Grant-in-Aid for Scientific Research (B)
We have improved the proof of the existence of Lipschitz continuous invariant foliations for two dimensional dispersing billiards without eclipse and make it more constructive and more elementary. This may give a new progress in applications. We introduced the geometry of geodesies due to Busemann to study the billiard ball trajectories in their configuration spaces. In particular, we found out some relations between parallels and periodic trajectories. Moreover, we observed interesting phenomena of billiard systems in a closed curve given by |x/a|^r+|y/b|^r=1(r>1) with computer simulations. Its Poincare map gives a lot of information, by which those phenomena may be proved.In hyperbolic and intermittent maps regarded as toy models of classical periodic billiards, transports are found to be characterized by the spectra of the Frobenius-Perron operator. Furthermore, we researched the relationship between boundary elements method and scattering problems in quantum billiards.We studied the renormalization associated to indifferent periodic points of complex dynamics. We were able to define a new space of maps which is invariant under parabolic renormalization and its perturbations.We observed large deviations for countable to one piecewise invertible Markov systems. In particular, we showed the(level 2)upper large deviation bounds and exponential decreasing property under certain conditions. Moreover, we researched multifractal version of large deviation laws.Further, we have succeeded to prove the conjecture by Boyle and Maass. We can construct a one-parameter family of complex structures with critical point at the point the recurrence and the transience switch to the other. We showed that freedom of deformation to preserve recurrence at the point is relatively low. As related topics to Quantum Chaos and so on, we have been studying the theory of operator algebras itself and obtained several results in the subject.
Attribution of KAKENHI