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

M¨ obius number systems Alexandr Kazda, Petr K˚ urka M¨ obius trans- formations M¨ obius number systems Convergence M¨ obius number systems Alexandr Kazda, Petr K˚ urka Examples Existence Charles University, Prague theorem Conclusions Numeration Marseille, March 23–27, 2009

M¨ obius number Outline systems Alexandr Kazda, Petr K˚ urka 1 M¨ obius transformations M¨ obius trans- formations Convergence 2 Convergence M¨ obius number systems Examples 3 M¨ obius number systems Existence theorem 4 Examples Conclusions 5 Existence theorem 6 Conclusions

M¨ obius number systems Alexandr Kazda, Petr K˚ urka • Our goal: To use sequences of M¨ obius transformations to M¨ obius trans- represent points on R = R ∪ {∞} or the unit circle T . formations • A M¨ obius tranformation (MT) is any nonconstant function Convergence M : C ∪ {∞} → C ∪ {∞} of the form M¨ obius number systems M ( z ) = az + b Examples cz + d Existence theorem Conclusions • We will consider MTs that preserve the upper half-plane • or the unit disc D .

M¨ obius number systems Alexandr Kazda, Petr K˚ urka • Our goal: To use sequences of M¨ obius transformations to M¨ obius trans- represent points on R = R ∪ {∞} or the unit circle T . formations • A M¨ obius tranformation (MT) is any nonconstant function Convergence M : C ∪ {∞} → C ∪ {∞} of the form M¨ obius number systems M ( z ) = az + b Examples cz + d Existence theorem Conclusions • We will consider MTs that preserve the upper half-plane • or the unit disc D .

M¨ obius number systems Alexandr Kazda, Petr K˚ urka • Our goal: To use sequences of M¨ obius transformations to M¨ obius trans- represent points on R = R ∪ {∞} or the unit circle T . formations • A M¨ obius tranformation (MT) is any nonconstant function Convergence M : C ∪ {∞} → C ∪ {∞} of the form M¨ obius number systems M ( z ) = az + b Examples cz + d Existence theorem Conclusions • We will consider MTs that preserve the upper half-plane • or the unit disc D .

M¨ obius number systems Alexandr Kazda, Petr K˚ urka • Our goal: To use sequences of M¨ obius transformations to M¨ obius trans- represent points on R = R ∪ {∞} or the unit circle T . formations • A M¨ obius tranformation (MT) is any nonconstant function Convergence M : C ∪ {∞} → C ∪ {∞} of the form M¨ obius number systems M ( z ) = az + b Examples cz + d Existence theorem Conclusions • We will consider MTs that preserve the upper half-plane • or the unit disc D .

M¨ obius number R versus T systems Alexandr 8 -4 Kazda, Petr -3 4 3 K˚ urka -2 2 M¨ obius trans- formations Convergence 1 -1 M¨ obius -2 2 number systems Examples 1/2 -1/2 Existence 1/3 -1/3 theorem 1/4 -1/4 0 Conclusions • Using the stereometric projection, we have a one-to-one correspondence between the upper half-plane and unit disc. • This projection is itself an MT. • Therefore we can translate MTs that represent T to the ones that represent R .

M¨ obius number R versus T systems Alexandr 8 -4 Kazda, Petr -3 4 3 K˚ urka -2 2 M¨ obius trans- formations Convergence 1 -1 M¨ obius -2 2 number systems Examples 1/2 -1/2 Existence 1/3 -1/3 theorem 1/4 -1/4 0 Conclusions • Using the stereometric projection, we have a one-to-one correspondence between the upper half-plane and unit disc. • This projection is itself an MT. • Therefore we can translate MTs that represent T to the ones that represent R .

M¨ obius number R versus T systems Alexandr 8 -4 Kazda, Petr -3 4 3 K˚ urka -2 2 M¨ obius trans- formations Convergence 1 -1 M¨ obius -2 2 number systems Examples 1/2 -1/2 Existence 1/3 -1/3 theorem 1/4 -1/4 0 Conclusions • Using the stereometric projection, we have a one-to-one correspondence between the upper half-plane and unit disc. • This projection is itself an MT. • Therefore we can translate MTs that represent T to the ones that represent R .

M¨ obius number R versus T systems Alexandr 8 -4 Kazda, Petr -3 4 3 K˚ urka -2 2 M¨ obius trans- formations Convergence 1 -1 M¨ obius -2 2 number systems Examples 1/2 -1/2 Existence 1/3 -1/3 theorem 1/4 -1/4 0 Conclusions • Using the stereometric projection, we have a one-to-one correspondence between the upper half-plane and unit disc. • This projection is itself an MT. • Therefore we can translate MTs that represent T to the ones that represent R .

M¨ obius number R versus T systems Alexandr Kazda, Petr K˚ urka M¨ obius trans- formations Convergence M¨ obius • We will be mostly talking about representing the unit number systems circle. Examples • However, the example number systems represent R . Existence theorem • How to tell them apart: half-plane-preserving MTs have a Conclusions hat, disc-preserving MTs don’t.

M¨ obius number R versus T systems Alexandr Kazda, Petr K˚ urka M¨ obius trans- formations Convergence M¨ obius • We will be mostly talking about representing the unit number systems circle. Examples • However, the example number systems represent R . Existence theorem • How to tell them apart: half-plane-preserving MTs have a Conclusions hat, disc-preserving MTs don’t.

M¨ obius number R versus T systems Alexandr Kazda, Petr K˚ urka M¨ obius trans- formations Convergence M¨ obius • We will be mostly talking about representing the unit number systems circle. Examples • However, the example number systems represent R . Existence theorem • How to tell them apart: half-plane-preserving MTs have a Conclusions hat, disc-preserving MTs don’t.

M¨ obius number Disc M¨ obius transformations systems Alexandr M : D → D Kazda, Petr K˚ urka M¨ obius trans- formations Convergence • A direct calculation shows that all MTs that preserve D M¨ obius number must look like this: systems Examples • M ( z ) = α z + β Existence β z + α, theorem Conclusions • where | β | < | α | are complex numbers. • Examples follow.

M¨ obius number Disc M¨ obius transformations systems Alexandr M : D → D Kazda, Petr K˚ urka M¨ obius trans- formations Convergence • A direct calculation shows that all MTs that preserve D M¨ obius number must look like this: systems Examples • M ( z ) = α z + β Existence β z + α, theorem Conclusions • where | β | < | α | are complex numbers. • Examples follow.

M¨ obius number Disc M¨ obius transformations systems Alexandr M : D → D Kazda, Petr K˚ urka M¨ obius trans- formations Convergence • A direct calculation shows that all MTs that preserve D M¨ obius number must look like this: systems Examples • M ( z ) = α z + β Existence β z + α, theorem Conclusions • where | β | < | α | are complex numbers. • Examples follow.

M¨ obius number Disc M¨ obius transformations systems Alexandr M : D → D Kazda, Petr K˚ urka M¨ obius trans- formations Convergence • A direct calculation shows that all MTs that preserve D M¨ obius number must look like this: systems Examples • M ( z ) = α z + β Existence β z + α, theorem Conclusions • where | β | < | α | are complex numbers. • Examples follow.

M¨ obius number Examples of M¨ obius systems Alexandr transformations Kazda, Petr K˚ urka M¨ obius trans- formations 8 8 8 -4 -4 -4 -3 4 -3 4 -3 4 3 3 3 Convergence -2 -2 -2 2 2 2 M¨ obius number systems 1 1 1 -1 -1 -1 Examples Existence 1/2 1/2 1/2 -1/2 -1/2 -1/2 theorem -1/3 1/3 -1/3 1/3 -1/3 1/3 1/4 1/4 1/4 -1/4 -1/4 -1/4 Conclusions 0 0 0 M 1 ( z ) = (2 i +1) z +1 (7+2 i ) z + i M 0 ( z ) = 3 z − i M 2 ( z ) = iz − 3 2 i − 1 − iz +(7 − 2 i ) ˆ ˆ ˆ M 2 ( x ) = 4 x +1 M 0 ( x ) = x / 2 M 1 ( x ) = x + 1 3 − x hyperbolic parabolic elliptic

M¨ obius number Examples of M¨ obius systems Alexandr transformations Kazda, Petr K˚ urka M¨ obius trans- formations 8 8 8 -4 -4 -4 -3 4 -3 4 -3 4 3 3 3 Convergence -2 -2 -2 2 2 2 M¨ obius number systems 1 1 1 -1 -1 -1 Examples Existence 1/2 1/2 1/2 -1/2 -1/2 -1/2 theorem -1/3 1/3 -1/3 1/3 -1/3 1/3 1/4 1/4 1/4 -1/4 -1/4 -1/4 Conclusions 0 0 0 M 1 ( z ) = (2 i +1) z +1 (7+2 i ) z + i M 0 ( z ) = 3 z − i M 2 ( z ) = iz − 3 2 i − 1 − iz +(7 − 2 i ) ˆ ˆ ˆ M 2 ( x ) = 4 x +1 M 0 ( x ) = x / 2 M 1 ( x ) = x + 1 3 − x hyperbolic parabolic elliptic

M¨ obius number Examples of M¨ obius systems Alexandr transformations Kazda, Petr K˚ urka M¨ obius trans- formations 8 8 8 -4 -4 -4 -3 4 -3 4 -3 4 3 3 3 Convergence -2 -2 -2 2 2 2 M¨ obius number systems 1 1 1 -1 -1 -1 Examples Existence 1/2 1/2 1/2 -1/2 -1/2 -1/2 theorem -1/3 1/3 -1/3 1/3 -1/3 1/3 1/4 1/4 1/4 -1/4 -1/4 -1/4 Conclusions 0 0 0 M 1 ( z ) = (2 i +1) z +1 (7+2 i ) z + i M 0 ( z ) = 3 z − i M 2 ( z ) = iz − 3 2 i − 1 − iz +(7 − 2 i ) ˆ ˆ ˆ M 2 ( x ) = 4 x +1 M 0 ( x ) = x / 2 M 1 ( x ) = x + 1 3 − x hyperbolic parabolic elliptic