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 (rho) over bar (x) of these maps can be estimated using (rho) over bar (x)=lim(t ->infinity)t(1-alpha) rho(x, t), in agreement with earlier work of Thaler. Here rho(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 lambda(alpha) of nearby trajectories. For a particular map introduced by Thaler we use an analytical expression for the infinite invariant density to calculate lambda(alpha) exactly, which perfectly matches simulations without fitting. Misunderstanding which recently appeared in the literature is removed.