M obius number systems M obius trans- formations Convergence - PowerPoint PPT Presentation

M¨ obius number systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨ obius number systems M¨ obius trans- formations Convergence Alexandr Kazda M¨ obius number (with thanks to Petr K˚ urka) systems Examples Charles University, Prague Subshifts admitting a number NSAC system Novi Sad Conclusions August 17–21, 2009

M¨ obius number Outline systems Alexandr Kazda (with thanks to Petr K˚ urka) 1 M¨ obius transformations M¨ obius trans- formations Convergence 2 Convergence M¨ obius number systems 3 M¨ obius number systems Examples Subshifts admitting a 4 Examples number system Conclusions 5 Subshifts admitting a number system 6 Conclusions

M¨ obius number systems Alexandr Kazda • Our goal: To use sequences of M¨ (with thanks obius transformations to to Petr K˚ urka) represent points on R = R ∪ {∞} . M¨ obius trans- • A M¨ obius transformation (MT) is any nonconstant formations function M : C ∪ {∞} → C ∪ {∞} of the form Convergence M¨ obius number M ( z ) = az + b systems cz + d Examples Subshifts admitting a number • We will consider MTs that preserve the upper half-plane. system • These are precisely the MTs with a , b , c , d real and Conclusions ad − bc = 1.

M¨ obius number systems Alexandr Kazda • Our goal: To use sequences of M¨ (with thanks obius transformations to to Petr K˚ urka) represent points on R = R ∪ {∞} . M¨ obius trans- • A M¨ obius transformation (MT) is any nonconstant formations function M : C ∪ {∞} → C ∪ {∞} of the form Convergence M¨ obius number M ( z ) = az + b systems cz + d Examples Subshifts admitting a number • We will consider MTs that preserve the upper half-plane. system • These are precisely the MTs with a , b , c , d real and Conclusions ad − bc = 1.

M¨ obius number systems Alexandr Kazda • Our goal: To use sequences of M¨ (with thanks obius transformations to to Petr K˚ urka) represent points on R = R ∪ {∞} . M¨ obius trans- • A M¨ obius transformation (MT) is any nonconstant formations function M : C ∪ {∞} → C ∪ {∞} of the form Convergence M¨ obius number M ( z ) = az + b systems cz + d Examples Subshifts admitting a number • We will consider MTs that preserve the upper half-plane. system • These are precisely the MTs with a , b , c , d real and Conclusions ad − bc = 1.

M¨ obius number systems Alexandr Kazda • Our goal: To use sequences of M¨ (with thanks obius transformations to to Petr K˚ urka) represent points on R = R ∪ {∞} . M¨ obius trans- • A M¨ obius transformation (MT) is any nonconstant formations function M : C ∪ {∞} → C ∪ {∞} of the form Convergence M¨ obius number M ( z ) = az + b systems cz + d Examples Subshifts admitting a number • We will consider MTs that preserve the upper half-plane. system • These are precisely the MTs with a , b , c , d real and Conclusions ad − bc = 1.

M¨ obius number Classifying M¨ obius transformations systems Alexandr Kazda (with thanks to Petr K˚ urka) M 0 ( x ) = x / 2 M¨ obius trans- formations Convergence • Hyperbolic, two fixed points. M¨ obius number systems M 1 ( x ) = x + 1 Examples Subshifts admitting a number system • Parabolic, one fixed point. Conclusions 1 M 2 ( x ) = − x + 1 • Elliptic, no fixed points in R .

M¨ obius number Classifying M¨ obius transformations systems Alexandr Kazda (with thanks to Petr K˚ urka) M 0 ( x ) = x / 2 M¨ obius trans- formations Convergence • Hyperbolic, two fixed points. M¨ obius number systems M 1 ( x ) = x + 1 Examples Subshifts admitting a number system • Parabolic, one fixed point. Conclusions 1 M 2 ( x ) = − x + 1 • Elliptic, no fixed points in R .

M¨ obius number Classifying M¨ obius transformations systems Alexandr Kazda (with thanks to Petr K˚ urka) M 0 ( x ) = x / 2 M¨ obius trans- formations Convergence • Hyperbolic, two fixed points. M¨ obius number systems M 1 ( x ) = x + 1 Examples Subshifts admitting a number system • Parabolic, one fixed point. Conclusions 1 M 2 ( x ) = − x + 1 • Elliptic, no fixed points in R .

M¨ obius number Classifying M¨ obius transformations systems Alexandr Kazda (with thanks to Petr K˚ urka) M 0 ( x ) = x / 2 M¨ obius trans- formations Convergence • Hyperbolic, two fixed points. M¨ obius number systems M 1 ( x ) = x + 1 Examples Subshifts admitting a number system • Parabolic, one fixed point. Conclusions 1 M 2 ( x ) = − x + 1 • Elliptic, no fixed points in R .

M¨ obius number Classifying M¨ obius transformations systems Alexandr Kazda (with thanks to Petr K˚ urka) M 0 ( x ) = x / 2 M¨ obius trans- formations Convergence • Hyperbolic, two fixed points. M¨ obius number systems M 1 ( x ) = x + 1 Examples Subshifts admitting a number system • Parabolic, one fixed point. Conclusions 1 M 2 ( x ) = − x + 1 • Elliptic, no fixed points in R .

M¨ obius number Classifying M¨ obius transformations systems Alexandr Kazda (with thanks to Petr K˚ urka) M 0 ( x ) = x / 2 M¨ obius trans- formations Convergence • Hyperbolic, two fixed points. M¨ obius number systems M 1 ( x ) = x + 1 Examples Subshifts admitting a number system • Parabolic, one fixed point. Conclusions 1 M 2 ( x ) = − x + 1 • Elliptic, no fixed points in R .

M¨ obius number Classifying M¨ obius transformations systems Alexandr Kazda (with thanks to Petr K˚ urka) M 0 ( x ) = x / 2 M¨ obius trans- formations Convergence • Hyperbolic, two fixed points. M¨ obius number systems M 1 ( x ) = x + 1 Examples Subshifts admitting a number system • Parabolic, one fixed point. Conclusions 1 M 2 ( x ) = − x + 1 • Elliptic, no fixed points in R .

M¨ obius number Classifying M¨ obius transformations systems Alexandr Kazda (with thanks to Petr K˚ urka) M 0 ( x ) = x / 2 M¨ obius trans- formations Convergence • Hyperbolic, two fixed points. M¨ obius number systems M 1 ( x ) = x + 1 Examples Subshifts admitting a number system • Parabolic, one fixed point. Conclusions 1 M 2 ( x ) = − x + 1 • Elliptic, no fixed points in R .

M¨ obius number Classifying M¨ obius transformations systems Alexandr Kazda (with thanks to Petr K˚ urka) M 0 ( x ) = x / 2 M¨ obius trans- formations Convergence • Hyperbolic, two fixed points. M¨ obius number systems M 1 ( x ) = x + 1 Examples Subshifts admitting a number system • Parabolic, one fixed point. Conclusions 1 M 2 ( x ) = − x + 1 • Elliptic, no fixed points in R .

M¨ obius number Defining convergence systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨ obius trans- formations Convergence • A sequence M 1 , M 2 , . . . represents the number x if M¨ obius M n ( i ) → x for n → ∞ . number systems • Isn’t it a bit arbitrary? Examples • No. This definition is quite natural. Subshifts admitting a • For example, if M 1 , M 2 , . . . represents x then number system M n ( K ) → { x } for any K compact lying above the real line. Conclusions

M¨ obius number Defining convergence systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨ obius trans- formations Convergence • A sequence M 1 , M 2 , . . . represents the number x if M¨ obius M n ( i ) → x for n → ∞ . number systems • Isn’t it a bit arbitrary? Examples • No. This definition is quite natural. Subshifts admitting a • For example, if M 1 , M 2 , . . . represents x then number system M n ( K ) → { x } for any K compact lying above the real line. Conclusions

M¨ obius number Defining convergence systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨ obius trans- formations Convergence • A sequence M 1 , M 2 , . . . represents the number x if M¨ obius M n ( i ) → x for n → ∞ . number systems • Isn’t it a bit arbitrary? Examples • No. This definition is quite natural. Subshifts admitting a • For example, if M 1 , M 2 , . . . represents x then number system M n ( K ) → { x } for any K compact lying above the real line. Conclusions

M¨ obius number Defining convergence systems Alexandr Kazda (with thanks to Petr K˚ urka) M¨ obius trans- formations Convergence • A sequence M 1 , M 2 , . . . represents the number x if M¨ obius M n ( i ) → x for n → ∞ . number systems • Isn’t it a bit arbitrary? Examples • No. This definition is quite natural. Subshifts admitting a • For example, if M 1 , M 2 , . . . represents x then number system M n ( K ) → { x } for any K compact lying above the real line. Conclusions

M¨ obius number Preliminaries from Symbolic systems Alexandr dynamics Kazda (with thanks to Petr K˚ urka) M¨ obius trans- formations • Let A be finite alphabet. Let A ⋆ denote the set of all finite Convergence words over A , A ω the set of all one-sided infinite words. M¨ obius number • A ⋆ with the operation of concatenation is a monoid. systems Examples • Let w i denote the i -th letter of the word w . Subshifts admitting a • A set Σ ⊂ A ω is a subshift if Σ can be defined by a set of number system forbidden (finite) factors. Conclusions • For v = v 1 . . . v n a word, denote by F v the transformation F v 1 ◦ · · · ◦ F v n .