Abstract
Background:The properties of the groupPGL(2,C) on the Upper Poincar´e Half Plane have been analyzed.Objective:In particular, the classification of points and geodesics has been achieved by considering the solution to the free Hamiltonian associated problem.Methods:The free Hamiltonian associated problem implies to discard the symmetry sl(2,Z) for the definition of reduced geodesics. By means of the new definition and classification of reduced geodesics, new construction for tori, punctured tori, and the tessellation of the Upper Poincar´e Half Plane is found.Results:A definition of quadratic surds is proposed, for which the folding group corresponds to the tiling group, (also) for Hamiltonian systems on the Hyperbolic Plane (also realized as the Upper Poincar´e Half Plane (UPHP)).Conclusion:The initial conditions determine the result of the folding of the trajectories as tiling punctured tori and for tori.
Highlights
The analysis of the solutions of Hamiltonian systems on the Upper Poincare Half Plane (UPHP) allows to outline the differences between the possible tessellation groups of the system and the folding groups for · chaotic systems
For the congruence subgroups of PGL(2,C) defined on domains containing more than one point ui on the absolute at v = 0, rational geodesics are the degenerate geodesics with x± = ui, (i.e. u = const = ui), and the nondegenerate geodesics with x± = ui, the other endpoint parameterized such that the intersection with the group domain is not at v = 0, and those so transformed by the evolution of the equations of motion
Rational geodesics for the Hamiltonian system associated with the limiting process of tessellation of a torus are any geodesics with x+ = χ+ and x− on the potential sides; such a case is within the allowed initial conditions of the problem
Summary
The free Hamiltonian associated problem implies to discard the symmetry sl(2,Z) for the definition of reduced geodesics. By means of the new definition and classification of reduced geodesics, new construction for tori, punctured tori, and the tessellation of the Upper Poincare Half Plane is found
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