| Abstract: |
| Diophantine approximation is to find rational numbers that approximate irrationals. It is related to the two kinds of dynamical systems, dynamics on the parameter space and dynamics on the phase space. The Gauss map and geodesic flows are parameter space dynamics. The Diophantine approximation exponent quantifies the rate at which the geodesic approaches the cusp of the fundamental domain of the modular group. It also gives the rate of the recurrence time of irrational rotations and translations on torus which are dynamical systems of the phase space.
In this talk we generalize classical results on the Diophantine approximation to the Hecke group H_q. When q=4, the Diophantine approximation on H_4 corresponds to the approximation on the unit circle and it is related with continued fraction algorithms to find best rational approximations of given parities. We also discuss the dynamical systems on the phase space associated with the Diophantine approximation on H_4. |
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