Constructing an opposite map to a specified chaotic map

For a two-segmental complete chaotic map F : [0, 1] -> [0, 1] that preserves an invariant density phi and has a partitioning point at x(c), its opposite map (F) over tilde is defined to possess the following four characteristics: (i) (F) over tilde has the same metric structure; (ii) (F) over tilde preserves an invariant density (phi) over tilde (x) = phi(1 - x); (iii) both F and (F) over tilde have the same degree of chaoticity in the sense of identical Lyapunov exponent and (iv) the partitioning point of (F) over tilde is at, = 1 - (x) over tilde (c). An approach for constructing opposite maps analytically for all four types of two-segmental complete chaotic maps is provided. Meanwhile, a mutual implication relationship that is invariant with respect to conjugation (metric equivalence) is defined for all two-segmental complete chaotic maps that share an identical invariant measure, an identical Lyapunov exponent and an identical partitioning point. Through this relationship, a unique implied family of chaotic maps is formed so that as long as any member of this family is identified, the rest can be constructed analytically, which makes it possible for all known statistical properties originally established for a particular class of chaotic maps to be generalized to all two-segmental chaotic maps. Numerical simulations conducted are in good agreement with theoretical results.