Characterizations of turbulent one-dimensional mappings via 𝜔-limit sets

Abstract

The structure of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi>ω<!-- ω --></mml:mi> <mml:annotation encoding="application/x-tex">\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-limit sets for nonturbulent functions is studied, and various characterizations for turbulent and chaotic functions are obtained. In particular, it is proved that a continuous function mapping a compact interval into itself is turbulent if and only if there exists an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi>ω<!-- ω --></mml:mi> <mml:annotation encoding="application/x-tex">\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-limit set which is a unilaterally convergent sequence