Linear toral endomorphisms, exactness, Bernoulliness and generators
Abstract
Let T be a non-invertible surjective continuous endomorphism of the group ttfd := rrfd/zzfd. Then T preserves the Haar measure, is associated to some d × d matrix A with integer coefficients and is r-to-one, where r = |det A| ≥ 2. In~\cite{Krzyzewski}, Krzyżewski gives a necessary and sufficient condition for T to be exact, and shows that T -- viewed as a measure-preserving map -- is isomorphic to the direct product of some (possibly trivial) toral automorphism by some exact toral endomorphism. We re-prove these results by more elementary and constructive methods, whereas Krzyżewski actually works with endomorphisms of compact Abelian groups and relies on non-constructive theorems. Krzyżewski's exactness condition and Mihailescu's papers ~\cite{Mihailescu 2012} and~\cite{Mihailescu 2013} show that T is isomorphic to the one-sided uniform Bernoulli shift on odc 0,r−1 fdczzf_+ if and only if A is expanding (i.e. the modulus of each eigenvalue is >1). When A is not expanding and hyperbolic (i.e. all eigenvalues have modulus different from $1$), Mihailescu~\cite{Mihailescu 2012} shows that T cannot have a generating Rokhlin partition. More generally, when A is not expanding (whether hyperbolic or not), we show that the endomorphism T cannot have a smooth finite generator (whether independent or not).
Related
- Mixing Rates for Analytic Observables: Toral Endomorphisms and Anosov Blaschke Deformations
- Extreme values and logarithm laws for toral translations
- Sharp Bounds for Toral Eigenfunctions and Threefold Additive Energy on Circles
- Convergence Rate of Birkhoff Average for Toral Quasi-Periodic Rotations and Applications
- Quaternionic Extensions of Hyperbolic Toral Automorphisms
BibTeX
@misc{leuridan2026linear,
title = {{Linear toral endomorphisms, exactness, Bernoulliness and generators}},
author = {Christophe Leuridan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01642},
eprint = {2610.01642},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01642},
}
RIS
TY - UNPB
TI - Linear toral endomorphisms, exactness, Bernoulliness and generators
AU - Christophe Leuridan
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01642
UR - https://arxiv.org/abs/2610.01642
AB - Let T be a non-invertible surjective continuous endomorphism of the group ttf^d := rrf^d/zzf^d. Then T preserves the Haar measure, is associated to some d × d matrix A with integer coefficients and is r-to-one, where r = |det A| ≥ 2. In~\cite{Krzyzewski}, Krzyżewski gives a necessary and sufficient condition for T to be exact, and shows that T -- viewed as a measure-preserving map -- is isomorphic to the direct product of some (possibly trivial) toral automorphism by some exact toral endomorphism. We re-prove these results by more elementary and constructive methods, whereas Krzyżewski actually works with endomorphisms of compact Abelian groups and relies on non-constructive theorems. Krzyżewski's exactness condition and Mihailescu's papers ~\cite{Mihailescu 2012} and~\cite{Mihailescu 2013} show that T is isomorphic to the one-sided uniform Bernoulli shift on odc 0,r−1 fdc^zzf_+ if and only if A is expanding (i.e. the modulus of each eigenvalue is >1). When A is not expanding and hyperbolic (i.e. all eigenvalues have modulus different from $1$), Mihailescu~\cite{Mihailescu 2012} shows that T cannot have a generating Rokhlin partition. More generally, when A is not expanding (whether hyperbolic or not), we show that the endomorphism T cannot have a smooth finite generator (whether independent or not).
ER -