Reference - Fractal Geometry

Question

I am looking for textbooks or lecture notes about Fractal Geometry that reach an advance level on the topic and aren't just introductory.

Answer 1

This is an slightly edited and expanded version of my comment.

There are several ways to proceed, depending on your background and interests.

If you're mainly interested in a pure mathematics perspective, I suggest beginning with Falconer's The Geometry of Fractal Sets and then going to C. A. Rogers Hausdorff Measures (if you're more interested in general metric space considerations) or Mattila's Geometry of Sets and Measures in Euclidean Spaces (if you're more interested in geometric measure theory in ${\mathbb R}^n).$ In fact, if you're a "pure general topology sort of person", you would probably want to begin with Rogers' book.

If you're interested in a broader mathematical perspective, I suggest beginning with Falconer's Fractal Geometry: Mathematical Foundations and Applications followed by Falconer's Techniques in Fractal Geometry (more applied oriented) or Edgar's Integral, Probability, and Fractal Measures (more pure oriented).

The book by Rogers is probably the least well known, so I'll include a few comments about it. No detail is overlooked in Rogers' book (which makes it great for self-study), his treatment is in a metric space setting (and thus set-theoretic metric and topological issues not present in ${\mathbb R}^n$ come into play), and he deals with Hausdorff measures for general measure functions and not just for power functions. On the other hand, you won't see Julia sets or coastlines or the word "fractal" in Rogers' book (except in the 21 page and 94 item bibliography Forward by Falconer in the 2nd edition of the book).

Also, below are a couple of nice treatments I posted in sci.math about 5 years ago that I think are definitely worth looking at. I haven't looked for this material on the internet in several years, so there are probably other such items I don't know about that are worth looking at.

Mark Pollicott, Lectures on Fractals and Dimension Theory, April-May 2005, 106 pages.

David Worth, Construction of Geometric Outer-Measures and Dimension Theory, MS Thesis (University of New Mexico), December 2006, xi + 77 pages.