Self-similar fractals and arithmetic dynamics

ýThe concept of self-similarity on subsets of algebraic varietiesý ýis defined by considering algebraic endomorphisms of the varietyý ýas `similarity' mapsý. ýSelf-similar fractals are subsets of algebraic varietiesý ýwhich can be written as a finite and disjoint union ofý ý`similar' copiesý. ýFractals provide a framework in whichý, ýone caný ýunite some results and conjectures in Diophantine geometryý. ýWeý ýdefine a well-behaved notion of dimension for self-similar fractalsý. ýWe alsoý ýprove a fractal version of Roth's theorem for algebraic points oný ýa variety approximated by elements of a fractal subsetý. ýAs aý ýconsequenceý, ýwe get a fractal version of Siegel's theorem on finiteness of integral pointsý ýon hyperbolic curves and a fractal version of Falting's theorem ýon Diophantine approximation on abelian varietiesý.