Description
The Brownian continuum tree (CRT) is an important random metric space that was extensively investigated in the 1990s. It can be constructed by a change of metric from a Brownian excursion function on [0,1]. This change of metric can be applied to all continuous circle mappings to give a continuum tree associated with the function. In 2008, Picard proved that analytic properties of the function are connected to the dimension theory of its tree: the upper box dimension of the continuum tree coincides with the variation index of the contour function. We will provide a short and direct proof of Picard's theorem through the study of packings. The methods used will inspire different notions of variations and variation indices, and we will link the dimension theory of the tree with the dimension theory of the graph of its contour function. The title and abstract may be familiar to some of you who came to the one-day fractal meeting at BME about a year ago. This talk includes some updates and strengthened results. (Joint work with Maik Gröger.)