PreprintCC BY 4.0HAL
A cohomological equation related to the Pascal transformation
Abstract
Let P be the Pascal transformation on the space Omega={0,1}^N, endowed with the Bernoulli measure of parameters (p0,1-p0), where 0 < p0 < 1. A special cohomological equation associated to P is considered, and in particular the following result is proved: exp(2pi it) is an eigenvalue of P if and only if \sum_{n≥0} DZ(t.C(n,w_0+...+w_{n-1}))^2 < \infty for almost all (w_0,w_1,...) in Omega, where the C(n,p)=n!/p!(n-p)! are the binomial coefficients and DZ(x) the distance of x to the nearest integer.
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BibTeX
@misc{bousch2026cohomological,
title = {{A cohomological equation related to the Pascal transformation}},
author = {Thierry Bousch},
howpublished = {HAL},
year = {2026},
month = aug,
url = {https://hal.science/hal-05830238v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A cohomological equation related to the Pascal transformation
AU - Thierry Bousch
PY - 2026
DA - 2026/08/17
PB - HAL
UR - https://hal.science/hal-05830238v1
AB - Let P be the Pascal transformation on the space Omega={0,1}^N, endowed with the Bernoulli measure of parameters (p0,1-p0), where 0 < p0 < 1. A special cohomological equation associated to P is considered, and in particular the following result is proved: exp(2pi it) is an eigenvalue of P if and only if \sum_{n≥0} DZ(t.C(n,w_0+...+w_{n-1}))^2 < \infty for almost all (w_0,w_1,...) in Omega, where the C(n,p)=n!/p!(n-p)! are the binomial coefficients and DZ(x) the distance of x to the nearest integer.
ER -