  The aim of this paper is to show how extracting dynamical behavior and
ergodic properties from deterministic chaos with the assistance of exact
invariant measures. On the one hand, we provide an approach to deal with the
inverse problem of finding nonlinear interval maps from a given invariant
measure. Then, we show how to identify ergodic properties by means of
transitions along the phase space via exact measures. On the other hand, we
discuss quantitatively how infinite measures imply maps having subexponential
Lyapunov instability (weakly chaotic), as opposed to finite measure ergodic
maps, that are fully chaotic. In addition, we provide general solutions of maps
for which infinite invariant measures are exactly known throughout the interval
(a demand from this field). Finally, we give a simple proof that infinite
measure implies universal Mittag-Leffler statistics of observables, rather than
narrow distributions typically observed in finite measure ergodic maps.
