  A system presenting fractal structure in its thermodynamical functions is
introduced, and it is shown that Tsallis statistics is the correct framework
for describing the thermodynamical aspects of such fractal. Its Haussdorf
dimension and its Lipshitz-H\"older exponent are determined in terms of the
entropic index $q$. The connections with the intermittency in experimental data
is discussed. The thermodynamical aspects of the thermofractal is related to
the microscopic interaction of its components through the S-matrix.
