Inverse problem in chaotic map theory

Inverse problem in chaotic map theory is formulated. Solvable chaotic maps having the given invariant measure qðxÞ ¼ An= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 x2n p of n ¼ 2 and n ¼ 3 are obtained. They are derived systematically by using multiplication formulas of the elliptic functions. The Lyapunov number k is also computed exactly for each map function, which is expressed commonly by k ¼ logm in terms of the multiplication factor m.