Beta-expansions induced by general Parry polynomials and sharp iteration asymptotics of their transfer operators
Abstract
This paper generalizes previous results to any Parry polynomials pn(z) \cite{Pa} of finite degree n≥ 2. Let β>1 be the unique positive root of pn and consider the transfer operator 𝒫 induced by the map [0,1]∋ x↦ βx−⌊ βx⌋ ∈ [0,1). Suppose all other roots of pn have magnitude less than one. Then for any smooth function F on [0,1] we prove two term asymptotics in L∞ of the form (𝒫r(F))(·)= u(·) ∫01 F(x) d x +β−rv(·) (F(1)−F(0) )+o(β−r), r≥ 1, where u is piecewise constant and 𝒫u=u, while v is piecewise equal to a polynomial of degree one and 𝒫v= β−1v. When 𝒫 is restricted to functions of bounded variation $BV([0,1])$, it is known that | z |≤ β−1 contains its essential spectrum. Thus the contribution coming from v in the above estimate is not guaranteed if F is not smooth enough. Furthermore, we show that when 𝒫 is restricted to $BV([0,1])$ then any | z |<β−1 is an eigenvalue with infinite multiplicity. Moreover, we construct a meromorphic extension of (𝒫−z)−1 to the domain | z |>β−1, where the finite rank poles are numbers ζ with β−1<| ζ|≤ 1 such that βζ is a root of pn(z). If such poles ζ other than $1$ exist, then we construct eigenvectors of 𝒫* in BV* obeying 𝒫*Û=ζÛ, where Û(F) is given by series involving the values of F taken in some specially chosen, finer and finer partitions of [0,1]. When F is either a polynomial or a characteristic function like χ[0,x], then Û(F) can be explicitly computed.
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BibTeX
@misc{cornean2026betaexpansions,
title = {{Beta-expansions induced by general Parry polynomials and sharp iteration asymptotics of their transfer operators}},
author = {Horia D. Cornean and Ira W. Herbst and Kasper S. Sørensen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27844},
eprint = {2610.27844},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27844},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Beta-expansions induced by general Parry polynomials and sharp iteration asymptotics of their transfer operators
AU - Horia D. Cornean
AU - Ira W. Herbst
AU - Kasper S. Sørensen
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27844
UR - https://arxiv.org/abs/2610.27844
AB - This paper generalizes previous results to any Parry polynomials p_n(z) \cite{Pa} of finite degree n≥ 2. Let β>1 be the unique positive root of p_n and consider the transfer operator 𝒫 induced by the map [0,1]∋ x↦ βx−⌊ βx⌋ ∈ [0,1). Suppose all other roots of p_n have magnitude less than one. Then for any smooth function F on [0,1] we prove two term asymptotics in L^∞ of the form (𝒫^r(F))(·)= u(·) ∫_0^1 F(x) d x +β^−rv(·) (F(1)−F(0) )+o(β^−r), r≥ 1, where u is piecewise constant and 𝒫u=u, while v is piecewise equal to a polynomial of degree one and 𝒫v= β^−1v. When 𝒫 is restricted to functions of bounded variation $BV([0,1])$, it is known that | z |≤ β^−1 contains its essential spectrum. Thus the contribution coming from v in the above estimate is not guaranteed if F is not smooth enough. Furthermore, we show that when 𝒫 is restricted to $BV([0,1])$ then any | z |<β^−1 is an eigenvalue with infinite multiplicity. Moreover, we construct a meromorphic extension of (𝒫−z)^−1 to the domain | z |>β^−1, where the finite rank poles are numbers ζ with β^−1<| ζ|≤ 1 such that βζ is a root of p_n(z). If such poles ζ other than $1$ exist, then we construct eigenvectors of 𝒫^* in BV^* obeying 𝒫^*Û=ζÛ, where Û(F) is given by series involving the values of F taken in some specially chosen, finer and finer partitions of [0,1]. When F is either a polynomial or a characteristic function like χ_[0,x], then Û(F) can be explicitly computed.
ER -