COMPLEXITY OF CRITICAL FUNCTIONS FOR HAMILTONIAN-SYSTEMS

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.