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Let $U \subset M$ be an open subset of a Riemannian manifold. I’m trying to find or construct an example of a topologically transitive dynamical system $f : U \to U$ for which

  • $f : \Lambda \to \Lambda$ is topologically transitive, where $\Lambda = \overline{\bigcap_{n=0}^\infty f(U)}$ (i.e. there exists an orbit that is dense in $\Lambda$), and
  • $\Lambda$ admits an $f$-invariant Borel probability measure $\mu$ that is not ergodic (i.e. there exists a subset $A \subset \Lambda$ for which $f(A) \subset A$ and $0 < \mu(A) < 1$) and is a physical measure (i.e. the set $\left\{x \in \Lambda : \frac 1{n}\sum_{i=0}^{n-1}\phi(f^i(x)) \to \int \phi \, d\mu \: \forall \phi \in C(M)\right\}$ has positive Lebesgue measure)

It’s been shown that if $M$ is a surface and $f:M \to M$ is a $C^{1+\alpha}$ diffeomorphism, then there is only one such measure $\mu$ (in particular the unique SRB measure), and thus it must be ergodic (by the ergodic decomposition theorem). So in order for this to work, $\Lambda$ probably must be a Cantor set (or locally a product of a Cantor set with an interval), a horseshoe, or some other kind of hyperbolic attractor. Is there a known or classical example of such a dynamical system?

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  • $\begingroup$ What rules out the full shift, say? $\endgroup$
    – Ville Salo
    Jan 26, 2021 at 17:43
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    $\begingroup$ Your assertion about existence of a unique SRB measure is missing some hypotheses; perhaps you want to assume that f is Anosov, or at least Axiom A, in order to make that claim. But in that setting, there are lots of periodic points, hence lots of periodic orbit measures, and taking the average of one of those with the SRB gives you an invariant non-ergodic probability. In general as soon as there are multiple invariant measures you can just average two of them and get an invariant non-ergodic measure. So the only examples where all invariant measures are ergodic are the uniquely ergodic ones. $\endgroup$
    – Vaughn Climenhaga
    Jan 26, 2021 at 19:04
  • $\begingroup$ Maybe you wanted to add another condition on $\mu$, such as being physical? (ie., its set of generic points has positive Lebesgue measure.) Or having some kind of absolute continuity property? $\endgroup$
    – Vaughn Climenhaga
    Jan 26, 2021 at 19:06
  • $\begingroup$ I hadn't considered a measure supported on a periodic orbit; you're right, I should have specified that I'm interested in physical measures. I've edited the question accordingly. $\endgroup$
    – D. Ford
    Jan 26, 2021 at 22:05
  • $\begingroup$ I'm still wondering about the sentence describing the surface case. I guess the fact that there is at most one such measure in that case follows from Rodriguez Hertz, Rodriguez Hertz, Tahzibi, Ures (Comm. Math. Phys. 2011) but it's not immediately clear to me where existence comes from in general. How do you know that every topologically transitive $C^{1+\alpha}$ surface diffeomorphism has an SRB measure? $\endgroup$
    – Vaughn Climenhaga
    Jan 27, 2021 at 3:55

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