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Invariant measures and the compactness of the domain

Marian Jabłoński, Paweł Góra (1998)

Annales Polonici Mathematici

We consider piecewise monotonic and expanding transformations τ of a real interval (not necessarily bounded) into itself with countable number of points of discontinuity of τ’ and with some conditions on the variation V [ 0 , x ] ( 1 / | τ ' | ) which need not be a bounded function (although it is bounded on any compact interval). We prove that such transformations have absolutely continuous invariant measures. This result generalizes all previous “bounded variation” existence theorems.

Invariant measures for iterated function systems

Tomasz Szarek (2000)

Annales Polonici Mathematici

A new criterion for the existence of an invariant distribution for Markov operators is presented. Moreover, it is also shown that the unique invariant distribution of an iterated function system is singular with respect to the Hausdorff measure.

Invariant scrambled sets and maximal distributional chaos

Xinxing Wu, Peiyong Zhu (2013)

Annales Polonici Mathematici

For the full shift (Σ₂,σ) on two symbols, we construct an invariant distributionally ϵ-scrambled set for all 0 < ϵ < diam Σ₂ in which each point is transitive, but not weakly almost periodic.

Inverse limit of M -cocycles and applications

Jan Kwiatkowski (1998)

Fundamenta Mathematicae

For any m, 2 ≤ m < ∞, we construct an ergodic dynamical system having spectral multiplicity m and infinite rank. Given r > 1, 0 < b < 1 such that rb > 1 we construct a dynamical system (X, B, μ, T) with simple spectrum such that r(T) = r, F*(T) = b, and C ( T ) / w c l T n : n ∈ ℤ = ∞

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