0
$\begingroup$

We're referencing Yakov Pesin's "Dimension Theory in Dynamical Systems" in an effort to compute the Hausdorff dimension of a particular invariant set $\Lambda$ of a hyperbolic toral automorphism. Theorem 22.1 in the text gives the dimension of the set in terms of the measure-theoretic entropy with respect to the unique equilibrium measure. This equilibrium measure is guaranteed to be unique as the set $\Lambda$ is locally maximal.

The problem we're running into is that the sets we consider need not be locally maximal. However, again from Pesin theorem A2.2 we know that equilibrium measures will exist for our system. This leads to the question: Will the equation given in theorem 22.1 work if the integration is taken with respect to one of these measures?

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ This question is not useful in the form that it's given. You can reasonably assume that very few people reading it will know the statement of Pesin's Theorem 21. On the other hand, if you say what's in those theorems you have a better chance of reaching people with a general knowledge of the subject area who might be willing to help answer your question. $\endgroup$
    – Anthony Quas
    Commented Feb 15, 2011 at 21:47
  • $\begingroup$ I second Anthony's comment. I'm reasonably familiar with Pesin's book and imagine I might be able to say something useful, but I'm traveling and don't have it with me at the moment, so I'd need statements of the relevant results in order to be of any help. As a general rule, self-contained questions are nearly always better than questions that rely on outside references. $\endgroup$
    – Vaughn Climenhaga
    Commented Feb 15, 2011 at 22:59

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .