Do zeta functions for intermittent maps have branch points?

TL;DR

This paper provides numerical evidence that the dynamical zeta function for intermittent maps with a neutral fixed point exhibits branch point singularities at z=1, and introduces a resummed cycle expansion method for efficient zero computation near the branch point.

Contribution

It demonstrates the presence of branch point singularities in the zeta function of intermittent maps and develops a novel resummed cycle expansion technique for analyzing zeros near these singularities.

Findings

Zeta function has branch point at z=1 for intermittent maps.

Power series expansion computed up to order 20, involving 10^5 periodic orbits.

Resummed cycle expansion efficiently computes zeros near the branch point.

Abstract

We present numerical evidence that the dynamical zeta function and the Fredholm determinant of intermittent maps with a neutral fix point have branch point singularities at z=1 We consider the power series expansion of zeta function and the Fredholm determinant around z=0 with the fix point pruned. This power series is computed up to order 20, requiring 10^5periodic orbits. We also discuss the relation between correlation decay and the nature of the branch point. We conclude by demonstrating how zeros of zeta functions with thermodynamic weights that are close to the branch point can be efficiently computed by a resummed cycle expansion. The idea is quite similar to that of Pade\'{e} approximants, but the ansatz is a generalized series expansion around the branch point instead of a rational function.