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Is there any characterization for the support of T-invariant measures? where T is a C¹ expanding map of the circle i.e. T'(x)>Lambda>1 for all x in the circle.

I know there are periodic and total support measures but what about the rest of ergodic invariant measures?

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  • $\begingroup$ I don't understand... how can $T:\mathbb{S}^1\to \mathbb{S}^1$ have $T'(x)>1$ for all $x$ ? $\endgroup$
    – Pietro Majer
    Commented Apr 11, 2015 at 22:02
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    $\begingroup$ Even for the $\times p$ map, there are very wild sets who are the support for an invariant and ergodic measure, basically you can get Cantor sets of every possible dimension. $\endgroup$
    – Asaf
    Commented Apr 11, 2015 at 22:02
  • $\begingroup$ @PietroMajer: let $T$ have degree two or higher, for example $T \colon \mathbb{R}/\mathbb{Z} \to \mathbb{R}/\mathbb{Z}$ defined by $T(x):=2x \mod \mathbb{Z}$. $\endgroup$
    – Ian Morris
    Commented Apr 27, 2015 at 18:38
  • $\begingroup$ ah, ok thanks -I assumed T was a diffeo :) $\endgroup$
    – Pietro Majer
    Commented Apr 27, 2015 at 18:52
  • $\begingroup$ Excuse me,@IanMorris but I have another question: Is it truth that if x has a dense positive orbit and $x$ generates an ergodic measure $\mu_x=\lim \frac{1}{n}\sum \delta_{T^jx}$ then its support is total? $\endgroup$
    – chaut
    Commented May 11, 2015 at 14:13

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