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In the 1988 textbook Fractals Everywhere M. Barnsley introduced an algorithm for generating fractals through a random procedure which he called the chaos game. Using ideas from the classical theory of covering times of Markov chains we prove an asymptotic formula for the expected time taken by this procedure to generate a $delta$-dense subset of a given self-similar fractal satisfying the open set condition.
We prove distributional limit theorems for the length of the largest convex minorant of a one-dimensional random walk with independent identically distributed increments. Depending on the increment law, there are several regimes with different limit
We provide a proof of the (well-known) result that the Poincare exponent of a non-elementary Kleinian group is a lower bound for the upper box dimension of the limit set. Our proof only uses elementary hyperbolic and fractal geometry.
The convex grabbing game is a game where two players, Alice and Bob, alternate taking extremal points from the convex hull of a point set on the plane. Rational weights are given to the points. The goal of each player is to maximize the total weight
We study the attractor of Iterated Function Systems composed of infinitely many affine, homogeneous maps. In the special case of second generation IFS, defined herein, we conjecture that the attractor consists of a finite number of non-overlapping in
We prove that a minimal second countable ample groupoid has dynamical comparison if and only if its type semigroup is almost unperforated. Moreover, we investigate to what extent a not necessarily minimal almost finite groupoid has an almost unperfor