Infinite ergodicity that preserves the Lebesgue measure

In this study, we prove that a countably infinite number of one-parameterized one-dimensional dynamical systems preserve the Lebesgue measure and are ergodic for measure. The consider connect parameter region in which exact one almost all orbits diverge to infinity correspond critical points weak chaos tends occur (the Lyapunov exponent converging zero). These results generalization work by Adler Weiss. Using numerical simulation, show distributions normalized these obey Mittag–Leffler distribution order 1 / 2.