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Some measure rigidity and equidistribution results for β-maps
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 9 / September 2024
- Published online by Cambridge University Press:
- 23 October 2023, pp. 2581-2598
- Print publication:
- September 2024
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- Article
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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.
