Can someone suggest me a mathematically oriented introduction to fractals? What I am looking for should
- introduce Fractals as well-defined mathematical objects (classic maths text book style)
- with a mature notation.
My considerations so far:
I did not have the opportunity to skim through any of the existing textbooks, but I fear that Mandelbrot's works would not satisfy criterion 2 (as he was the first to tackle this topic in depth and I consider him an "out-of-the-box mathematician"). Moreover, as this topic has been popular beyond mathematics, I fear that many other textbooks are too much focused on nice pictures and hands-on experience, so they might not satisfy criterion 1.
Maybe I am wrong with my assumptions, but I am sure some of you can give good advices in this regard.
Best Answer
I second YuiTo Cheng's (now deleted) suggestion of Edgar's book (and, as an added bonus, Edgar himself seems to be pretty active on MSE). It is very approachable text which doesn't rely on a great deal of background (i.e. a clever undergraduate with some knowledge of analysis should be able to get through it without too much difficulty). I am citing the first edition below, though there is also a second edition (which is likely a mathematical improvement, but the proofs no longer end with smiley faces, so something is lost).
I would argue that the "bible" of fractal geometry (for undergrads) is Falconer's Fractal Geometry: Mathematical foundations and applications. This text is a little more advanced, but it doesn't really assume any more background than Edgar's book (though if you have seen measure theory before, it will help). Falconer has also written a volume on fractals for the Oxford "Very Short Introductions" series, as well as a more advanced (graduate level) text for the Cambridge "Tracts in Mathematics" series. All three are worth having if you are working on fractals.
Falconer, Kenneth, Fractal geometry. Mathematical foundations and applications, Hoboken, NJ: John Wiley & Sons (ISBN 978-1-119-94239-9/hbk). xxx, 368 p. (2014). ZBL1285.28011.
Falconer, K. J., The geometry of fractal sets, Cambridge Tracts in Mathematics, 85. Cambridge etc.: Cambridge University Press. XIV, 162 p. £ 17.50; {$} 32.50 (1985). ZBL0587.28004.
Falconer, Kenneth, Fractals. A very short introduction, Very Short Introductions 367. Oxford: Oxford University Press (ISBN 978-0-19-967598-2/pbk). xv, 132 p. (2013). ZBL1279.28001.
Finally, you should read Hutchinson's 1981 paper on self-similar sets. This is a foundational paper in the field, but is also very approachable (again, a clever advanced undergraduate should be able to digest it with some work[1]).
[1] A little over a year ago, as a graduate student, I was given the opportunity to co-advise an undergraduate research project. Our main goal was to was to introduce these students to fractal zeta functions, but we started by reading Hutchinson's paper. It took about a month to get through the really meaty portions of the paper (some of the latter sections were not prerequisite for the rest of the project, so we skipped them), and the undergrads involved seemed to have a pretty solid understanding of the basic outlines. I will, once again, recommend this paper as an important and worthwhile read.