Abstract
We give a constructive proof of the following result: in all aperiodic dynamical system, for all sequences (an)n∈ℕ ⊂ ℝ+ such that an ↗ ∞ and [Formula: see text] as n → ∞, there exists a set [Formula: see text] having the property that the sequence of the distributions of [Formula: see text] is dense in the space of all probability measures on ℝ. This extends a result of O. Durieu and D. Volný, Ergod. Th. Dynam. Syst. to the non-ergodic case.
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