Journal de l’École polytechnique — Mathématiques· 2026Q1
Hyperbolic absolutely continuous invariant measures for C r one-dimensional maps
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- 2026year
Short summary
C r maps (r > 1) with a Lyapunov exponent greater than 1/r log ||f'||∞ on a set A of positive Lebesgue measure admit hyperbolic ergodic invariant measures absolutely continuous with respect to Lebesgue measure.
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Key points
- C r maps (r > 1) with a specific Lyapunov exponent condition admit hyperbolic ergodic invariant measures.
- These invariant measures are absolutely continuous with respect to the Lebesgue measure.
- The basins of these measures cover the set A (where the condition holds) Lebesgue-almost everywhere.
- The proof relies on the Ledrappier–Young entropy characterization of SRB measures.
AI-generated from the title and abstract; the full text is not read.
Abstract
For r > 1 , we show, using the Ledrappier–Young entropy characterization of SRB measures for non-invertible maps, that if a C r map f of the interval or the circle has its Lyapunov exponent greater than 1 r log ∥ f ′ ∥ ∞ on a set A of positive Lebesgue measure, then it admits hyperbolic ergodic invariant measures that are absolutely continuous with respect to the Lebesgue measure. We also show that the basins of these measures cover A Lebesgue-almost everywhere.
The authors' abstract, as published at the source. Journal de l’École polytechnique — Mathématiques, 2026 · DOI ↗
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Field: Mathematical Physics
Mathematical PhysicsMathematics