x I with x x we have that l k ηx l k ηx for all k 0. We see that the iage l 0 ay be covered by not ore than 3 0 s 3 intervals of the length s 0 3. Hence we conclude that for any interval I s sup # { x I l x = y } 3 0 s 3 < 3s y R Since by Lea 5.0 any interval of the partition contains at ost two ain branches, the set Ω E is a union of not ore than two intervals, which ay be covered by + s s disoint intervals of the length s. Therefore ax y R #{ x Ω i l x = y} 3s 3 s s < s s. Corollary. In the canonical bases of Φ Ω and Φ Ω the atrix of the operator T satisfies condition D: ax τ i + s s. Proof. Recall the definition of the operator T : T φx: = sgn d l yφy. 33 Then for φ = χ Ω T φx: = i Z we have τ i χ Ω i x = i Z y l x y l x sgn d l yχ Ω yχ Ω i x = = i Z y l x Ω sgn d l yχ Ω i x; 36 6

27 O. KOZLOVSKI & P. VYTOVA therefore τ i = y l Ω i Ω sgn d l y. The definition of the ap l guarantees that τ i are well-defined; in particular τ i #{x Ω l x = y Ω i }, and the right hand side is independent on the choice of y. Obviously, sup x #{x Ω l x = y Ω i } sup #{x Ω l x = y Ω i }, x Corollary. We have the following upper bound for a total nuber of preiages of a point x R : { sup y R l y = x } s ; 37 x R s { y R l y = x } s sup x R s. 38 Proof. By definition of a partition of the class G, the interval [, ] contains not ore than intervals of the partition; and intervals [ δ; ] and [; + δ] contain not ore than δ intervals of the partition each. Finally, both aps are biections on the copleent to [ δ, + δ]. Lea 5.3. Let Ω be the canonical partition of the ap l. Let Ω be another partition of the class G. Then # { i, Ω i [, ] l Ω E } δs Proof. We shall prove that Ω i D in l a E a s δ; then the Lea will fro fro the lower bound on the size of the eleents of partition. Indeed, by induction one can show that k la k a k E k sk k s δ. 7

28 FAST DYAMO O THE REAL LIE The case k = is trivial. Then we proceed k la k a k E k k la k a k E k + k la k a k E k \E k s δ sk k + k δ = sk k s s δ. Corollary. Let Ω be the canonical partition of the ap l. Let Ω be another partition of the class G; and let l be a ap defined as above on p. 4. Then #{i, Ω i [, ] l Ω E } δs Proof. The inequality for the ap l follows fro the fact that iages of all branches under adusted ap l are shorter than the iages of the sae branches under the original ap l. Proposition 5.3. In the canonical bases of Φ Ω and Φ Ω the operator T defined by 33 is a generalised toy dynao. Proof. We have checked the condition D already. We should verify the following conditions. where D #D in 3 δs ; D3 for any pair i, Sp we have that τ i = 0 whenever i > δ ; D4 #Ar. Ar: = { { l... r } #{i { l... r } τ i = } δ } ; D in : = { i, { l... r } { l... r } τ i }. To verify the condition D: #D in 3s δ we shall show that D in Ω i 3s δ and then taking into account Ω i s we get the result. Let E be the -escaping set as defined by 34 above. We introduce three subsets of the set D in. D in : = {i, D in Ω i [, ] \ l Ω \ E } copleent to the iages of the ain branches; D in : = {i, D in Ω i [, ] l Ω E } 8

29 O. KOZLOVSKI & P. VYTOVA iage of the points that were apped outside [, ] and back; D in 3 : = {i, D in Ω i [, ] l Ω \ E } iage of the points that were inside [, ] in first iterations. We clai that D in = D in D in D in 3 : indeed, for any pair of indices i, D in we have that Ω E. We shall show that D in Ω i s δ, D in Ω i s δ, and #D in 3 s δ. We start with D in and recall the original partition a by the points of discontinuity of the ap l. Let JD in be the union of intervals with indices corresponding to D in : JD in : = : i, D in Ω \ E. We ay write then D in Ω i #{a JD in } a JD in l a + a [,]\E l a ; and we shall show by induction in k that k+ k la k a k [,]\E k < s k k kα, where kα. The case k = is trivial. Let b k be the canonical partition of the ap l k 0. This partition has k eleents in [, ]. There exists a correspondence between the sets of indices τ : {i Z a k i, } { k,..., k } that satisfies dl k = dl k a k 0 k b τ. and τ τ for all. In particular, sgn d l k = sgn d l k a k 0 k b τ We split the intervals a k into two groups: B k : = { { k,..., k } = τi for soe i Z}; B k : = { k,..., k } \ B k. 9

30 FAST DYAMO O THE REAL LIE We also see that l k 0b k = [, ] for any interval of the partition b k. k+ k a k = + l k 0b k τ l a k [,]\E k = B k s s B k l k 0b k \ l k a k + B k l k 0 b k k B k s k k δ + k δ s k kδ. Therefore we deduce that + B k B k l k 0b k \ l k a k τ + k δ + l k a k τ + k δ k la k a k [,]\E k B k k la k a k [,]\E k l0 k b k < s α, where kα. = Since there are not ore than ain branches, and the length of intervals of the partition Ω is bounded Ω i s + s, we get + k < s α + s + s δs ; a k l a k [,]\E k provided s < < s are chosen such that s s > +α, which is possible. The inequality for the set D in follows fro of Lea 5.3. Finally, for the set D 3 in we observe that i, D 3 in if and only if there exist two ain branches a, a Ω such that for any k < we have l k a [, ] and l k a [, ], and l a l a Ω i. Since both a and a belong to the sae eleent of the partition we conclude that either l a s or l a s. By Lea 5.5 there are at ost α ain branches with this property. Without loss of generality we assue the latter. Then by definition of l we have l a s + s + s. Hence #D 3 in α δ. It follows that as required. #D in = #D in + #D in + #D in 3 s δ + α δ δ δ 3s δ, The condition D3 follows fro the fact that the ap l is linear and on the coplient R\[ δ, +δ] in other words, the copleent consists of two pieces of continuity, and, oreover, it is given by l x = x + b on these set. Therefore, 30 are

31 O. KOZLOVSKI & P. VYTOVA τ i = 0 whenever i > b δ δ. Obviously, b δ, so we get τ i = 0 whenever i > δ. ow it only reains to show that the generalised toy dynao, constructed fro the ap l, is a good approxiation to the operator l. Theore 3. Let Ω be a partition of the class G. Consider a sequence l R with α and let Ω be the canonical partition of the ap l. Then for the operator T = l : Φ Φ defined by 33 and for any essentially bounded integrable function g L R we have l s 3 T W δ g g. /+α s Proof. Let g Ω = and let f = W δ g. Then f g /, since g = ax Ω Z gx dx, / sup gx. Ω x R By definition, we write T fx = l fx = y l x y l x sgn l yfy; 39 sgn l yfy. 40 We begin with weighted L -nor. l f l f = Ω Z T fx l fx dx Ω δ + T + fx l fx dx+ 4 + s r + = l δ Ω +δ +δ + Ω T fx l fx dx+ 4 T fx l fx dx. 43 We estiate all three ters separately. By the very definition, on the infinite intervals +δ, + and, δ the ap l is given by l x = x+. 3 =

32 FAST DYAMO O THE REAL LIE Therefore, the ap l + δ, + ; oreover, is one to one on each of the intervals, δ and l, δ + δ, +,, +. Observe that for the last point a R of the last point of discontinuity of the ap l we have, using Lea 6.3: + a fx dx = + a W δ gx dx = + = Ω The first difference we estiate by the su of absolute values. + +δ y l x sgn l yfy = + +δ a = y l x sgn l Ω W δ gx dx W δ g δ s δ. yfy dx = f l x fl x dx + a a + a fx dx, + fx dx = where a and a are the first and the last points of discontinuity of the ap l. Suing up, + +δ Siilarly, l fx T fx dx 4δ sup f + 4 f 4 δ δ / + δ s δ g 8 δ s δ. 44 l fx T fx dx 8 δ s δ. 45 Suing up 44 and 45, and taking into account that f =, we get an upper bound for the first ter 4: δ + + +δ l fx T fx dx 6 δ s δ. 46 ow we use a rough upper bound to estiate the second ter. Since by Corollary of Lea 5. any point has at ost s s preiages with respect to l or l ; 3

33 O. KOZLOVSKI & P. VYTOVA and taking into account f /. +δ l T fx dx sup x +δ l fx + T fx dx l fx + T fx s δ 3 + δ f s 3/ α s s. Therefore we get an upper bound for the second ter 4: s +δ l + T fx dx / α s. 47 δ s The third ter 43 is a little ore coplicated. We split the su into two ters: long branches and all other intervals. Let a be a partition of R by the points of discontinuity cf. 3 and let a n = a n, a n+ be its intervals. Let a nl =, a n l + and a nr = a n r, be the ost left and the ost right intervals of the partition inside the interval [, ]. Let E be the -escaping set as defined by 34 above. By definition of l and T, r i= l Ω i l fx T fx dx = Ω i = r i= l Ω i Ω i y l x sgn l yfy ŷ l x sgn l ŷfŷ dx. Let us introduce two functions h, x: Z R R; h, x = y l x sgn l yχa yfy; and ĥ, x: Z R R; ĥ, x = ŷ l x sgn l ŷχa ŷfŷ. Then we see that h, x = Z y l x sgn l yfy; 33

34 FAST DYAMO O THE REAL LIE and ĥ, x = Z ŷ l x sgn l ŷfŷ; both sus are well-defined, because they have finite nuber of non-zero ters, since by Corollary of Lea 5. the total nuber of preiages of a point is not ore than 3 + s s. Therefore we ay write r i= l Ω i l fx T fx dx = Ω i = = <n l <n l + + n r =n l + >n r + + a E a E First we estiate the finite sus: <n l + for all s. >n r l r i= l k= l δ Ω i + Ω i r + δ k= r r i= l r i= l >n r Ω i Ω i r i= l Ω i h, x ĥ, x dx a Ω k r i= l Ω i Z h, x ĥ, x dx = h, x ĥ, x dx = Ω i Ω i h, x ĥ, x dx. 48 Ω i h, x + ĥ, x dx Ω i δ sup τ i sup f δ s s g s 3 /+α s ; 49 Observe that for any doain of a ain branch a E and y, ŷ a, such that l y = l ŷ we have that sgnl y = sgn l ŷ = and l y > s. As before, let a ŷ y = a, a +. Then inf l ax l a l a, l a + l a + Hence for any f W δ L R we see that fŷ fy s supw δ g g s 34 δ /. δs s.

35 O. KOZLOVSKI & P. VYTOVA Suing up, since the total nuber of ain branches is not ore than, we get for the first ter of 48: a r E i=l Ω i h, x ĥ, x dx Ω i inf Ω i To estiate the last ter, we introduce two sets of indices 3/ dx δs D def = { s, t Ω s [, ] l Ω t E } ; D def = { s, t Ω s [, ] l Ω t E }. s /+α. s 50 By Lea 5.3 and its Corollary, we see #D δs and # D δs. Observe that i, { a Ω i a E, Ω i l a [, ] } { Ω t Ω s s, t, Ω s [, ] l Ω t E }. along with i, { a Ω i a E, Ω i l a [, ] } { Ω t Ω s s, t, Ω s [, ] l Ω t E }.. Hence we calculate an upper bound for the second ter of 48: a r E i=l sup f sup g h, x ĥ, x dx Ω i Ω i a r E i=l i, D D Ω i Ω i Ω i g s δs s Ω i y l x χ a y + ŷ l x χ a ŷ dx τ i dx sup g sup τ i #D D 35 s 3 /+α s. 5

36 FAST DYAMO O THE REAL LIE ow we collect the four estiates 46, 47, 49, 50, and 5 together and get for any function g with g = : l W δ g l W δ g L,Ω 6 δ / α s δ + s s /+α + s s + s 3 /+α s 3 s 3. 5 /+α s for large enough and s < < s chosen such that s s /+α. ow we turn our attention to the supreu nor. We ay write sup l fx l fx = sup x x sup x sup x i Z l δ + sup x + r + δ l δ y l x Ω i + l δ y l x sgn l yfy y l x Ω i y l x Ω i sgn l yfy y l x Ω i sgn l yfy sgn l yfy y l x Ω i y l x sgn l y l x Ω i sgn l sgn l yfy yfy sgn l yfy + yfy. 53 Observe that sup x r + δ l δ r + δ sup x l δ y l x Ω i #{y l sgn l yfy x Ω i } sup Ω i y l x Ω i sgn l r + δ fy sup x l δ yfy r + δ sup τ i l δ τ i sup fy Ω i sup fy. 54 Ω i 36

37 O. KOZLOVSKI & P. VYTOVA Our goal is to estiate the last su fro above via weighted L -nor. Recall that f = W δ g. By definition of the weighted L nor, we see W δ g in particular, r + δ l δ r + δ l δ Ω i W δ g = Ω i = sup Ω i r + δ l δ Ω i Ω i W δ g W δ g + r + δ W δ g sup W δ g + Ω i l δ r + δ l δ sup Ω i sup W δ g ; Ω i W δ g Ω i W δ g. 55 Ω i We know that for any bounded, continuous, absolutely integrable, and piecewise differentiable function f : R R and any finite interval I Therefore < r + δ l δ sup Ω i W δ g Ω i Ω i sup I f I I W δ g r + δ l δ f Ω i I f. d Wδ gx dx < dx d Wδ gx d w δ x t gt dt dx dw δx t gt dtdx = dx R dx R R R dx = gt dw δx t dxdt gt dt dx πδ δ sup Ω g L,Ω. R R Z Ω 56 Hence, substituting 56 to 55, and using Lea 6.3 r + δ l δ sup Ω i W δ g W δ g + sup Ω i δ g δ s δ + g s δ + δ g s. 57 δ 37

38 FAST DYAMO O THE REAL LIE Finally, taking into account g Ω second ter of 53 =, we substitute 57 to 54 and get for the / sup x r + δ l δ y l x Ω i sgn l yfy y l x Ω i sgn l yfy sup τ i /+ δ s δ δ δ / s. 58 s Let us define Ax def = x s, x + s. We have the following upper bound for the first su in 53: sup x sup x R l δ sup + y,y Ax + l δ y l x Ω i fy fy sgn l yfy sup y y s Suing up 58 and 59, we get in 53 y l x Ω i fy fy sgn l sup f δs yfy / δs. 59 / sup l fx l fx δ / s + x δ s δs by straightforward calculation. s 3, 3 60 /+α s 6. Invariant cone in Φ. In this section we construct an invariant cone in the space of essentially bounded and absolutely integrable functions Φ for the operator W δ l W, which is independent δ of the choice of δ. We exploit the properties of the Weierstrass transfor that we prove below. 6.. Discretization and the Weierstrass transfor toolbox. Here we prove a few estiates showing that the iage of the Weierstrass transfor with Gaussian kernel of a large variance copared to the size of eleents of a partition ay be very well approxiated by a step function on the partition. 38

39 O. KOZLOVSKI & P. VYTOVA Definition. Given a partition Ω of the class G we define a linear discretization operator D Ω : D Ω : L R L R Φ Ω L R L R; D Ω : f Z d χ Ω, d = ax fx + in fx. 6 Ω Ω Definition. The Weierstrass transfor W δ kernel with variance δ is a convolution with the Gaussian W δ : f w δ f, where w δ x = πδ e x δ. 6 Lea 6.. Let f : R R be a differentiable function. Then Proof. Indeed, by straightforward calculation, f D Ω f Ω,L = k Z f D Ω f Ω,L dfx dx. 63 dx fx D Ω fx dx Ω k Ω k Ω k ax fx in fx dx = k Z Ω Ω k Ω k k ax fx in fx Ω k Ω k k Z k Z = dfx dx. dx R R Ω k dfx dx = dx Lea 6.. Let Ω and Ω be two partitions of the class G, δ, s, s. Let D Ω a discretization operator and let W δ be the Weierstrass transfor defined above. Then for any bounded integrable function f D Ω W δ f W δ f axsup Ω, sup Ω f δ s δ f. Reark 5. The dispersion δ in the Gaussian kernel is the sae δ as in the definition of a partition of the class G. 39 be

40 FAST DYAMO O THE REAL LIE Proof. We begin with estiation of the L -nor. Let D Ω W δ f = Z d χ Ω. Then D Ω W δ f W δ f = sup w δ x tftdt d χ Ω x = x R R Z = sup ax w δ x tftdt in w δ x tftdt k Z Ω k R sup Ω k ax d w δ x tftdt k Z Ω dx k R sup Ω d k sup x R R dx w δx tftdt sup Ω k sup fx sup e x t δ x e x t δ x R πδ t= Ω k R t=+ sup Ω k δ f. ow we proceed to the weighted L -nor. Using Lea 6. we get D Ω W δ f W δ f d R dx W δfx dx = = d w δ x tftdt dx dw δx t ft dtdx = R dx R R R dx = ft dw δx t dxdt = sup Ω ft dt f Ω. dx πδ δ Ω R R Lea 6.3. Let Ω and Ω be two partitions of the class Gδ, s, s. Then an upper bound of the nor of the Weierstrass transfor is given by Z W δ f sup Ω δ δ f δ s δ f. 64 Proof. We estiate the nor of the operator W δ on step functions first. Let φ Φ Ω be a step function on Ω. Assue that φ = Z c χ Ω and φ Ω =, that is ax c, / sup c = ; Z x which iplies Then c, sup c /. Z W δ φx = Z c w δ x tdt. Ω 40

41 O. KOZLOVSKI & P. VYTOVA So we calculate W δ φ Ω,L = Ω k Z k c w δ x tdt dx Ω k Z Ω c Ω k Z Z k w δ x tdtdx = Ω k Ω = c Z Ω Ω k Z k w δ x tdxdt = Ω k = c Z Ω Ω k Ω k Ω >δ + Ω k Ω <δ Ω k w δ x tdxdt. We know that Ω k w δ x tdx < Ω δ. k We also observe that for any t Ω t δ Ω Ω k Ω >δ k w δ x tdx Ω inf Ω k k w δ x t+ Therefore, taking into account that Z c, W δ φ Ω,L k Z Z + w δ x tdx dx e inf Ω k. t+δ c e inf Ω k + δ e Ω δ sup Ω inf Ω k + δ. δ ow we consider arbitrary function f L R L R with f =. Then W δ f Ω,L = Ω k Z k w δ x tftdt dx Ω k Z Ω ft Z Ω Ω k Z k w δ x tdxdt = Ω k = ft + Z Ω Ω Ω k Ω >δ Ω k Ω <δ k w δ x tdxdt Ω k e ft Z Ω inf Ω k + 3 δ dt δ e sup Ω inf Ω k + 3 δ. δ In the last inequality we take into account that f Ω,L = Ω Z Ω ft dt. 4