Anybody can help me to have an idea about an example showing the difference of a Physical measur with compare to an SRB measure?
By a Physical measure i mean in the sense of $\nu$ a hyperbolic(non-atomic) measure then having positive Lebesgue Basin of attraction.
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By an SRB measure i mean in the sense that, for any measurable partitions sub-ordinate to W^{u}-manifolds and up to an invariant hyperbolic measure(non-atomic), then almost everywhere, the conditional measures be Absolutely continuse with respect to induced Riemanian Lebesgue.
Thanks,
Pooh
Best Answer
Physical measure always means that the basin has positive Lebesgue measure.
SRB is simetimes synonymous to physical measure, and simetimes used to mean that it is hyperbolic and has a desintegration along unstable manifolds which is absolutely continuous wrt leaf volume. If you further suppose that such a measure be ergodic (either through definition or as an extra hypothesis), then it is a physical measure in the above sense. (You require $C^{1+\alpha}$ regularity for this to be true.) It was first written down in "Ergodic Attractors" by Pugh and Shub.
There is no other relation between the two. Physical measure do not have to be neither hyperbolic nor ergodic, but even if they are, they do not need to have an absolutely continuous disintegration along unstable leaves. Think of the time-1 map of the "8-attractor". This is a flow on a surface with a fixed hyperbolic point. The stable and unstable manifolds coincide, so that it looks like the number "8" when you draw it. Put to repelling fixed points in each of the loops of the number "8". The dynamics has a spiralling behaviour from the inside towards the edges of these loops. The Dirac measure at the fixed hyperbolic point is an ergodic hyperbolic measure, but not SRB in the sense you describe.