On `observable' Li-Yorke tuples for interval maps

In this paper we study the set of Li-Yorke $d$-tuples and its $d$-dimensional Lebesgue measure for interval maps $T\colon [0,1] \to [0,1]$. If a topologically mixing $T$ preserves an absolutely continuous probability measure 9with respect to Lebesgue), then the $d$-tuples have Lebesgue full measure, but if $T$ preserves an infinite absolutely continuous measure, the situation becomes more interesting. Taking the family of Manneville-Pomeau maps as example, we show that for any $d \ge 2$, it is possible that the set of Li-Yorke $d$-tuples has full Lebesgue measure, but the set of Li-Yorke $d+1$-tuples has zero Lebesgue measure

Topologically mixing maps and the pseudoarc

It is known that the pseudoarc can be constructed as the inverse limit of the copies of [0, 1] with one bonding map f which is topologically exact. On the other hand, the shift homeomorphism σ f is topologically mixing in this case. Thus, it is natural to ask whether f can be only mixing or must be exact. It has been recently observed that, in the case of some hereditarily indecomposable continua (e.g., pseudocircles) the property of mixing of a bonding map implies its exactness. The main aim of the present article is to show that the indicated kind of forcing of recurrence is not the case for the bonding map defining the pseudoarc.Відомо, що псевдодугу можна отримати як обернену границю копій відрізка [0,1] з єдиним зв'язуючим відображенням f, що є топологічно точним. З іншого боку, в цьому випадку зсувний гомеоморфізм σf є топологічно перемішуючим. Таким чином, природно запитати чи може f бути тільки перемішуючим, чи воно обов'язково повинно бути точним. Нещодавно було встановлено, що для деяких спадково нерозкладних континуумів (наприклад, псевдокіл) точність зв'язуючого відображення є наслідком властивості перемішування. В даній статті показано, що розглянутий тип примусового повернення не реалізується для зв'язуючого відображення, що визначає псевдодугу

A Compact Minimal Space Whose Cartesian Square Is Not Minimal

A compact metric space X is called minimal if it admits a minimal homeomorphism; i.e. a homeomorphism h:X→ X such that the forward orbit {hn(x):n=1, 2, ...} is dense in X, for every x ∈ X. In my talk I shall outline a construction of a family of 1-dimensional minimal spaces from A compact minimal space Y such that its square YxY is not minimal whose existence answer the following long standing problem in the negative. Problem. Is minimality preserved under Cartesian product in the class of compact spaces? Note that for the fixed point property this question had been resolved in the negative already 50 years ago by Lopez, and a similar counterexample does not exist for flows, as shown by Dirbák