N. Buric et al., COMPLEXITY OF CRITICAL FUNCTIONS FOR HAMILTONIAN-SYSTEMS, Journal of physics. A, mathematical and general, 27(15), 1994, pp. 5201-5208
Transition to predominantly chaotic motion in Hamiltonian systems with
two degrees of freedom is described by a complicated function of freq
uency. which is called the critical function. A graph of this function
is a fractal set with the local structure, which is believed to depen
d only on the arithmetic nature of the frequency. We calculated numeri
cally fractal dimensions of this function for a few typical systems us
ing the method of modular smoothing and an efficient algorithm for com
putation of the fractal dimensions. The dimensions which measure the c
omplexity of the fractal are indeed the same, within the error bounds,
and are equal to the dimension of the exponent of the Brjuno function
, which is a purely arithmetic function.