Analysis Seminar

I will define conformal iterated function systems $S$ over a countable alphabet $E$ and their limit sets (attractors) $J_E$. I will discuss the formula for the Hausdorff dimension of this limit set, commonly referred to as a version of Bowen's formula, involving topological pressure. The main focus will be on the set $$ Sp(E)=\{HD(J_F): F\subset E\}, $$ called the dimension spectrum of the systenm $S$. I will prove that always $$ Sp(E)\supset (0,\theta_E), $$ where $theta_E$ is the finiteness parameter of $S$ (will be defined). I will also construct a system for which $Sp(E)$ is a proper subset of $(0,HD(J_E)]$. I will then discuss the property that $$ Sp(S)=(0,HD(J_E)], $$ called the full spectrum dimension property. In particular, I will discuss the conformal iterated function systems and their various subsystems, generated by real and complex continued fraction algorithms, and will show that many of them (subsystems) enjoy the full spectrum dimension property.