  We proved that for the countably infinite number of one-parameterized one
dimensional dynamical systems, they preserve the Lebesgue measure and they are
ergodic for the measure (infinite ergodicity). Considered systems connect the
parameter region in which dynamical systems are exact and the parameter region
in which systems are dissipative, and correspond to the critical points of the
parameter in which weak chaos occurs (the Lyapunov exponent converges to zero).
These results are the generalization of the work by R. Adler and B. Weiss. We
show that the distributions of normalized Lyapunov exponent for these systems
obey the Mittag-Leffler distribution of order $1/2$ by numerical simulation.
