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<p>We prove a general measurable Livšic regularity theorem for real-valued cocycles over non-invertible dynamical systems using only abstract hypotheses on an associated transfer operator. As illustrative applications we derive measurable Livšic regularity results in the analytic regularity class for cocycles over real-analytic expanding maps, in the bounded-variation regularity class for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\beta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>-transformations, and in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(C^\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mi>α</mi> </msup> </math></EquationSource> </InlineEquation> regularity over the class of virtually expanding maps recently introduced by M. Tsujii.</p>

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General Real Measurable Livšic Regularity via Transfer Operators

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  • 出版模式：

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  • 发表日期：

    2026年8月1日

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  • Ian D. Morris

摘要

We prove a general measurable Livšic regularity theorem for real-valued cocycles over non-invertible dynamical systems using only abstract hypotheses on an associated transfer operator. As illustrative applications we derive measurable Livšic regularity results in the analytic regularity class for cocycles over real-analytic expanding maps, in the bounded-variation regularity class for \(\beta \) β -transformations, and in \(C^\alpha \) C α regularity over the class of virtually expanding maps recently introduced by M. Tsujii.