RESONANCES AND TRANSPORT IN THE SAWTOOTH MAP

We study transport in a completely chaotic Hamiltonian system, the hyperbolic sawtooth map. Analytical expressions are obtained for its cantori and resonances. We show that resonances give a complete partition of phase space. The flux leaking out of a resonance is given by its turnstiles, whose form and areas are obtained analytically. When the total flux out of a resonance becomes one third the area of an island, the topology of the turnstiles changes. At the same parameter value, a horseshoe is formed corresponding to the orbits trapped within the resonance. Based on this, a coding scheme for the trapped orbits is introduced and expressions for trapped ordered orbits are obtained. The partial flux transferred from one resonance to another is determined by the degree of overlap of their turnstiles. We calculate the survival probability within a resonance using the Markov model; the results are compared with results obtained numerically and from periodic-orbit theory. © 1990.