Matricial representation of period doubling cascade

Abstract

Starting from the cycle permutation sigma_(2^k) associated with the (2^k)-periodic orbit of the period doubling cascade we obtain the inverse permutation (sigma_(2^k))^-1. Then we build a matrix permutation related to (sigma_(2^k))^-1, which includes the visiting order of the (2^k)-periodic orbit points. After some manipulations a recurrence relation of matricial representation of period doubling cascade is obtained. Finally the explicit matricial representation is reached.