Hausdorff dimension via ergodic theory This is definitely a soft question, but it was recently mentioned to me that one can study the dimension of fractals via ergodic methods. I'm familiar with ergodic theory on about the level of Einseidler and Ward, but googling for references is just bringing up papers that I can't follow or books that I don't have access to (without loaning from another library, but the semester is almost finished.) Does anyone know of any accessible references? I'm specifically wondering if it's possible to find (or get bounds on) the Hausdorff or Minkowski dimensions of a Brownian path or graph, or perhaps other "well known" sets like a variation on a cantor set.
Thanks in advance. 
 A: I would like to add some references to the very good ones given by Jonas. If you want to learn more about ergodic theory and its relation with dimension theory, then mandatory readings are the books by Pesin, "Dimension Theory in Dynamical Systems" in which he develops a wonderful generalized notion of dimension and notes that Bowen's Formula (one of the fundamental tools to compute the Hausdorff dimension of certain fractal sets) is a version of a more general phenomena for this generalized dimension. Some works by Luis Barreira exploit this construction, and some of them are presented in his book "Dimension and Recurrence in Hyperbolic Dynamics".
A: Your question is a bit broad and it is difficult to reply without having more information. Since you mention Einseidler and Ward, you really should be able to read some of the papers in the area, since many give definitions and summarize the key results.
Basically ergodic theory is level 0 of what you ask. After that you should learn at least pieces of the thermodynamic formalism, but since you seem to have a tight schedule right now, my best recommendation is that you have a careful look at Falconer's popular book, really a work of art. If you want to grasp a bit of the thermodynamic formalism you can have a look at his second "popular", although  from the dynamical systems point of view none of them have real content. So, all boils down to what you know and to what you want to know, and our question doesn't give enough information.
It is nevertheless true that to some extent the theory may be developed pretending that the thermodynamic formalism does not exist. Again this is the view of Falconer's book.
