SLYRB measures: natural invariant measures for chaotic systems

In many applications it is useful to consider not only the set that constitutes an attractor but also (if it exists) the asymptotic distribution of a typical trajectory converging to the attractor. Indeed, in the physics literature such a distribution is often assumed to exist. When it exists, it is called a “natural invariant measure”. The results by Lasota and Yorke, and by Sinai, Ruelle and Bowen represent two approaches both of which establish the existence of an invariant measure. The goal of this paper is to relate the “Lasota–Yorke measure” for chaotic attractors in one-dimensional maps and the “Sinai–Ruelle–Bowen measure” for chaotic attractors in higher-dimensional dynamical systems. We introduce the notion of “SLYRB measure”. (We pronounce the term “SLYRB” as a single word “slurb”.) The SRB concept of measure can be motivated by asking how a trajectory from a typical initial point is distributed asymptotically. Similarly the SLYRB concept of measure can be motivated by asking what the average distribution is for trajectories of a large collection of initial points in some region not necessarily restricted to a single basin. The latter is analogous to ask where all the rain drops from a rain storm go and the former asks about where a single rain drop goes, perhaps winding up distributed throughout a particular lake.