Characteristics of (α,β)-Mean Li–Yorke Chaos of Linear Operators on Banach Spaces

In this paper, we introduce a new concept, [Formula: see text]-mean Li–Yorke chaotic operator, which includes the standard mean Li–Yorke chaotic operators as special cases. We show that when [Formula: see text] or [Formula: see text], [Formula: see text]-mean Li–Yorke chaotic dynamics is strictly stronger than the ones that appeared in mean Li–Yorke chaos. When [Formula: see text] or [Formula: see text], it has completely different characteristics from the mean Li–Yorke chaos. We prove that no finite-dimensional Banach space can support [Formula: see text]-mean Li–Yorke chaotic operators. Moreover, we show that an operator is [Formula: see text]-mean Li–Yorke chaos if and only if there exists an [Formula: see text]-mean semi-irregular vector for the underlying operator, and if and only if there exists an [Formula: see text]-mean irregular vector when [Formula: see text], which generalizes the recent results by Bernardes et al. given in 2018. When [Formula: see text], we construct a counterexample in which it is an [Formula: see text]-mean Li–Yorke chaotic operator but does not admit an [Formula: see text]-mean irregular vector. In addition, we show that an operator with dense generalized kernel is [Formula: see text]-mean Li–Yorke chaotic if and only if there exists a residual set of [Formula: see text]-mean irregular vectors, and if and only if there exists an [Formula: see text]-mean unbounded orbit.

ON REGULAR GROUPS AND FIELDS

Regular groups and fields are common generalizations of minimal and quasi-minimal groups and fields, so the conjectures that minimal or quasi-minimal fields are algebraically closed have their common generalization to the conjecture that each regular field is algebraically closed. Standard arguments show that a generically stable regular field is algebraically closed. LetKbe a regular field which is not generically stable and letpbe its global generic type. We observe that ifKhas a finite extensionLof degreen, thenP(n)has unbounded orbit under the action of the multiplicative group ofL.Known to be true in the minimal context, it remains wide open whether regular, or even quasi-minimal, groups are abelian. We show that if it is not the case, then there is a counter-example with a unique nontrivial conjugacy class, and we notice that a classical group with one nontrivial conjugacy class is not quasi-minimal, because the centralizers of all elements are uncountable. Then, we construct a group of cardinality ω1with only one nontrivial conjugacy class and such that the centralizers of all nontrivial elements are countable.