On λ-power distributional n-chaos

Abstract

For each real number λ ∈ 2 [0, 1], λ-power distributional chaos has been introduced and studied via Furstenberg families recently. The chaoticity gets stronger and stronger as λ varies from 1 to 0, where 1-power distributional chaos is exactly the usual distributional chaos. As a generalization of distributional n-chaos, λ-power distributional n-chaos is defined similarly. Lots of classic results on distributional chaos can be improved to be the versions of λ-power distributional n-chaos accordingly. A practical method for distinguishing 0-power distributional n-chaos is given. A transitive system is constructed to be 0-power distributionally n-chaotic but without any distributionally (n + 1)-scrambled tuples. For each λ ∈ 2 [0, 1], λ-power distributional n-chaos can still appear in minimal systems with zero topological entropy.

Keywords

Furstenberg family / λ-power distributional n-chaos / Minimal system / Topological entropy

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