Discontinuity of physical measures

Abstract

Abstract In general, transformations from a space into itself have many ergodic invariant measures. A problem of great interest is to determine which ones are physically important. We consider a simple transformation τ of the unit interval for which we can identify two ergodic measures both of which have [0,1] as its support: the physical measure known as the SRB measure and a continuous measure which is singular with respect to the SRB measure. Computational results show that orbits of τ exhibit the physical measure, but certain small perturbations of τ result in orbits which exhibit the continuous measure that is singular with respect to the physical measure. This discontinuity in asymptotic statistical behavior has important consequences.