For interval maps with an odd-periodic point, transitivity is equivalent to transitivity of the induced map on probability measures, and several counterexamples separate other dynamical properties.
Ruette, Chaos on the interval, University Lecture Se ries, 67
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Dynamics of continuous maps induced on the space of probability measures
For interval maps with an odd-periodic point, transitivity is equivalent to transitivity of the induced map on probability measures, and several counterexamples separate other dynamical properties.