Accelerating Cycle Expansions by Dynamical Conjugacy

Periodic orbit theory provides two important functions—the dynamical zeta function and the spectral determinant for the calculation of dynamical averages in a nonlinear system. Their cycle expansions converge rapidly when the system is uniformly hyperbolic but greatly slow down in the presence of non-hyperbolicity. We find that the slow convergence can be attributed to singularities in the natural measure. A properly designed coordinate transformation may remove these singularities and results in a dynamically conjugate system where fast convergence is restored. The technique is successfully demonstrated on several examples of one-dimensional maps and some remaining challenges are discussed.