Generalized thermostatistics based on multifractal phase space
  We consider the self-similar phase space with reduced fractal dimension $d$ being distributed within domain $0<d<1$ with spectrum $f(d)$. Related thermostatistics is shown to be governed by the Tsallis' formalism of the non-extensive statistics, where role of the non-additivity parameter plays inverted value ${\bar\tau}(q)\equiv 1/\tau(q)>1$ of the multifractal function $\tau(q)= qd(q)-f(d(q))$, being the specific heat, $q\in(1,\infty)$ is multifractal parameter. In this way, the equipartition law is shown to take place. Optimization of the multifractal spectrum $f(d)$ derives the relation between the statistical weight and the system complexity. 
A.I. Olemskoi