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

Exact invariant measures: How the strength of measure settles the intensity of chaos