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Anybody can help me to have an idea about an example showing the difference of a physical measure 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 continuous with respect to induced Riemannian Lebesgue.

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    $\begingroup$ 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. $\endgroup$
    – Pengfei
    Commented Mar 8, 2014 at 16:17
  • $\begingroup$ Thanks, and if i do not have hyperbolic measures, then what will be the example? $\endgroup$
    – Po0oh
    Commented Mar 8, 2014 at 16:42
  • $\begingroup$ 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. $\endgroup$
    – Po0oh
    Commented Mar 8, 2014 at 17:28
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    $\begingroup$ 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). $\endgroup$
    – Pengfei
    Commented Mar 9, 2014 at 7:06
  • $\begingroup$ 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. $\endgroup$
    – Po0oh
    Commented Mar 9, 2014 at 19:51

2 Answers 2

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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.

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Maybe the best reference to clarify that is L.S. Young's paper: What are SRB Measures, and Which Dynamical Systems have them. Several autors mention them to be the same. In the comments, Pengfei give you a good example when they are different. I don't know if it works if measure is not Dirac(atomic) nor a factor of a Dirac measure.

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    $\begingroup$ 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). $\endgroup$
    – Pengfei
    Commented Mar 10, 2014 at 5:26
  • $\begingroup$ Thanks Pengfei yeah i liked this example and would remember it. I just had Kan example in my mind which has two physical measures with positive Leb basins but also Hyperbolic. Where SRB and Phyisical measures become the same as u said. $\endgroup$
    – Po0oh
    Commented Mar 13, 2014 at 21:12

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