Observed measures and fluctuations in dissipative infinite ergodic systems: Randomization theory for the infinite-modal maps with ant-lion property

Universal aspects of a certain class of infinite ergodic systems are studied by using the infi nite-modal maps with a special interest to their observed measures. It is shown that the ant-lion (AL) property is an important nature to realize the infinite ergodicity in the dissipative dynamics. The AL-property, which seems to be a little bit paradoxical one, is characterized by the monotonical relaxation of mean orbits into a singular stable point, but it causes the divergence of the Lyapunov exponent as well as the emergence of a number of absolutely-continuous invariant measures. Our main concern is to characterize the unique observed measure in those many admissible ergodic measures. To this end, firstly the randomization formulae are developed on the basis of the uniform distribution theorem by Weyl, to derive the stochastic aspects of the AL-property. Actually, it is shown that the statistical natures of the infinite-modal maps are well explained by the randomization formulae. Furthermore, it is shown that the observed measure derived from the randomization formulae is universal, and that the asymptotic form obeys the power law with the exponent ?1, in agreement with numerical simulations.