Non-Uniform Invariant Measure For Some Hyperbolic Maps.

Abstract

The study of the circumstances that suggest that a dynamical system has at least one physical measure is a natural approach to the wider study of the presence (or lack thereof) of physical measures. The topic of what kind of measures can constitute physical measures is related to this approach. When T-invariant, ergodic and ν << m, this is the simplest example of physical measure. Then according to the concept of absolutely continuity, ν(Bν ) = 1 implies m(Bν ) > 0 (where Bν is the the basin of attraction of ν) is physical follows immediately from Birkhoff’s ergodic theorem in this situation. In this instance, the existence of physical measures is thus reduced to the following. Does T admit absolutely continuous invariant ergodic measure. The major theme we address in this thesis is determining a.c.i.p for the LSV Map...