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.