BME Fractal Geometry Seminar

In this talk, we will discuss aspects of thermodynamic formalism for matrix cocycles, with a particular focus on the existence and uniqueness of equilibrium measures. Given a matrix cocycle \(A\), we consider the family of potentials \(\varphi_t(x)=t\log\|A(x)\|\), as \(t\) ranges over the real numbers, and study how the equilibrium measures associated with these potentials depend on the parameter \(t\).

We present an example of a matrix cocycle for which the equilibrium measure is unique for every \(t>-2\), while for \(t\leq -2\) there are multiple equilibrium measures. This provides an example of a phase transition characterized by the loss of uniqueness of equilibrium measures.

This is joint work with Anthony Quas.

In order to prove the existence of a fractal induced by a given mapping family, we have to show that the induced invariance operator has a fixed point in the fractal space. Hutchinson handled this problem by showing that the invariance operator becomes a hyper-contraction provided that the underlying family consists of contractions. Then the Blaschke Completeness Theorem and the Banach Contraction Principle complete the proof.

In this talk, we are going to present an alternative approach to fractals by completely avoiding the use of the fractal space. A main tool of our method is a shrinking technique, which allows us to replace bounded domains by totally bounded ones. As applications, we present fixed point and fractal results for various kinds of mapping families, and illustrate the method with such examples that are not covered by the Hutchinson Theorem. In the proofs, the Knaster--Tarski Fixed Point Theorem, the Kantorovich Iteration, and the Brouwer Fixed Point Theorem play the key role.