Multiplicities of Periodic Orbit Lengths for Non-Arithmetic Models
نویسندگان
چکیده
Multiplicities of periodic orbit lengths for non-arithmetic Hecke triangle groups are discussed. It is demonstrated both numerically and analytically that at least for certain groups the mean multiplicity of periodic orbits with exactly the same length increases exponentially with the length. The main ingredient used is the construction of joint distribution of periodic orbits when group matrices are transformed by field isomorphisms. The method can be generalized to other groups for which traces of group matrices are integers of an algebraic field of finite degree.
منابع مشابه
Prey-Predator System; Having Stable Periodic Orbit
The study of differential equations is useful in to analyze the possible past or future with help of present information. In this paper, the behavior of solutions has been analyzed around the equilibrium points for Gause model. Finally, some results are worked out to exist the stable periodic orbit for mentioned predator-prey system.
متن کاملSome Studies on Arithmetical Chaos in Classical and Quantum Mechanics
Several aspects of classical and quantum mechanics applied to a class of strongly chaotic systems are studied. The latter consists of single particles moving without external forces on surfaces of constant negative Gaussian curvature whose corresponding fundamental groups are supplied with an arithmetic structure. It is shown that the arithmetical features of the considered systems lead to exce...
متن کاملPeriodic Orbits in Arithmetical Chaos
1 Length spectra of periodic orbits are investigated for some chaotic dynamical systems whose quantum energy spectra show unexpected statistical properties and for which the notion of arithmetical chaos has been introduced recently. These systems are defined as the uncon-strained motions of particles on two dimensional surfaces of constant negative curvature whose fundamental groups are given b...
متن کاملThe Correlation between Multiplicities of Closed Geodesics on the Modular Surface
One of the connections between Number Theory and Mathematical Physics that emerged in recent years is Arithmetical Quantum Chaos. On the physical side we have the problem of how the notion of chaos in classical dynamical systems can be transfered to quantum mechanical systems. Here Gutzwiller’s trace formula is a useful quantitative tool which connects the lengths of closed orbits in the classi...
متن کاملGeodesics and commensurability classes of arithmetic hyperbolic 3-manifolds
This sharpens [10], where it was shown that the complex length spectrum of M determines its commensurability class. Suppose M ′ is an arithmetic hyperbolic 3-manifold which is not commensurable to M . Theorem 1.1 implies QL(M) 6= QL(M ′), though by Example 2.1 below it is possible that one of QL(M ′) or QL(M) contains the other. By the length formulas recalled in §2.1 and §2.2, each element of ...
متن کامل