Markov finite approximation of Frobenius-Perron operator

Abstract

THE PROBLEM of existence and computation of absolutely continuous invariant measures for nonsingular transformations on measure spaces is one of the main concerns in the modern ergodic theory [3]. Lasota and Yorke have established the existence of the invariant measures for a class of nonsingular measurable transformations S from [0, l] to itself [4]. Later in [6], Li and Yorke gave a sufficient condition for the uniqueness of this invariant density and thus for the ergodicity of the mapping. The classical Birkhoff’s Individual Ergodic Theorem says that, if ,U is an ergodic invariant probability measure under S, then for any measurable set A C [0, 11, the time average lim .+,(1/n) C~:AX~(S~(X)), which measures the “average time” spent in A under iteration of S, exists to be &I) for p-almost all x, where xA is the characteristic function of A (= 1 on A and = 0 off A). This seems to suggest that we may use the Cesaro sum of the above form to calculate the invariant measure. A simple and important example in [5], however, shows that the computer round-off error can completely dominate the calculation and make the implementation difficult. In order to overcome this difficulty, Li [5] proposed a rigorous numerical procedure which can be implemented on a computer with negligible round-off error problem. A mapping S: [0, l] + [0, l] is called piecewise C2, if there is a partition 0 = a,, < a, < ... < a, = 1 of [0, l] such that for k = 1, . . . , r, the restriction Sk of S on (uk_, , a,) is a C2-function which can be extended to the closed interval [ak_i, ak] as a C2-function. S need not be continuous at the point ak. For A c [0, 11, we write S-‘(A) for (x: S(x) E A). Throughout the paper we assume S is a nonsingular measurable transformation, i.e. S is measurable and for any measurable subset A c [0, l] with m(A) = 0, m(S-‘(A)) = 0. Here, m denotes the Lebesgue measure on (0, 11. The operator Ps: L'(0, 1) + L’(0, 1) defined by