A number theoretical method for calculating high order correlation functions of the Tchebyscheff maps

Abstract:

Correlation functions play a key role in the statistical description of chaotic maps. The main concern of this paper is the calculation of correlation functions of the Tchebyscheff maps, which is traditionally handled by using the graph theoretical method introduced by Beck in 1991. However, this method has poor efficiency when the orders of map and correlation function are large. To overcome this problem, we introduce a number theoretical method based on the definition of correlation functions of the Tchebyscheff maps. In this method, the calculation is transformed into solving a class of Diophantine equations with strictly increasing exponentials, which can be solved in a hierarchical way. Then we obtain all non-vanishing correlation functions with an order not more the order of map and show that the value of non-vanishing correlation functions is independent of the order of map as well as the number of non-vanishing correlation functions is closely related to the Stirling numbers of the second kind. As an application, we calculate all non-vanishing 12-order correlation functions of the Tchebyscheff maps with an order no less than 12.