Numerical estimate of infinite invariant densities: Application to Pesin-type identity

Weakly chaotic maps with unstable fixed points are investigated in the regime where the invariant density is non-normalizable. We propose that the infinite invariant density ρ̄(x) of these maps can be estimated using ρ̄ (x) = lim t→∞t^1-1ρ(x; t), in agreement with earlier work of Thaler. Here λ(x; t) is the normalized density of particles. This definition uniquely determines the infinite density and is a valuable tool for numerical estimations. We use this density to estimate the sub-exponential separation λ of nearby trajectories. For a particular map introduced by Thaler we use an analytical expression for the infinite invariant density to calculate λ exactly, which perfectly matches simulations without fitting. Misunderstanding which recently appeared in the literature is removed.