Geometry, dynamics and fractals

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 g...

Statistical Properties of Random Fractals: Geometry, Growth and Interface Dynamics

This thesis comprises analytic and numerical studies of static, geometrical properties of fractals and dynamical processes in them. First, we have numerically estimated the subset fractal dimensions DS describing the scaling of some subsets S of the fractal cluster with the linear cluster size R in the q-state Potts models. These subsets include the total mass of the cluster, the hull, the exte...

Fractals in complexity and geometry

Some Fractals in Goldpoint Geometry

Problems of interest in goldpoint geometry [1] arise from study of tile-figures that are obtained when goldpoints are marked on sides of triangles, squares, pentagons, etc. and joined by lines in various ways. Many combinatoric problems arise naturally in the course of such studies. Another type of problem is to determine how to combine collections of golden tiles in jig-saw fashion, so that th...

Spectral triples and the geometry of fractals

We construct spectral triples for the Sierpinski gasket as infinite sums of unbounded Fredholm modules associated with the holes in the gasket and investigate their properties. For each element in the K-homology group we find a representative induced by one of our spectral triples. Not all of these triples, however, will have the right geometric properties. If we want the metric induced by the ...