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arXiv · 1504.04214

Regularity of absolutely continuous invariant measures for piecewise expanding unimodal maps

Fabian Contreras·Dmitry Dolgopyat
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Abstract

Let $f : [0, 1] \to [0, 1]$ be a piecewise expanding unimodal map of class $C^{k+1}$, with $k \geq 1$, and $μ= ρdx$ the (unique) SRB measure associated to it. We study the regularity of $ρ$. In particular, points $\mathcal{N}$ where $ρ$ is not differentiable has zero Hausdorff dimension, but is uncountable if the critical orbit of $f$ is dense. This improves on a work of Szewc (1984). We also obtain results about higher orders of differentiability of $ρ$ in the sense of Whitney.

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Fabian Contreras, Dmitry Dolgopyat. 2016-04-12. Regularity of absolutely continuous invariant measures for piecewise expanding unimodal maps. https://doi.org/10.1088/0951-7715%2F29%2F9%2F2775

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