SELF-SIMILAR FRACTALS AND ARITHMETIC DYNAMICS

The concept of self-similarity on subsets of algebraic varieties is defined by considering algebraic endomorphisms of the variety as 'similarity' maps. Self-similar fractals are subsets of algebraic varieties which can be written as a finite and disjoint union of 'similar' copies. Fractals provide a framework in which, one can unite some results and conjectures in Diophantine geometry. We define a well-behaved notion of dimension for self-similar fractals. We also prove a fractal version of Roth's theorem for algebraic points on a variety approximated by elements of a fractal subset. As a consequence, we get a fractal version of Siegel's theorem on finiteness of integral points on hyperbolic curves and a fractal version of Faltings' theorem on Diophantine approximation on abelian varieties.