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

Big Conjugacies Conjugation of two CIC, γ 1 and γ 2 : G γ 1 ,γ 2 = g γ 1 ◦ g − 1 γ 2 H γ 1 ,γ 2 = h γ 1 ◦ h − 1 γ 2

H¨ OLDER REGULARITY For κ = n + ξ with n ∈ Z and ξ ∈ (0 , 1): The function K : T → R has global H¨ older exponent κ ( K ∈ Λ κ ( T )) when K is n time differentiable and, for some constant C > 0: | D n K ( θ 1 ) − D n K ( θ 0 ) | ≤ C | θ 1 − θ 0 | ξ κ ( K ) := Is the H¨ older regularity of K

H¨ OLDER REGULARITY For κ = n + ξ with n ∈ Z and ξ ∈ (0 , 1): The function K : T → R has global H¨ older exponent κ ( K ∈ Λ κ ( T )) when K is n time differentiable and, for some constant C > 0: | D n K ( θ 1 ) − D n K ( θ 0 ) | ≤ C | θ 1 − θ 0 | ξ κ ( K ) := Is the H¨ older regularity of K

H¨ OLDER REGULARITY For κ = n + ξ with n ∈ Z and ξ ∈ (0 , 1): The function K : T → R has global H¨ older exponent κ ( K ∈ Λ κ ( T )) when K is n time differentiable and, for some constant C > 0: | D n K ( θ 1 ) − D n K ( θ 0 ) | ≤ C | θ 1 − θ 0 | ξ κ ( K ) := Is the H¨ older regularity of K

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 ⇒ Universal property renormalization operator 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 )

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 ⇒ Universal property renormalization operator 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 )

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 ⇒ Universal property renormalization operator 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 )

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 ⇒ Universal property renormalization operator 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 )

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 ⇒ Universal property renormalization operator 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 )

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 ⇒ Universal property renormalization operator 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 )

Poisson kernel method Poisson kernel (periodic case): s | k | e 2 πikx � P s ( x ) = k ∈ Z 1 − s 2 = 1 − 2 s cos 2 πx + s 2 , s ∈ [0 , 1) e − t √− ∆ h � � � � ( x ) = P exp( − 2 πt ) ∗ h ( x ) h k e − 2 πt | k | e 2 πikx . � ˆ = k ∈ Z Theorem (“Poisson kernel method”): h ∈ Λ α ( T ) if and only if ∀ η ≥ 0 � ∂ � η � e − t √− ∆ h � L ∞ ≤ C t α − η . � � � � ∂t � �

Poisson kernel method Poisson kernel (periodic case): s | k | e 2 πikx � P s ( x ) = k ∈ Z 1 − s 2 = 1 − 2 s cos 2 πx + s 2 , s ∈ [0 , 1) e − t √− ∆ h � � � � ( x ) = P exp( − 2 πt ) ∗ h ( x ) h k e − 2 πt | k | e 2 πikx . � ˆ = k ∈ Z Theorem (“Poisson kernel method”): h ∈ Λ α ( T ) if and only if ∀ η ≥ 0 � ∂ � η � e − t √− ∆ h � L ∞ ≤ C t α − η . � � � � ∂t � �

Poisson kernel method Poisson kernel (periodic case): s | k | e 2 πikx � P s ( x ) = k ∈ Z 1 − s 2 = 1 − 2 s cos 2 πx + s 2 , s ∈ [0 , 1) e − t √− ∆ h � � � � ( x ) = P exp( − 2 πt ) ∗ h ( x ) h k e − 2 πt | k | e 2 πikx . � ˆ = k ∈ Z Theorem (“Poisson kernel method”): h ∈ Λ α ( T ) if and only if ∀ η ≥ 0 � ∂ � η � e − t √− ∆ h � L ∞ ≤ C t α − η . � � � � ∂t � �

Advantages of the “Poisson kernel method” � ∂ � η � e − t √− ∆ h � � � log L ∞ ≤ const + ( α − η ) log t � � ∂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.

Advantages of the “Poisson kernel method” � ∂ � η � e − t √− ∆ h � � � log L ∞ ≤ const + ( α − η ) log t � � ∂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.

Advantages of the “Poisson kernel method” � ∂ � η � e − t √− ∆ h � � � log L ∞ ≤ const + ( α − η ) log t � � ∂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.

Numerical computation of CIC Area Preserving Twist Maps (APTM) Let X ω be an orbit with rotation number ω then: Birkhoff: For any rational number ω ∈ [ ρ 1 , ρ 2 ] exists at least a pair of periodic orbits with rotation number ω . Aubry Mather: Let { ω i } ∞ i =0 , ω i ∈ Q , s.t. i →∞ ω i = ρ lim then the limit set of { X ω i } ∞ i =0 converges to an IC ρ (or a Cantorus)

Numerical computation of CIC Area Preserving Twist Maps (APTM) Let X ω be an orbit with rotation number ω then: Birkhoff: For any rational number ω ∈ [ ρ 1 , ρ 2 ] exists at least a pair of periodic orbits with rotation number ω . Aubry Mather: Let { ω i } ∞ i =0 , ω i ∈ Q , s.t. i →∞ ω i = ρ lim then the limit set of { X ω i } ∞ i =0 converges to an IC ρ (or a Cantorus)

Numerical computation of CIC Area Preserving Twist Maps (APTM) Let X ω be an orbit with rotation number ω then: Birkhoff: For any rational number ω ∈ [ ρ 1 , ρ 2 ] exists at least a pair of periodic orbits with rotation number ω . Aubry Mather: Let { ω i } ∞ i =0 , ω i ∈ Q , s.t. i →∞ ω i = ρ lim then the limit set of { X ω i } ∞ i =0 converges to an IC ρ (or a Cantorus)

Numerical computation of CIC Area Preserving Twist Maps (APTM) Let X ω be an orbit with rotation number ω then: Birkhoff: For any rational number ω ∈ [ ρ 1 , ρ 2 ] exists at least a pair of periodic orbits with rotation number ω . Aubry Mather: Let { ω i } ∞ i =0 , ω i ∈ Q , s.t. i →∞ ω i = ρ lim then the limit set of { X ω i } ∞ i =0 converges to an IC ρ (or a Cantorus)

Greene’s residues method Greene criterion to determine CIC with rotation number ρ : Let R i be the residue of an hyperbolic periodic orbits { X ω i } ∞ i =0 , such that i →∞ ω i = ρ lim X ω i are the approximants of an IC ρ If lim i →∞ R i �→ 0 then ∃ IC ρ If lim i →∞ R i �→ −∞ then � ∃ IC ρ (Cantorus) If lim i →∞ R i �→ − 0 . 25542 . . . then IC ρ is critical

Greene’s residues method Greene criterion to determine CIC with rotation number ρ : Let R i be the residue of an hyperbolic periodic orbits { X ω i } ∞ i =0 , such that i →∞ ω i = ρ lim X ω i are the approximants of an IC ρ If lim i →∞ R i �→ 0 then ∃ IC ρ If lim i →∞ R i �→ −∞ then � ∃ IC ρ (Cantorus) If lim i →∞ R i �→ − 0 . 25542 . . . then IC ρ is critical

Greene’s residues method Greene criterion to determine CIC with rotation number ρ : Let R i be the residue of an hyperbolic periodic orbits { X ω i } ∞ i =0 , such that i →∞ ω i = ρ lim X ω i are the approximants of an IC ρ If lim i →∞ R i �→ 0 then ∃ IC ρ If lim i →∞ R i �→ −∞ then � ∃ IC ρ (Cantorus) If lim i →∞ R i �→ − 0 . 25542 . . . then IC ρ is critical

Greene’s residues method Greene criterion to determine CIC with rotation number ρ : Let R i be the residue of an hyperbolic periodic orbits { X ω i } ∞ i =0 , such that i →∞ ω i = ρ lim X ω i are the approximants of an IC ρ If lim i →∞ R i �→ 0 then ∃ IC ρ If lim i →∞ R i �→ −∞ then � ∃ IC ρ (Cantorus) If lim i →∞ R i �→ − 0 . 25542 . . . then IC ρ is critical

Greene’s residues method Greene criterion to determine CIC with rotation number ρ : Let R i be the residue of an hyperbolic periodic orbits { X ω i } ∞ i =0 , such that i →∞ ω i = ρ lim X ω i are the approximants of an IC ρ If lim i →∞ R i �→ 0 then ∃ IC ρ If lim i →∞ R i �→ −∞ then � ∃ IC ρ (Cantorus) If lim i →∞ R i �→ − 0 . 25542 . . . then IC ρ is critical

Greene’s residues method Greene criterion to determine CIC with rotation number ρ : Let R i be the residue of an hyperbolic periodic orbits { X ω i } ∞ i =0 , such that i →∞ ω i = ρ lim X ω i are the approximants of an IC ρ If lim i →∞ R i �→ 0 then ∃ IC ρ If lim i →∞ R i �→ −∞ then � ∃ IC ρ (Cantorus) If lim i →∞ R i �→ − 0 . 25542 . . . then IC ρ is critical

Numerical Experiments  y n +1 = y n + λV ( x n )  We studied six APTM: x n +1 = x n + y n +1  Standard map: V ( x ) = sin(2 πx ) Two harmonics map: V ( x ) = sin(2 πx ) + 0 . 03 sin(6 πx ) Critical map: V ( x ) = sin(2 πx ) − 0 . 5 sin(4 πx )

Numerical Experiments  y n +1 = y n + λV ( x n )  We studied six APTM: x n +1 = x n + y n +1  Standard map: V ( x ) = sin(2 πx ) Two harmonics map: V ( x ) = sin(2 πx ) + 0 . 03 sin(6 πx ) Critical map: V ( x ) = sin(2 πx ) − 0 . 5 sin(4 πx )

Numerical Experiments  y n +1 = y n + λV ( x n )  We studied six APTM: x n +1 = x n + y n +1  Standard map: V ( x ) = sin(2 πx ) Two harmonics map: V ( x ) = sin(2 πx ) + 0 . 03 sin(6 πx ) Critical map: V ( x ) = sin(2 πx ) − 0 . 5 sin(4 πx )

Numerical Experiments  y n +1 = y n + λV ( x n )  We studied six APTM: x n +1 = x n + y n +1  Standard map: V ( x ) = sin(2 πx ) Two harmonics map: V ( x ) = sin(2 πx ) + 0 . 03 sin(6 πx ) Critical map: V ( x ) = sin(2 πx ) − 0 . 5 sin(4 πx )

Analytical map: sin(2 πx ) V ( x ) = β = 0 . 2 , 0 . 4 1 − β cos(2 πx ) Tent map:  j +1 4 ( − 1) j odd 17 2  π 2 j 2  � V ( x ) = c j sin(2 πjx ) c j = j =1  0 j even  V(x) 1.5 V(x) 1 0.5 0 x x -0.5 -1 -1.5 0 1 2 3 4 5 6

Analytical map: sin(2 πx ) V ( x ) = β = 0 . 2 , 0 . 4 1 − β cos(2 πx ) Tent map:  j +1 4 ( − 1) j odd 17 2  π 2 j 2  � V ( x ) = c j sin(2 πjx ) c j = j =1  0 j even  V(x) 1.5 V(x) 1 0.5 0 x x -0.5 -1 -1.5 0 1 2 3 4 5 6

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 → 2 20 points CPL algorithm test: η = 1 , 2 , 3 , 4 , 5

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 → 2 20 points CPL algorithm test: η = 1 , 2 , 3 , 4 , 5

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 → 2 20 points CPL algorithm test: η = 1 , 2 , 3 , 4 , 5

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 → 2 20 points CPL algorithm test: η = 1 , 2 , 3 , 4 , 5

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 → 2 20 points CPL algorithm test: η = 1 , 2 , 3 , 4 , 5

CIC: R ( θ ) Std → thin solid. 2Har → thick solid. CritMp → dotted. Ana2 → thin dashed. Ana4 → thick dashed. Tent → dotted dashed.

Advance map: g ( θ ) Std → thin solid. 2Har → thick solid. CritMp → dotted. Ana2 → thin dashed. Ana4 → thick dashed. Tent → dotted dashed.

Hull map: h ( θ ) Std → thin solid. 2Har → thick solid. CritMp → dotted. Ana2 → thin dashed. Ana4 → thick dashed. Tent → dotted dashed.

Inverse hull map: h − 1 ( θ ) Std → thin solid. 2Har → thick solid. CritMp → dotted. Ana2 → thin dashed. Ana4 → thick dashed. Tent → dotted dashed.

Big conjugacies: H ( θ )

Self similarity of h

Self similarity of h – Fourier spectrum

CLP analysis � ∂ � η � � e − t √− ∆ K � � � � � � log 10 versus log 10 ( t ) � � � � ∂t � � � � L ∞ ( T )

H¨ older regularities − → Numerical results κ ( h − 1 ) Map κ ( R ) κ ( g ) κ ( h ) Standart 1 . 83 ± 0 . 09 1 . 83 ± 0 . 09 0 . 772 ± 0 . 001 0 . 92 ± 0 . 01 Two har- 1 . 79 ± 0 . 06 1 . 75 ± 0 . 09 0 . 721 ± 0 . 001 0 . 92 ± 0 . 01 monics Critical 1 . 83 ± 0 . 04 1 . 84 ± 0 . 09 0 . 724 ± 0 . 002 0 . 93 ± 0 . 02 Analytic 1 . 86 ± 0 . 08 1 . 86 ± 0 . 08 0 . 722 ± 0 . 001 0 . 92 ± 0 . 01 0.2 Analytic 1 . 85 ± 0 . 05 1 . 85 ± 0 . 05 0 . 724 ± 0 . 002 0 . 93 ± 0 . 01 0.4 Tent 1 . 85 ± 0 . 15 1 . 88 ± 0 . 12 0 . 726 ± 0 . 003 0 . 93 ± 0 . 02

H¨ older regularities of ”Big” Conjugacies We compute the regularities of all big conjugacies H between each of the six functions h i We have thirty functions H Applying CLP method: κ ( H ) = 1 . 80 ± 0 . 15

H¨ older regularities of ”Big” Conjugacies We compute the regularities of all big conjugacies H between each of the six functions h i We have thirty functions H Applying CLP method: κ ( H ) = 1 . 80 ± 0 . 15

H¨ older regularities of ”Big” Conjugacies We compute the regularities of all big conjugacies H between each of the six functions h i We have thirty functions H Applying CLP method: κ ( H ) = 1 . 80 ± 0 . 15

H¨ older regularities for rotation number silver mean Silver mean = σ S = [2 , 2 , 2 , 2 , . . . ] Maps: Standard and Two harmonics κ ( R S ) = 1 . 70 ± 0 . 15 κ ( g S ) = 1 . 75 ± 0 . 15 κ ( h S ) = 0 . 715 ± 0 . 015 κ ( h − 1 S ) = 0 . 87 ± 0 . 02 κ ( H S ) = 1 . 80 ± 0 . 15

H¨ older regularities for rotation number silver mean Silver mean = σ S = [2 , 2 , 2 , 2 , . . . ] Maps: Standard and Two harmonics κ ( R S ) = 1 . 70 ± 0 . 15 κ ( g S ) = 1 . 75 ± 0 . 15 κ ( h S ) = 0 . 715 ± 0 . 015 κ ( h − 1 S ) = 0 . 87 ± 0 . 02 κ ( H S ) = 1 . 80 ± 0 . 15

H¨ older regularity and scaling factors Shenker & Kadanoff (82): Let θ den ∈ T stand the value around which the iterates of the function G are most dense. Iteration of p den = ( θ den , R ( θ den )) are more dense around p den . Asymptotic invariant behaviour: ∆ i θ := g F n +3 ( θ den ) − θ den Fibonacci and where F i = numbers ∆ i r := R ( g F n +3 ( θ den )) − R ( θ den ) ∆ i +3 θ ∆ i +3 r ∼ α − 1 ∼ β − 1 3 3 ∆ i θ ∆ i r where α 3 ∼ − 4 . 84581 and β 3 ∼ − 16 . 8597

H¨ older regularity and scaling factors Shenker & Kadanoff (82): Let θ den ∈ T stand the value around which the iterates of the function G are most dense. Iteration of p den = ( θ den , R ( θ den )) are more dense around p den . Asymptotic invariant behaviour: ∆ i θ := g F n +3 ( θ den ) − θ den Fibonacci and where F i = numbers ∆ i r := R ( g F n +3 ( θ den )) − R ( θ den ) ∆ i +3 θ ∆ i +3 r ∼ α − 1 ∼ β − 1 3 3 ∆ i θ ∆ i r where α 3 ∼ − 4 . 84581 and β 3 ∼ − 16 . 8597

H¨ older regularity and scaling factors Shenker & Kadanoff (82): Let θ den ∈ T stand the value around which the iterates of the function G are most dense. Iteration of p den = ( θ den , R ( θ den )) are more dense around p den . Asymptotic invariant behaviour: ∆ i θ := g F n +3 ( θ den ) − θ den Fibonacci and where F i = numbers ∆ i r := R ( g F n +3 ( θ den )) − R ( θ den ) ∆ i +3 θ ∆ i +3 r ∼ α − 1 ∼ β − 1 3 3 ∆ i θ ∆ i r where α 3 ∼ − 4 . 84581 and β 3 ∼ − 16 . 8597

H¨ older regularity and scaling factors Shenker & Kadanoff (82): Let θ den ∈ T stand the value around which the iterates of the function G are most dense. Iteration of p den = ( θ den , R ( θ den )) are more dense around p den . Asymptotic invariant behaviour: ∆ i θ := g F n +3 ( θ den ) − θ den Fibonacci and where F i = numbers ∆ i r := R ( g F n +3 ( θ den )) − R ( θ den ) ∆ i +3 θ ∆ i +3 r ∼ α − 1 ∼ β − 1 3 3 ∆ i θ ∆ i r where α 3 ∼ − 4 . 84581 and β 3 ∼ − 16 . 8597

H¨ older regularity and scaling factors Shenker & Kadanoff (82): Let θ den ∈ T stand the value around which the iterates of the function G are most dense. Iteration of p den = ( θ den , R ( θ den )) are more dense around p den . Asymptotic invariant behaviour: ∆ i θ := g F n +3 ( θ den ) − θ den Fibonacci and where F i = numbers ∆ i r := R ( g F n +3 ( θ den )) − R ( θ den ) ∆ i +3 θ ∆ i +3 r ∼ α − 1 ∼ β − 1 3 3 ∆ i θ ∆ i r where α 3 ∼ − 4 . 84581 and β 3 ∼ − 16 . 8597

H¨ older regularity and scaling factors Shenker & Kadanoff (82): Let θ den ∈ T stand the value around which the iterates of the function G are most dense. Iteration of p den = ( θ den , R ( θ den )) are more dense around p den . Asymptotic invariant behaviour: ∆ i θ := g F n +3 ( θ den ) − θ den Fibonacci and where F i = numbers ∆ i r := R ( g F n +3 ( θ den )) − R ( θ den ) ∆ i +3 θ ∆ i +3 r ∼ α − 1 ∼ β − 1 3 3 ∆ i θ ∆ i r where α 3 ∼ − 4 . 84581 and β 3 ∼ − 16 . 8597

H¨ older regularity and scaling factors Shenker & Kadanoff (82): Let θ den ∈ T stand the value around which the iterates of the function G are most dense. Iteration of p den = ( θ den , R ( θ den )) are more dense around p den . Asymptotic invariant behaviour: ∆ i θ := g F n +3 ( θ den ) − θ den Fibonacci and where F i = numbers ∆ i r := R ( g F n +3 ( θ den )) − R ( θ den ) ∆ i +3 θ ∆ i +3 r ∼ α − 1 ∼ β − 1 3 3 ∆ i θ ∆ i r where α 3 ∼ − 4 . 84581 and β 3 ∼ − 16 . 8597

H¨ older regularity and scaling factors → | ∆ r | ∼ | ∆ θ | κ H¨ older regularity of R − Asymptotical scaling: | β 3 ∆ r | ∼ | α 3 ∆ θ | κ k ( R ) ≤ log( β 3 ) log( α 3 ) ∼ 1 . 7901 This bound is saturated.

H¨ older regularity and scaling factors → | ∆ r | ∼ | ∆ θ | κ H¨ older regularity of R − Asymptotical scaling: | β 3 ∆ r | ∼ | α 3 ∆ θ | κ k ( R ) ≤ log( β 3 ) log( α 3 ) ∼ 1 . 7901 This bound is saturated.

H¨ older regularity and scaling factors → | ∆ r | ∼ | ∆ θ | κ H¨ older regularity of R − Asymptotical scaling: | β 3 ∆ r | ∼ | α 3 ∆ θ | κ k ( R ) ≤ log( β 3 ) log( α 3 ) ∼ 1 . 7901 This bound is saturated.

H¨ older regularity and scaling factors → | ∆ r | ∼ | ∆ θ | κ H¨ older regularity of R − Asymptotical scaling: | β 3 ∆ r | ∼ | α 3 ∆ θ | κ k ( R ) ≤ log( β 3 ) log( α 3 ) ∼ 1 . 7901 This bound is saturated.

Conclusions We accurately compute de golden critical invariant circles of six twist maps older regularity of R , g , h , h − 1 and H We obtain the 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

Conclusions We accurately compute de golden critical invariant circles of six twist maps older regularity of R , g , h , h − 1 and H We obtain the 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

Conclusions We accurately compute de golden critical invariant circles of six twist maps older regularity of R , g , h , h − 1 and H We obtain the 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

Conclusions We accurately compute de golden critical invariant circles of six twist maps older regularity of R , g , h , h − 1 and H We obtain the 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

Conclusions We accurately compute de golden critical invariant circles of six twist maps older regularity of R , g , h , h − 1 and H We obtain the 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