# Physical Measure Vs. SRB measures

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. & 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

• In my opinion, $m(B_f(\mu))>0$ (the basin has positive Lebesgue measure) is the only requirement for a physical measure. They are the same if the physical measure is required to be hyperbolic. – Pengfei Mar 8 '14 at 16:17
• Thanks, and if i do not have hyperbolic measures, then what will be the example? – Po0oh Mar 8 '14 at 16:42
• Actually i was in doubt if the difference between them means: if they are the same then m(B_{f}(\mu))=1, not only bigger than zero. – Po0oh Mar 8 '14 at 17:28
• Consider a monotone map $f:[0,1]\to [0,1]$ such that $f(0)=0$, $f(1)=1$, $f(x)<x$ for all $x\in(0,1)$. If $f'(0)=f'(1)=1$, then we can identify the two endpoints and get a smooth map on $\mathbb{T}$, say $g$. Then $o$ is an indifferent fixed point of $g$ (so the Dirac measure $\delta_o$ is not hyperbolic), and $B_g(\delta_o)=\mathbb{T}$ (full Lebesague measure and hence physical). – Pengfei Mar 9 '14 at 7:06
• Yes. Does it works if measure is not Dirac(atomic) nor a factor of Dirac for example? (by factor i mean for example in product measure case, $\mu\neq\mu_{1}*\delta_{x}$)- Specially i like to know about such example. – Po0oh Mar 9 '14 at 19:51

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.
• If we are not that picky (^-^) about the regularity, then it suffices to consider Denjoy's non-transitive circle diffeomorphisms with irrational rotation numbers, which can be made $C^{1+\alpha}$. Then the unique invariant measure $\mu$ supported on the minimal set $C\subset\mathbb{T}$ is a physical measure with $B_f(\mu)=\mathbb{T}$ (again, not hyperbolic and hence not SRB). – Pengfei Mar 10 '14 at 5:26