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We give a new proof of Moeckels result that for any finite index subgroup of the modular group, almost every real number has its regular continued fraction approximants equidistributed into the cusps of the subgroup according to the weighted cusp widths. Our proof uses a skew product over a cross-section for the geodesic flow on the modular surface. Our techniques show that the same result holds true for approximants found by Nakadas alpha-continued fractions, and also that the analogous result holds for approximants that are algebraic numbers given by any of Rosens lambda-continued fractions, related to the infinite family of Hecke triangle Fuchsian groups.
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We adjust Arnouxs coding, in terms of regular continued fractions, of the geodesic flow on the modular surface to give a cross section on which the return map is a double cover of the natural extension for the alpha-continued fractions, for each $alpha$ in (0,1]. The argument is sufficiently robust to apply to the Rosen continued fractions and their recently introduced alpha-variants.
This paper is devoted to searching for Riemannian metrics on 2-surfaces whose geodesic flows admit a rational in momenta first integral with a linear numerator and denominator. The explicit examples of metrics and such integrals are constructed. Few superintegrable systems are found having both a polynomial and a rational integrals which are functionally independent of the Hamiltonian.
The problem of the existence of an additional (independent on the energy) first integral, of a geodesic (or magnetic geodesic) flow, which is polynomial in momenta is studied. The relation of this problem to the existence of nontrivial solutions of stationary dispersionless limits of two-dimensional soliton equations is demonstrated. The nonexistence of an additional quadratic first integral is established for certain classes of magnetic geodesic flows.
Let $Q$ be a closed manifold admitting a locally-free action of a compact Lie group $G$. In this paper we study the properties of geodesic flows on $Q$ given by Riemannian metrics which are invariant by such an action. In particular, we will be interested in the existence of geodesics which are closed up to the action of some element in the group $G$, since they project to closed magnetic geodesics on the quotient orbifold $Q/G$.
Consider the geodesic flow on a real-analytic closed hypersurface $M$ of $mathbb{R}^n$, equipped with the standard Euclidean metric. The flow is entirely determined by the manifold and the Riemannian metric. Typically, geodesic flows are perturbed by varying the metric. In the present paper, however, only the Euclidean metric is used, and instead the manifold $M$ is perturbed. In this context, analogues of the following theorems are proved: the bumpy metric theorem; a theorem of Klingenberg and Takens regarding generic properties of $k$-jets of Poincare maps along geodesics; and the Kupka-Smale theorem. Moreover, the proofs presented here are valid in the real-analytic topology. Together, these results imply the following two main theorems: if $M$ is a real-analytic closed hypersurface in $mathbb{R}^n$ (with $n geq 3$) on which the geodesic flow with respect to the Euclidean metric has a nonhyperbolic periodic orbit, then $C^{omega}$-generically the geodesic flow on $M$ with respect to the Euclidean metric has a hyperbolic periodic orbit with a transverse homoclinic orbit; and there is a $C^{omega}$-open and dense set of real-analytic, closed, and strictly convex surfaces $M$ in $mathbb{R}^3$ on which the geodesic flow with respect to the Euclidean metric has a hyperbolic periodic orbit with a transverse homoclinic orbit. The methods used here also apply to the classical setting of perturbations of metrics on a Riemannian manifold to obtain real-analyt
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