This image shows the behavior of a certain function (basically, the "inverse temperature modulo a timescale") associated to various "greedily refined" Markov partitions for the geodesic flow on a g-torus as a function of the (log-) number of partition refinements. When I saw that not only the limit but also even the oscillatory behavior was essentially identical for different genera, I was convinced that there was true physical relevance for this very abstract quantity. There was no reason (other than physics!) to expect such uniformity. Details are in http://arxiv.org/abs/1009.2127