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Some measure rigidity and equidistribution results for β-maps
Published online by Cambridge University Press: 23 October 2023
Abstract
We prove $\times a \times b$ measure rigidity for multiplicatively independent pairs when
$a\in \mathbb {N}$ and
$b>1$ is a ‘specified’ real number (the b-expansion of
$1$ has a tail or bounded runs of
$0$s) under a positive entropy condition. This is done by proving a mean decay of the Fourier series of the point masses average along
$\times b$ orbits. We also prove a quantitative version of this decay under stronger conditions on the
$\times a$ invariant measure. The quantitative version together with the
$\times b$ invariance of the limit measure is a step toward a general Host-type pointwise equidistribution theorem in which the equidistribution is for Parry measure instead of Lebesgue. We show that finite memory length measures on the a-shift meet the mentioned conditions for mean convergence. Our main proof relies on techniques of Hochman.
Keywords
MSC classification
- Type
- Original Article
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- Copyright
- © The Author(s), 2023. Published by Cambridge University Press
References
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