[PPT] - Regularity Properties of Critical Invariant Circles of Twist Maps PowerPoint Presentation

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Regularity Properties of Critical Invariant Circles of Twist Maps

Nikola P. Petrov, University of Oklahoma Arturo Olvera, IIMAS-UNAM November 28, 2008

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Global Stability of Mechanical Systems

Two degree of freedom Hamiltonian System (2DFHS): KAM tori − → Topological barrier in the phase space − → Global stability Hamiltonian flow ⇐⇒ Area Preserving Twist Map (APTM) Poincar´ e section: (2DFHS) − →(APTM) KAM torus (2DFHS) − →Invariant Circle (APTM) One parameter family of APTM Border of stability :⇒ Critical Invariant Circle

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Global Stability of Mechanical Systems

Two degree of freedom Hamiltonian System (2DFHS): KAM tori − → Topological barrier in the phase space − → Global stability Hamiltonian flow ⇐⇒ Area Preserving Twist Map (APTM) Poincar´ e section: (2DFHS) − →(APTM) KAM torus (2DFHS) − →Invariant Circle (APTM) One parameter family of APTM Border of stability :⇒ Critical Invariant Circle

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Global Stability of Mechanical Systems

Two degree of freedom Hamiltonian System (2DFHS): KAM tori − → Topological barrier in the phase space − → Global stability Hamiltonian flow ⇐⇒ Area Preserving Twist Map (APTM) Poincar´ e section: (2DFHS) − →(APTM) KAM torus (2DFHS) − →Invariant Circle (APTM) One parameter family of APTM Border of stability :⇒ Critical Invariant Circle

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Global Stability of Mechanical Systems

Two degree of freedom Hamiltonian System (2DFHS): KAM tori − → Topological barrier in the phase space − → Global stability Hamiltonian flow ⇐⇒ Area Preserving Twist Map (APTM) Poincar´ e section: (2DFHS) − →(APTM) KAM torus (2DFHS) − →Invariant Circle (APTM) One parameter family of APTM Border of stability :⇒ Critical Invariant Circle

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Global Stability of Mechanical Systems

Two degree of freedom Hamiltonian System (2DFHS): KAM tori − → Topological barrier in the phase space − → Global stability Hamiltonian flow ⇐⇒ Area Preserving Twist Map (APTM) Poincar´ e section: (2DFHS) − →(APTM) KAM torus (2DFHS) − →Invariant Circle (APTM) One parameter family of APTM Border of stability :⇒ Critical Invariant Circle

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Global Stability of Mechanical Systems

Two degree of freedom Hamiltonian System (2DFHS): KAM tori − → Topological barrier in the phase space − → Global stability Hamiltonian flow ⇐⇒ Area Preserving Twist Map (APTM) Poincar´ e section: (2DFHS) − →(APTM) KAM torus (2DFHS) − →Invariant Circle (APTM) One parameter family of APTM Border of stability :⇒ Critical Invariant Circle

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Global Stability of Mechanical Systems

Two degree of freedom Hamiltonian System (2DFHS): KAM tori − → Topological barrier in the phase space − → Global stability Hamiltonian flow ⇐⇒ Area Preserving Twist Map (APTM) Poincar´ e section: (2DFHS) − →(APTM) KAM torus (2DFHS) − →Invariant Circle (APTM) One parameter family of APTM Border of stability :⇒ Critical Invariant Circle

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Global Stability of Mechanical Systems

Two degree of freedom Hamiltonian System (2DFHS): KAM tori − → Topological barrier in the phase space − → Global stability Hamiltonian flow ⇐⇒ Area Preserving Twist Map (APTM) Poincar´ e section: (2DFHS) − →(APTM) KAM torus (2DFHS) − →Invariant Circle (APTM) One parameter family of APTM Border of stability :⇒ Critical Invariant Circle

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Critical Invariant Circles (CIC) exhibit remarkable scaling properties at the boundary of chaos (Shenker & Kadanoff, 82) Renormalization Group Analysis explains scaling properties (MacKay, 83) Scale invariances determine the regularity of the CIC CIC − →Fractional regularity − →Universal Property

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Critical Invariant Circles (CIC) exhibit remarkable scaling properties at the boundary of chaos (Shenker & Kadanoff, 82) Renormalization Group Analysis explains scaling properties (MacKay, 83) Scale invariances determine the regularity of the CIC CIC − →Fractional regularity − →Universal Property

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Critical Invariant Circles (CIC) exhibit remarkable scaling properties at the boundary of chaos (Shenker & Kadanoff, 82) Renormalization Group Analysis explains scaling properties (MacKay, 83) Scale invariances determine the regularity of the CIC CIC − →Fractional regularity − →Universal Property

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Critical Invariant Circles (CIC) exhibit remarkable scaling properties at the boundary of chaos (Shenker & Kadanoff, 82) Renormalization Group Analysis explains scaling properties (MacKay, 83) Scale invariances determine the regularity of the CIC CIC − →Fractional regularity − →Universal Property

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Our goal: Compute the regularity of CIC

Regularity of the conjugation:

CIC − →rigid rotation

x y θ CIC

Regularity between the conjugations:

CIC1 − →CIC2

y CIC x CIC y x

1 2

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Our goal: Compute the regularity of CIC

Regularity of the conjugation:

CIC − →rigid rotation

x y θ CIC

Regularity between the conjugations:

CIC1 − →CIC2

y CIC x CIC y x

1 2

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Our goal: Compute the regularity of CIC

Regularity of the conjugation:

CIC − →rigid rotation

x y θ CIC

Regularity between the conjugations:

CIC1 − →CIC2

y CIC x CIC y x

1 2

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Our goal: Compute the regularity of CIC

Regularity of the conjugation:

CIC − →rigid rotation

x y θ CIC

Regularity between the conjugations:

CIC1 − →CIC2

y CIC x CIC y x

1 2

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How to compute fractional regularity?

De la Llave and Petrov used Harmonic Analysis Methods to determine the regularity of Critical Circles Maps, T → T (Llave & Petrov,02) We extend this methodology to the study of CIC of APTM

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How to compute fractional regularity?

De la Llave and Petrov used Harmonic Analysis Methods to determine the regularity of Critical Circles Maps, T → T (Llave & Petrov,02) We extend this methodology to the study of CIC of APTM

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Area Preserving Twist Maps (APTM)

One parameter family of APTM Fλ : T × R → T × R: yn+1 = yn + λV (xn) xn+1 = xn + yn+1 where V (x) = V (x + 1) and has zero-average. Rotation number: ρ = lim

n→±∞

xn − x0 n

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Area Preserving Twist Maps (APTM)

One parameter family of APTM Fλ : T × R → T × R: yn+1 = yn + λV (xn) xn+1 = xn + yn+1 where V (x) = V (x + 1) and has zero-average. Rotation number: ρ = lim

n→±∞

xn − x0 n

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Area Preserving Twist Maps (APTM)

Invariant Circle of rotation number ρ, ICρ is the graph of a Lipschitz function (Birkhoff,)

If ρ is a Diophantine number − →ICρ depends analytically

n λ

Golden ICρ − →ρ = σG = [1, 1, 1, . . . ]

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Area Preserving Twist Maps (APTM)

Invariant Circle of rotation number ρ, ICρ is the graph of a Lipschitz function (Birkhoff,)

If ρ is a Diophantine number − →ICρ depends analytically

n λ

Golden ICρ − →ρ = σG = [1, 1, 1, . . . ]

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Area Preserving Twist Maps (APTM)

Invariant Circle of rotation number ρ, ICρ is the graph of a Lipschitz function (Birkhoff,)

If ρ is a Diophantine number − →ICρ depends analytically

n λ

Golden ICρ − →ρ = σG = [1, 1, 1, . . . ]

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Existence of Invariant Circles

If λ supx |V (x)| > 1 − → ∃ any ICρ If λ > 4/3 − → ∃ Golden ICρ Conjecture: For Diophantine ρ exists ¯ λρ such that: ∃ ICρ if |λ| < ¯ λρ and ∃ ICρ if |λ| > ¯ λρ Critical Invariant Circle (CIC) λ − →¯ λρ = ⇒ ICρ − →CIC

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Existence of Invariant Circles

If λ supx |V (x)| > 1 − → ∃ any ICρ If λ > 4/3 − → ∃ Golden ICρ Conjecture: For Diophantine ρ exists ¯ λρ such that: ∃ ICρ if |λ| < ¯ λρ and ∃ ICρ if |λ| > ¯ λρ Critical Invariant Circle (CIC) λ − →¯ λρ = ⇒ ICρ − →CIC

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Existence of Invariant Circles

If λ supx |V (x)| > 1 − → ∃ any ICρ If λ > 4/3 − → ∃ Golden ICρ Conjecture: For Diophantine ρ exists ¯ λρ such that: ∃ ICρ if |λ| < ¯ λρ and ∃ ICρ if |λ| > ¯ λρ Critical Invariant Circle (CIC) λ − →¯ λρ = ⇒ ICρ − →CIC

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Existence of Invariant Circles

If λ supx |V (x)| > 1 − → ∃ any ICρ If λ > 4/3 − → ∃ Golden ICρ Conjecture: For Diophantine ρ exists ¯ λρ such that: ∃ ICρ if |λ| < ¯ λρ and ∃ ICρ if |λ| > ¯ λρ Critical Invariant Circle (CIC) λ − →¯ λρ = ⇒ ICρ − →CIC

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Description of CIC:

R : T → R is the graph of ICρ Advance Map g : T → T defined by F(x, R(x)) = (g(x), R ◦ g(x))

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Description of CIC:

R : T → R is the graph of ICρ Advance Map g : T → T defined by F(x, R(x)) = (g(x), R ◦ g(x))

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Description of CIC:

Hull Map Ψ : T → T × R such that: F ◦ Ψ(x) = Ψ(x + ρ) Conjugation function h : π1 ◦ Ψ : T → T where: g ◦ h(x) = h(x + ρ)

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Description of CIC:

Hull Map Ψ : T → T × R such that: F ◦ Ψ(x) = Ψ(x + ρ) Conjugation function h : π1 ◦ Ψ : T → T where: g ◦ h(x) = h(x + ρ)

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Description of CIC:

Conjugating g to a rotation by σG: g ◦ h(x) = h(x + σG) (g = thick line, h = thin line)

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Big Conjugacies

Conjugation of two CIC, γ1 and γ2: Gγ1,γ2 = gγ1 ◦ g−1

γ2

Hγ1,γ2 = hγ1 ◦ h−1

γ2

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H¨ OLDER REGULARITY

For κ = n + ξ with n ∈ Z and ξ ∈ (0, 1): The function K : T → R has global H¨

lder exponent κ

(K ∈ Λκ(T)) when K is n time differentiable and, for some constant C > 0: |DnK(θ1) − DnK(θ0)| ≤ C |θ1 − θ0|ξ κ(K) := Is the H¨

lder regularity of K

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H¨ OLDER REGULARITY

For κ = n + ξ with n ∈ Z and ξ ∈ (0, 1): The function K : T → R has global H¨

lder exponent κ

(K ∈ Λκ(T)) when K is n time differentiable and, for some constant C > 0: |DnK(θ1) − DnK(θ0)| ≤ C |θ1 − θ0|ξ κ(K) := Is the H¨

lder regularity of K

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H¨ OLDER REGULARITY

For κ = n + ξ with n ∈ Z and ξ ∈ (0, 1): The function K : T → R has global H¨

lder exponent κ

(K ∈ Λκ(T)) when K is n time differentiable and, for some constant C > 0: |DnK(θ1) − DnK(θ0)| ≤ C |θ1 − θ0|ξ κ(K) := Is the H¨

lder regularity of K

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CIC and Universality

Universality: A characteristic is universal when it takes the same value in a open set of functions Conjectures:

∃! Nontrivial fixed point of renormalization operator ⇒ Universal property The regularity of R is a universal number (κ(R)) The regularity of g, h and h−1 are universal numbers Regularity of ”Big” conjugacies: κ(h) < κ(R) κ(h) < κ(H) κ(g) < κ(G)

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CIC and Universality

Universality: A characteristic is universal when it takes the same value in a open set of functions Conjectures:

∃! Nontrivial fixed point of renormalization operator ⇒ Universal property The regularity of R is a universal number (κ(R)) The regularity of g, h and h−1 are universal numbers Regularity of ”Big” conjugacies: κ(h) < κ(R) κ(h) < κ(H) κ(g) < κ(G)

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CIC and Universality

Universality: A characteristic is universal when it takes the same value in a open set of functions Conjectures:

∃! Nontrivial fixed point of renormalization operator ⇒ Universal property The regularity of R is a universal number (κ(R)) The regularity of g, h and h−1 are universal numbers Regularity of ”Big” conjugacies: κ(h) < κ(R) κ(h) < κ(H) κ(g) < κ(G)

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CIC and Universality

Universality: A characteristic is universal when it takes the same value in a open set of functions Conjectures:

∃! Nontrivial fixed point of renormalization operator ⇒ Universal property The regularity of R is a universal number (κ(R)) The regularity of g, h and h−1 are universal numbers Regularity of ”Big” conjugacies: κ(h) < κ(R) κ(h) < κ(H) κ(g) < κ(G)

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CIC and Universality

Universality: A characteristic is universal when it takes the same value in a open set of functions Conjectures:

∃! Nontrivial fixed point of renormalization operator ⇒ Universal property The regularity of R is a universal number (κ(R)) The regularity of g, h and h−1 are universal numbers Regularity of ”Big” conjugacies: κ(h) < κ(R) κ(h) < κ(H) κ(g) < κ(G)

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CIC and Universality

Universality: A characteristic is universal when it takes the same value in a open set of functions Conjectures:

∃! Nontrivial fixed point of renormalization operator ⇒ Universal property The regularity of R is a universal number (κ(R)) The regularity of g, h and h−1 are universal numbers Regularity of ”Big” conjugacies: κ(h) < κ(R) κ(h) < κ(H) κ(g) < κ(G)

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Poisson kernel method

Poisson kernel (periodic case): Ps(x) =

k∈Z

s|k| e2πikx = 1 − s2 1 − 2s cos 2πx + s2 , s ∈ [0, 1)

e−t√−∆ h
(x)

=

Pexp(−2πt) ∗ h
(x)

=

k∈Z

ˆ hk e−2πt|k| e2πikx . Theorem (“Poisson kernel method”): h ∈ Λα(T) if and only if ∀ η ≥ 0

∂

∂t η e−t√−∆ h

L∞ ≤ C tα−η .

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Poisson kernel method

Poisson kernel (periodic case): Ps(x) =

k∈Z

s|k| e2πikx = 1 − s2 1 − 2s cos 2πx + s2 , s ∈ [0, 1)

e−t√−∆ h
(x)

=

Pexp(−2πt) ∗ h
(x)

=

k∈Z

ˆ hk e−2πt|k| e2πikx . Theorem (“Poisson kernel method”): h ∈ Λα(T) if and only if ∀ η ≥ 0

∂

∂t η e−t√−∆ h

L∞ ≤ C tα−η .

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Poisson kernel method

Poisson kernel (periodic case): Ps(x) =

k∈Z

s|k| e2πikx = 1 − s2 1 − 2s cos 2πx + s2 , s ∈ [0, 1)

e−t√−∆ h
(x)

=

Pexp(−2πt) ∗ h
(x)

=

k∈Z

ˆ hk e−2πt|k| e2πikx . Theorem (“Poisson kernel method”): h ∈ Λα(T) if and only if ∀ η ≥ 0

∂

∂t η e−t√−∆ h

L∞ ≤ C tα−η .

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Advantages of the “Poisson kernel method”

log

∂

∂t η e−t√−∆ h

L∞ ≤ const + (α − η) log t

the number of values of t is not limited; all known Fourier coefficients taken into account in calculating each point; different η values → numerical tests.

SLIDE 48

Advantages of the “Poisson kernel method”

log

∂

∂t η e−t√−∆ h

L∞ ≤ const + (α − η) log t

the number of values of t is not limited; all known Fourier coefficients taken into account in calculating each point; different η values → numerical tests.

SLIDE 49

Advantages of the “Poisson kernel method”

log

∂

∂t η e−t√−∆ h

L∞ ≤ const + (α − η) log t

the number of values of t is not limited; all known Fourier coefficients taken into account in calculating each point; different η values → numerical tests.

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Numerical computation of CIC

Area Preserving Twist Maps (APTM) Let Xω be an orbit with rotation number ω then: Birkhoff: For any rational number ω ∈ [ρ1, ρ2] exists at least a pair of periodic orbits with rotation number ω. Aubry Mather: Let {ωi}∞

i=0, ωi ∈ Q, s.t.

lim

i→∞ ωi = ρ

then the limit set of {Xωi}∞

i=0 converges to an ICρ (or a

Cantorus)

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Numerical computation of CIC

Area Preserving Twist Maps (APTM) Let Xω be an orbit with rotation number ω then: Birkhoff: For any rational number ω ∈ [ρ1, ρ2] exists at least a pair of periodic orbits with rotation number ω. Aubry Mather: Let {ωi}∞

i=0, ωi ∈ Q, s.t.

lim

i→∞ ωi = ρ

then the limit set of {Xωi}∞

i=0 converges to an ICρ (or a

Cantorus)

SLIDE 52

Numerical computation of CIC

Area Preserving Twist Maps (APTM) Let Xω be an orbit with rotation number ω then: Birkhoff: For any rational number ω ∈ [ρ1, ρ2] exists at least a pair of periodic orbits with rotation number ω. Aubry Mather: Let {ωi}∞

i=0, ωi ∈ Q, s.t.

lim

i→∞ ωi = ρ

then the limit set of {Xωi}∞

i=0 converges to an ICρ (or a

Cantorus)

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Numerical computation of CIC

Area Preserving Twist Maps (APTM) Let Xω be an orbit with rotation number ω then: Birkhoff: For any rational number ω ∈ [ρ1, ρ2] exists at least a pair of periodic orbits with rotation number ω. Aubry Mather: Let {ωi}∞

i=0, ωi ∈ Q, s.t.

lim

i→∞ ωi = ρ

then the limit set of {Xωi}∞

i=0 converges to an ICρ (or a

Cantorus)

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Greene’s residues method

Greene criterion to determine CIC with rotation number ρ: Let Ri be the residue of an hyperbolic periodic orbits {Xωi}∞

i=0, such that

lim

i→∞ ωi = ρ

Xωi are the approximants of an ICρ If lim

i→∞ Ri → 0 then ∃

ICρ If lim

i→∞ Ri → −∞ then ∃

ICρ (Cantorus) If lim

i→∞ Ri → −0.25542 . . . then ICρ is critical

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Greene’s residues method

Greene criterion to determine CIC with rotation number ρ: Let Ri be the residue of an hyperbolic periodic orbits {Xωi}∞

i=0, such that

lim

i→∞ ωi = ρ

Xωi are the approximants of an ICρ If lim

i→∞ Ri → 0 then ∃

ICρ If lim

i→∞ Ri → −∞ then ∃

ICρ (Cantorus) If lim

i→∞ Ri → −0.25542 . . . then ICρ is critical

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Greene’s residues method

Greene criterion to determine CIC with rotation number ρ: Let Ri be the residue of an hyperbolic periodic orbits {Xωi}∞

i=0, such that

lim

i→∞ ωi = ρ

Xωi are the approximants of an ICρ If lim

i→∞ Ri → 0 then ∃

ICρ If lim

i→∞ Ri → −∞ then ∃

ICρ (Cantorus) If lim

i→∞ Ri → −0.25542 . . . then ICρ is critical

SLIDE 57

Greene’s residues method

Greene criterion to determine CIC with rotation number ρ: Let Ri be the residue of an hyperbolic periodic orbits {Xωi}∞

i=0, such that

lim

i→∞ ωi = ρ

Xωi are the approximants of an ICρ If lim

i→∞ Ri → 0 then ∃

ICρ If lim

i→∞ Ri → −∞ then ∃

ICρ (Cantorus) If lim

i→∞ Ri → −0.25542 . . . then ICρ is critical

SLIDE 58

Greene’s residues method

Greene criterion to determine CIC with rotation number ρ: Let Ri be the residue of an hyperbolic periodic orbits {Xωi}∞

i=0, such that

lim

i→∞ ωi = ρ

Xωi are the approximants of an ICρ If lim

i→∞ Ri → 0 then ∃

ICρ If lim

i→∞ Ri → −∞ then ∃

ICρ (Cantorus) If lim

i→∞ Ri → −0.25542 . . . then ICρ is critical

SLIDE 59

Greene’s residues method

Greene criterion to determine CIC with rotation number ρ: Let Ri be the residue of an hyperbolic periodic orbits {Xωi}∞

i=0, such that

lim

i→∞ ωi = ρ

Xωi are the approximants of an ICρ If lim

i→∞ Ri → 0 then ∃

ICρ If lim

i→∞ Ri → −∞ then ∃

ICρ (Cantorus) If lim

i→∞ Ri → −0.25542 . . . then ICρ is critical

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Numerical Experiments

We studied six APTM:    yn+1 = yn + λV (xn) xn+1 = xn + yn+1 Standard map: V (x) = sin(2πx) Two harmonics map: V (x) = sin(2πx) + 0.03 sin(6πx) Critical map: V (x) = sin(2πx) − 0.5 sin(4πx)

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Numerical Experiments

We studied six APTM:    yn+1 = yn + λV (xn) xn+1 = xn + yn+1 Standard map: V (x) = sin(2πx) Two harmonics map: V (x) = sin(2πx) + 0.03 sin(6πx) Critical map: V (x) = sin(2πx) − 0.5 sin(4πx)

SLIDE 62

Numerical Experiments

We studied six APTM:    yn+1 = yn + λV (xn) xn+1 = xn + yn+1 Standard map: V (x) = sin(2πx) Two harmonics map: V (x) = sin(2πx) + 0.03 sin(6πx) Critical map: V (x) = sin(2πx) − 0.5 sin(4πx)

SLIDE 63

Numerical Experiments

We studied six APTM:    yn+1 = yn + λV (xn) xn+1 = xn + yn+1 Standard map: V (x) = sin(2πx) Two harmonics map: V (x) = sin(2πx) + 0.03 sin(6πx) Critical map: V (x) = sin(2πx) − 0.5 sin(4πx)

SLIDE 64

Analytical map: V (x) = sin(2πx) 1 − β cos(2πx) β = 0.2, 0.4 Tent map: V (x) =

17

j=1

cj sin(2πjx) cj =      (−1)

j+1 2

4 π2j2

j odd j even

1.5
1
0.5

0.5 1 1.5 1 2 3 4 5 6

x V(x) V(x) x

SLIDE 65

Analytical map: V (x) = sin(2πx) 1 − β cos(2πx) β = 0.2, 0.4 Tent map: V (x) =

17

j=1

cj sin(2πjx) cj =      (−1)

j+1 2

4 π2j2

j odd j even

1.5
1
0.5

0.5 1 1.5 1 2 3 4 5 6

x V(x) V(x) x

SLIDE 66

Numerical results

Rotation number(CIC) = Golden mean Rotation number of the approximants ρ = 832040/1346269 CIC max error: 10−23, Residue max diff: 10−10 Fourier uniformly spaced grid → 220 points CPL algorithm test: η = 1, 2, 3, 4, 5

SLIDE 67

Numerical results

Rotation number(CIC) = Golden mean Rotation number of the approximants ρ = 832040/1346269 CIC max error: 10−23, Residue max diff: 10−10 Fourier uniformly spaced grid → 220 points CPL algorithm test: η = 1, 2, 3, 4, 5

SLIDE 68

Numerical results

Rotation number(CIC) = Golden mean Rotation number of the approximants ρ = 832040/1346269 CIC max error: 10−23, Residue max diff: 10−10 Fourier uniformly spaced grid → 220 points CPL algorithm test: η = 1, 2, 3, 4, 5

SLIDE 69

Numerical results

Rotation number(CIC) = Golden mean Rotation number of the approximants ρ = 832040/1346269 CIC max error: 10−23, Residue max diff: 10−10 Fourier uniformly spaced grid → 220 points CPL algorithm test: η = 1, 2, 3, 4, 5

SLIDE 70

Numerical results

Rotation number(CIC) = Golden mean Rotation number of the approximants ρ = 832040/1346269 CIC max error: 10−23, Residue max diff: 10−10 Fourier uniformly spaced grid → 220 points CPL algorithm test: η = 1, 2, 3, 4, 5

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CIC: R(θ)

Std → thin solid. 2Har → thick solid. CritMp → dotted. Ana2 → thin dashed. Ana4 → thick dashed. Tent → dotted dashed.

SLIDE 72

Advance map: g(θ)

Std → thin solid. 2Har → thick solid. CritMp → dotted. Ana2 → thin dashed. Ana4 → thick dashed. Tent → dotted dashed.

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Hull map: h(θ)

Std → thin solid. 2Har → thick solid. CritMp → dotted. Ana2 → thin dashed. Ana4 → thick dashed. Tent → dotted dashed.

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Inverse hull map: h−1(θ)

Std → thin solid. 2Har → thick solid. CritMp → dotted. Ana2 → thin dashed. Ana4 → thick dashed. Tent → dotted dashed.

SLIDE 75

Big conjugacies: H(θ)

SLIDE 76

Self similarity of h

SLIDE 77

Self similarity of h – Fourier spectrum

SLIDE 78

CLP analysis

log10

∂

∂t η e−t√−∆K

L∞(T)

versus log10(t)

SLIDE 79

H¨

lder regularities −

→ Numerical results

Map κ(R) κ(g) κ(h) κ(h−1) Standart 1.83 ± 0.09 1.83 ± 0.09 0.772 ± 0.001 0.92 ± 0.01 Two har- monics 1.79 ± 0.06 1.75 ± 0.09 0.721 ± 0.001 0.92 ± 0.01 Critical 1.83 ± 0.04 1.84 ± 0.09 0.724 ± 0.002 0.93 ± 0.02 Analytic 0.2 1.86 ± 0.08 1.86 ± 0.08 0.722 ± 0.001 0.92 ± 0.01 Analytic 0.4 1.85 ± 0.05 1.85 ± 0.05 0.724 ± 0.002 0.93 ± 0.01 Tent 1.85 ± 0.15 1.88 ± 0.12 0.726 ± 0.003 0.93 ± 0.02

SLIDE 80

H¨

lder regularities of ”Big” Conjugacies

We compute the regularities of all big conjugacies H between each of the six functions hi We have thirty functions H Applying CLP method: κ(H) = 1.80 ± 0.15

SLIDE 81

H¨

lder regularities of ”Big” Conjugacies

We compute the regularities of all big conjugacies H between each of the six functions hi We have thirty functions H Applying CLP method: κ(H) = 1.80 ± 0.15

SLIDE 82

H¨

lder regularities of ”Big” Conjugacies

We compute the regularities of all big conjugacies H between each of the six functions hi We have thirty functions H Applying CLP method: κ(H) = 1.80 ± 0.15

SLIDE 83

H¨

lder regularities for rotation number silver

mean

Silver mean = σS = [2, 2, 2, 2, . . . ] Maps: Standard and Two harmonics κ(RS) = 1.70 ± 0.15 κ(gS) = 1.75 ± 0.15 κ(hS) = 0.715 ± 0.015 κ(h−1

S )

= 0.87 ± 0.02 κ(HS) = 1.80 ± 0.15

SLIDE 84

H¨

lder regularities for rotation number silver

mean

Silver mean = σS = [2, 2, 2, 2, . . . ] Maps: Standard and Two harmonics κ(RS) = 1.70 ± 0.15 κ(gS) = 1.75 ± 0.15 κ(hS) = 0.715 ± 0.015 κ(h−1

S )

= 0.87 ± 0.02 κ(HS) = 1.80 ± 0.15

SLIDE 85

H¨

lder regularity and scaling factors

Shenker & Kadanoff (82):

Let θden ∈ T stand the value around which the iterates of the function G are most dense. Iteration of pden = (θden, R(θden)) are more dense around pden. Asymptotic invariant behaviour:

∆iθ := gFn+3(θden) − θden and ∆ir := R(gFn+3(θden)) − R(θden) where Fi = Fibonacci numbers

∆i+3θ ∆iθ ∼ α−1

3

∆i+3r ∆ir ∼ β−1

3

where α3 ∼ −4.84581 and β3 ∼ −16.8597

SLIDE 86

H¨

lder regularity and scaling factors

Shenker & Kadanoff (82):

Let θden ∈ T stand the value around which the iterates of the function G are most dense. Iteration of pden = (θden, R(θden)) are more dense around pden. Asymptotic invariant behaviour:

∆iθ := gFn+3(θden) − θden and ∆ir := R(gFn+3(θden)) − R(θden) where Fi = Fibonacci numbers

∆i+3θ ∆iθ ∼ α−1

3

∆i+3r ∆ir ∼ β−1

3

where α3 ∼ −4.84581 and β3 ∼ −16.8597

SLIDE 87

H¨

lder regularity and scaling factors

Shenker & Kadanoff (82):

Let θden ∈ T stand the value around which the iterates of the function G are most dense. Iteration of pden = (θden, R(θden)) are more dense around pden. Asymptotic invariant behaviour:

∆iθ := gFn+3(θden) − θden and ∆ir := R(gFn+3(θden)) − R(θden) where Fi = Fibonacci numbers

∆i+3θ ∆iθ ∼ α−1

3

∆i+3r ∆ir ∼ β−1

3

where α3 ∼ −4.84581 and β3 ∼ −16.8597

SLIDE 88

H¨

lder regularity and scaling factors

Shenker & Kadanoff (82):

Let θden ∈ T stand the value around which the iterates of the function G are most dense. Iteration of pden = (θden, R(θden)) are more dense around pden. Asymptotic invariant behaviour:

∆iθ := gFn+3(θden) − θden and ∆ir := R(gFn+3(θden)) − R(θden) where Fi = Fibonacci numbers

∆i+3θ ∆iθ ∼ α−1

3

∆i+3r ∆ir ∼ β−1

3

where α3 ∼ −4.84581 and β3 ∼ −16.8597

SLIDE 89

H¨

lder regularity and scaling factors

Shenker & Kadanoff (82):

Let θden ∈ T stand the value around which the iterates of the function G are most dense. Iteration of pden = (θden, R(θden)) are more dense around pden. Asymptotic invariant behaviour:

∆iθ := gFn+3(θden) − θden and ∆ir := R(gFn+3(θden)) − R(θden) where Fi = Fibonacci numbers

∆i+3θ ∆iθ ∼ α−1

3

∆i+3r ∆ir ∼ β−1

3

where α3 ∼ −4.84581 and β3 ∼ −16.8597

SLIDE 90

H¨

lder regularity and scaling factors

Shenker & Kadanoff (82):

Let θden ∈ T stand the value around which the iterates of the function G are most dense. Iteration of pden = (θden, R(θden)) are more dense around pden. Asymptotic invariant behaviour:

∆iθ := gFn+3(θden) − θden and ∆ir := R(gFn+3(θden)) − R(θden) where Fi = Fibonacci numbers

∆i+3θ ∆iθ ∼ α−1

3

∆i+3r ∆ir ∼ β−1

3

where α3 ∼ −4.84581 and β3 ∼ −16.8597

SLIDE 91

H¨

lder regularity and scaling factors

Shenker & Kadanoff (82):

Let θden ∈ T stand the value around which the iterates of the function G are most dense. Iteration of pden = (θden, R(θden)) are more dense around pden. Asymptotic invariant behaviour:

∆iθ := gFn+3(θden) − θden and ∆ir := R(gFn+3(θden)) − R(θden) where Fi = Fibonacci numbers

∆i+3θ ∆iθ ∼ α−1

3

∆i+3r ∆ir ∼ β−1

3

where α3 ∼ −4.84581 and β3 ∼ −16.8597

SLIDE 92

H¨

lder regularity and scaling factors

H¨

lder regularity of R −

→ |∆r| ∼ |∆θ|κ Asymptotical scaling: |β3∆r| ∼ |α3∆θ|κ k(R) ≤ log(β3)

log(α3) ∼ 1.7901

This bound is saturated.

SLIDE 93

H¨

lder regularity and scaling factors

H¨

lder regularity of R −

→ |∆r| ∼ |∆θ|κ Asymptotical scaling: |β3∆r| ∼ |α3∆θ|κ k(R) ≤ log(β3)

log(α3) ∼ 1.7901

This bound is saturated.

SLIDE 94

H¨

lder regularity and scaling factors

H¨

lder regularity of R −

→ |∆r| ∼ |∆θ|κ Asymptotical scaling: |β3∆r| ∼ |α3∆θ|κ k(R) ≤ log(β3)

log(α3) ∼ 1.7901

This bound is saturated.

SLIDE 95

H¨

lder regularity and scaling factors

H¨

lder regularity of R −

→ |∆r| ∼ |∆θ|κ Asymptotical scaling: |β3∆r| ∼ |α3∆θ|κ k(R) ≤ log(β3)

log(α3) ∼ 1.7901

This bound is saturated.

SLIDE 96

Conclusions

We accurately compute de golden critical invariant circles

f six twist maps

We obtain the H¨

lder regularity of R, g, h, h−1 and H

Our numerical experiments lend credibility to our Conjetures concerning the universality of the regularities of R, g, h, h−1 and H Our results seem to indicate that the regularities of R, h, h−1 saturate the upper bounds coming from previous studies of scaling exponents κ(H) is greater than κ(h) and κ(h−1) by a confortable margin

SLIDE 97

Conclusions

We accurately compute de golden critical invariant circles

f six twist maps

We obtain the H¨

lder regularity of R, g, h, h−1 and H

Our numerical experiments lend credibility to our Conjetures concerning the universality of the regularities of R, g, h, h−1 and H Our results seem to indicate that the regularities of R, h, h−1 saturate the upper bounds coming from previous studies of scaling exponents κ(H) is greater than κ(h) and κ(h−1) by a confortable margin

SLIDE 98

Conclusions

We accurately compute de golden critical invariant circles

f six twist maps

We obtain the H¨

lder regularity of R, g, h, h−1 and H

Our numerical experiments lend credibility to our Conjetures concerning the universality of the regularities of R, g, h, h−1 and H Our results seem to indicate that the regularities of R, h, h−1 saturate the upper bounds coming from previous studies of scaling exponents κ(H) is greater than κ(h) and κ(h−1) by a confortable margin

SLIDE 99

Conclusions

We accurately compute de golden critical invariant circles

f six twist maps

We obtain the H¨

lder regularity of R, g, h, h−1 and H

Our numerical experiments lend credibility to our Conjetures concerning the universality of the regularities of R, g, h, h−1 and H Our results seem to indicate that the regularities of R, h, h−1 saturate the upper bounds coming from previous studies of scaling exponents κ(H) is greater than κ(h) and κ(h−1) by a confortable margin

SLIDE 100

Conclusions

We accurately compute de golden critical invariant circles

f six twist maps

We obtain the H¨

lder regularity of R, g, h, h−1 and H

Our numerical experiments lend credibility to our Conjetures concerning the universality of the regularities of R, g, h, h−1 and H Our results seem to indicate that the regularities of R, h, h−1 saturate the upper bounds coming from previous studies of scaling exponents κ(H) is greater than κ(h) and κ(h−1) by a confortable margin

SLIDE 101