# Self-similar fractals and arithmetic dynamics

Document Type: Research Paper

Author

1 Sharif University of Technology,Tehran‎, ‎Iran

2 ‎Institute for Advanced Study‎, ‎Princeton‎, ‎USA.

Abstract

‎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.

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### History

• Receive Date: 02 December 2016
• Revise Date: 03 March 2018
• Accept Date: 04 March 2018