Looking for fractals texts suitable for guided self-study.

[tiohn]

I have finally convinced one of my professors to sponsor an honors research/reading course over the summer, but I need to find a suitable book. I own quite a number of older layperson's books on fractals including the standards by Mandelbrot and Barnsley and even Peitgen/Jürgens/Saupe's Chaos and Fractals. However, I need something much more mathematically advanced than what I own. I'm looking at Barnsley's Superfractals, since it appears to be much more aesthetically appealing (hey, fractals are beautiful), but I'm unsure about whether or not it is as rigorous as something like Falconer's work.

Of course, I really need to use several books, so I guess the real question is, what books should I be looking at? I'm working at an advanced undergraduate level with a firm grasp on analysis and will be starting graduate coursework next fall. In addition to the mathematics, I'm also looking for something to inspire an interesting project.

 

[Crosson]

Maybe you are interested in the kinds of fractals that are generated by nonlinear dynamical systems (smooth differential equations and/or discrete mappings), also known as Chaos theory. There is a lot of work that can be done with the fractal phase portraits of these systems.

The book everyone uses for this subject at your level is 'Nonlinear Dynamics' by Strogatz, who is a professor at MIT. Characteristic of the difficulty of this subject there are 6 preliminary chapters and and six more chapters about chaos and fractals. A more advanced book that goes straight into Chaos and deals more with discrete mappings is written by Ott called 'Chaos in Dynamical Systems'.