Matching geometric and expansion characteristics of wild chaotic attractors

Abstract

Wild chaotic attractors exhibit chaotic dynamics with a robustness property that cannot be destroyed with small perturbations. We consider a discrete-time system with the smallest possible dimension, namely, defined by a non-invertible map on the complex plane. For this map, wild chaos has been proven to exist in a small parameter region. Recently, it was conjectured to exist in a much larger region of parameter space, past a so-called backward critical tangency, at which a sequence of pre-images of a critical point converges to a saddle fixed point. Geometrically, a backward critical tangency leads to an abundance of homoclinic and heteroclinic tangencies between invariant manifolds of different dimensions, generating precisely what are believed to be the necessary ingredients for wild chaos. In this paper, we present corroborating evidence for this conjecture by computing Lyapunov exponents associated with the attractor. When the sum of the two (largest) Lyapunov exponents is positive, the dynamics is wild chaotic for this non-invertible map. We find that the zero-sum locus matches the locus of backward critical tangency, confirming its role as a boundary of existence of wild chaos.