Hyperbolic surfaces, cutting sequences, and continued fractions - PowerPoint PPT Presentation

Hyperbolic surfaces, cutting sequences, and continued fractions Claire Merriman October 21, 2019 The Ohio State University merriman.72@osu.edu

OCF Animation First frame of the animiation.

Regular Continued Fractions Way to represent x > 0 as 1 • x = a 0 + 1 a 1 + a 2 + . . .

Regular Continued Fractions Way to represent x > 0 as 1 • x = a 0 + 1 a 1 + a 2 + . . . 1 • π = 3 + 1 7 + 1 15 + 1 1 + 292 + . . .

Dynamics Define T : [0 , 1] → [0 , 1] by   � � � � 1 1 1 k +1 , 1 1 x − if x � = 0 x − k for x ∈   x k T ( x ) = = . 0 if x = 0 0 if x = 0   1 1 �→ 1 1 a 1 + a 2 + 1 1 a 2 + a 3 + a 3 + . . . a 4 + . . .

Gauss map 1 ... 1 1 1 1 1 1 1 1 1 9 8 7 6 5 4 3 2

Natural Extension T : [0 , 1) 2 → [0 , 1) 2 by Define ¯  � � � � 1 1 k +1 , 1 1 x − k , for x ∈  ¯ y + k k T ( x , y ) = . (0 , y ) if x = 0 

Natural Extension T : [0 , 1) 2 → [0 , 1) 2 by Define ¯  � � � � 1 1 k +1 , 1 1 x − k , for x ∈  ¯ y + k k T ( x , y ) = . (0 , y ) if x = 0      1 1 1 1 , ,     1 1 1 1  �→      a 0 + a − 1 +   a 1 + a 0 +   a 1 + . . . a 2 + . . .  a 2 + . . . a 1 + . . . 

Natural extension domain Plot of ¯ T n ( x , 0) for 1500 values of x , 1 ≤ n ≤ 200

Nakada α -continued fractions Nakada (1981) introduced the α -continued fractions. Define T α on [ α − 1 , α ]: � 1 T α ( x ) = 1 � | x | − | x | + 1 − α = ǫ � 1 1 � x − a 1 for ǫ x ∈ a 1 − 1 + α, a 1 + α ǫ 1 x = . ǫ 2 a 1 + a 2 + . . .

Nakada α -continued fractions Nakada (1981) introduced the α -continued fractions. Define T α on [ α − 1 , α ]: � 1 T α ( x ) = 1 � | x | − | x | + 1 − α = ǫ � 1 1 � x − a 1 for ǫ x ∈ a 1 − 1 + α, a 1 + α ǫ 1 1 When α = 1 x = . 2 , π = 3 + ǫ 2 1 a 1 + 7 + a 2 + . . . 1 16 − 293 + . . .

Gauss map ... ... ( 2, - 1 ) ( 3, - 1 ) ( 3, + 1 ) ( 2, + 1 ) ( 1, + 1 ) ' ' ' ' ' ' - 1 1 1 1 α - 1 α α + 3 α + 3 α + 2 α + 1

Natural extension Natural extension defined on [ α − 1 , α ) × R α . � ǫ 1 � � 1 1 � ( x , y ) �→ x − a 1 , for ǫ x ∈ a 1 − 1 + α, a 1 + ǫ y a 1 + α     ǫ 0 1 ǫ 1 1 , ,     ǫ 1 ǫ − 1 ǫ 2 ǫ 0  �→      a 0 + a − 1 +   a 1 + a 0 +   a 1 + . . . a 2 + . . .  a 3 + . . . a 1 + . . . 

RCF Animation Frame of the animation of the natural extension domain where α = . 5.

α -odd continued fractions Boca-M (2019) introduced the α -odd continued fractions. Define ϕ α on [ α − 2 , α ] ϕ α ( x ) = ǫ � 1 1 � x − 2 a 1 + 1 for ǫ x ∈ 2 a 1 + 1 + α, 2 a 1 − 1 + α

α -odd continued fractions ... ... ( 1, - 1 ) ( 3, - 1 ) ( 3, + 1 ) ( 1, + 1 ) ' ' ' ' - 1 1 1 α - 2 α α + 3 α + 3 α + 1

OCF Animation First frame of the animation of the natural extension domain with √ α = 1+ 5 .

Two stills from the animationof the natural extension domain,

Farey Tessellation H := { x + iy | y > 0 } q ′ iff pq ′ − p ′ q = ± 1. Connect two rational numbers p q , p ′ - 5 / 3 - 3 / 2 - 4 / 3 - 1 - 2 / 3 - 1 / 2 - 1 / 3 0 1 / 3 1 / 2 2 / 3 1 4 / 3 3 / 2 5 / 3 2

Geodesics Let S be the set of geodesics γ with endpoints • γ −∞ ∈ ( − 1 , 0) , γ ∞ ≥ 1 • γ −∞ ∈ (0 , 1) , γ ∞ ≤ − 1 η γ ξ γ - 1 - 2 / 3 - 1 / 2 γ - 0 1 / 3 1 / 2 2 / 3 1 4 / 3 3 / 2 5 / 3 2 7 / 3 5 / 2 γ + 3

Some segments of type L Some segments of type R

Example L L η γ ξ γ R R R L - 1 - 2 / 3 - 1 / 2 γ - 0 1 / 3 1 / 2 2 / 3 1 4 / 3 3 / 2 5 / 3 2 7 / 3 5 / 2 γ + 3 Cutting sequence . . . RR ξ γ L 2 R 1 L 3 . . .

Theorem (Series, ’85) A geodesic from γ −∞ to γ ∞ has two options: • γ −∞ ∈ ( − 1 , 0) , γ ∞ ∈ (1 , ∞ ) . This geodesic has the coding . . . L n − 2 R n − 1 ξ γ L n 0 R n 1 L n 2 . . . γ −∞ = − [ n − 1 , n − 2 , . . . ] and γ ∞ = n 0 + [ n 1 , n 2 , . . . ] • γ −∞ ∈ (0 , 1) , γ ∞ ∈ ( −∞ , − 1) . This geodesic has the coding . . . L n − 2 L n − 1 ξ γ R n 0 L n 1 R n 2 . . . � � γ −∞ = [ n − 1 , n − 2 , . . . ] and γ ∞ = − n 0 + [ n 1 , n 2 , . . . ] .

Action on Upper Half Plane Case 1, γ ∞ > 1. 1 1 Define ρ on S by ( x , y ) �→ ( a 1 − x , a 1 − y ). ξ ρ ( γ ) R η ρ ( γ ) L L L L R - 2 - 5 / 3 - 3 / 2 ρ ( γ + ) - 1 - 2 / 3 - 1 / 2 0 1 / 3 ρ ( γ - ) 1 / 2 2 / 3 1 . . . L 1 R 2 ξ γ L 2 η γ R 1 L 3 · · · �→ L 1 R 2 L 2 ξ ρ ( γ ) R 1 η ρ ( γ ) L 3 . . .

Lehner expansions Lehner (1994) defined continued fractions x ∈ [1 , 2] e o x = a 0 + e 1 a 1 + a 2 + . . . ( a i , e i ) = (1 , +1) , (2 , − 1).

Lehner expansions Lehner (1994) defined continued fractions x ∈ [1 , 2] e o x = a 0 + e 1 a 1 + a 2 + . . . ( a i , e i ) = (1 , +1) , (2 , − 1). Define L : [1 , 2] → [1 , 2] by  � � 1 1 , 3 if x ∈   2 − x 2 L ( x ) = � � 1 3 if x ∈ 2 , 2  x − 1 

Tent map 2 ( 2, - 1 ) ( 1, + 1 ) 3 2 2

Dajani and Kraaikamp (2000) introduced the Farey expansions for y ∈ [ − 1 , ∞ ) f 0 y = = � � ( f 0 / b 0 )( f 1 / b 1 )( f 2 / b 2 ) . . . � � f 1 b 0 + b 1 + . . . ( f i / b i ) = (+1 / 1) , ( − 1 / 2). � (1 / 1)( − 1 / 2) 3 (1 / 1)( − 1 / 2) 6 (1 / 1)( − 1 / 2) 14 . . . � π = � �

Natural extension L : [1 , 2) × [ − 1 , ∞ ) → [1 , 2) × [ − 1 , ∞ ) by  � � � � x − 2 , − 1 − 1 1 , 3 x ∈  � � e 0 e 0  y +2 2 , = � � � � x − a 0 y + a 0 1 1 3 x − 1 , x ∈ 2 , 2 .  y +1      ǫ 1 1 ǫ 2 1 a 0 + , a 1 + a 3 + . . .,     ǫ 2 ǫ − 1 ǫ 0  �→      a 1 + a − 1 +   a 0 +   a 2 + . . . a 2 + . . .  a 1 + . . . 

Geodesics Connect backwards endpoint γ −∞ to forward endpoint γ ∞ with γ Either • γ −∞ < 1 , 1 < γ ∞ < 2 • γ −∞ − 1 , − 2 < γ ∞ < − 1 L L R R ξ γ  L L η γ  - 2 γ - ∞ - 1 0 1 3 / 2 γ ∞ 2

Example L L R R ξ γ  L L η γ  - 2 γ - ∞ - 1 0 1 3 / 2 γ ∞ 2 Cutting sequence . . . LRL 2 R ξ γ L η γ R . . .

Converting to Lehner and Farey expansions Read Lehner expansion of γ ∞ starting at ξ γ . Farey expansion of γ −∞ from right to left starting at ξ γ . If the letter is the same as the previous (letter to the left), the digit is (2 , − 1), if it is different than the previous letter, the digit is (1 , +1).

Example L L R R ξ γ  L L η γ  - 2 γ - ∞ - 1 0 1 3 / 2 γ ∞ 2 Cutting sequence . . . RRL 2 R ξ γ L η γ R . . . R ξ γ LR . . . ❀ [ [(1 , +1) , (1 , +1) , . . . ] ] . . . LRLLR ξ γ ❀ � � (+1 / 1)( − 1 / 2)(+1 / 1)( − 1 / 2) . . . � �

Action on Upper Half Plane Case 1, 1 < γ ∞ < 2. � � 1 1 Define ρ on ± ((1 , 2) × ( −∞ , 1)) by ( x , y ) �→ a 1 − x , . a 1 − y L L R R ξ γ ρ ( η γ )= ξ ρ ( γ )   L η γ  - 2 γ - ∞ ρ ( γ ∞ ) - 1 0 ρ ( γ - ∞ ) 1 3 / 2 γ ∞ 2 . . . LRL 2 R ξ γ L η γ R · · · �→ . . . LRL 2 RL ξ ρ ( γ ) R η ρ ( γ ) . . .

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