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Some new cases of realization of spectral multiplicity function for ergodic transformations

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Katok A., Lemańczyk M.

Fundamenta Mathematicae

2009

tom 206

nr 1

185-215

Given a countable Abelian group 𝔾, its automorphism w for which $w^{M} = Id$, and a subgroup 𝔽 ⊂ 𝔾 we define $M(𝔾,w,𝔽) = {♯({w^{i}χ: i ∈ ℤ ∩ 𝔽): χ ∈ 𝔽∖{0}}$. We prove that each finite set of the form M(𝔾,w,𝔽) ∪ {2} is realized as the set of essential values of the multiplicity function of the Koopman operator of some weakly mixing automorphism.

Entropy and growth of expanding periodic orbits for one-dimensional maps

100%

Katok A., Mezhirov A.

Fundamenta Mathematicae

1998

tom 157

nr 2-3

245-254

Let f be a continuous map of the circle $S^1$ or the interval I into itself, piecewise $C^1$, piecewise monotone with finitely many intervals of monotonicity and having positive entropy h. For any ε > 0 we prove the existence of at least $e^{(h-ε)n_k}$ periodic points of period $n_k$ with large derivative along the period, $|(f^{n_k})'| > e^{(h-ε)n_k}$ for some subsequence ${n_k}$ of natural numbers. For a strictly monotone map f without critical points we show the existence of at least $(1-ε) e^{hn}$ such points.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Katok A.

1 Lemańczyk M.

1 Mezhirov A.

1 2009

1 1998

Wyniki wyszukiwania

Some new cases of realization of spectral multiplicity function for ergodic transformations

Entropy and growth of expanding periodic orbits for one-dimensional maps