Ergodicity of isometries mapping a sphere into itself in the field of p-adic numbers

Sattarov, Iskandar

Ўзбекистон Миллий университети хабарлари · 2026-yil

Annotatsiya

In this paper we study measure-preserving dynamical systems generated by isometries actingon spheres in the field of p-adic numbers. We focus on the boundary case ρ(r) = |f(x)−x| p = r, which is notcovered by the proper invariant-ball argument. Anashin’s finite-level criterion is formulated for 1-Lipschitzself-maps of Z p and therefore applies, after affine normalization, to dynamical systems on invariant balls.For p ≥ 3, an invariant sphere is not a ball: it is a disjoint union of p − 1 balls of the next smaller radius.Hence the criterion on Z p is not directly applicable to a map preserving the whole sphere. We establish anintrinsic finite-factor criterion for isometries of arbitrary p-adic spheres: the system is ergodic if and onlyif the permutations induced on all canonical partitions of the sphere consist of single cycles. We furtherprove a metric cyclic-lifting criterion: once the first finite factor consists of a single cycle, cyclicity at everysubsequent level is equivalent to the return equalities |f jN n (x ∗ )−x ∗ | p = r/p n , j = 1,...,p−1, at one pointand at every level, where N n = (p − 1)p n−1 . We apply these criteria to the borderline invariant spheres of(2,2)-rational functions with two fixed points and of (2,1)-rational functions with a unique fixed point. Inboth families we give explicit ergodic examples over Q 3 . The resulting ergodic systems are also minimal anduniquely ergodic.

Maqola ma’lumotlari
MualliflarSattarov, Iskandar
JurnalЎзбекистон Миллий университети хабарлари
NashriyotЎзбекистон Миллий университети
ISSN2181-7324, 3093-8996
Nashr sanasi2026-10-07
Jild2
Son2.3
Betlar202-220
TilIngliz
Indeksga qo‘shilgan2026-10-08

Maqolaning asl sahifasi

Kalit so‘zlar

p-adic numbers, sphere, isometry, Haar measure, ergodicity, finite dynamical system.

Iqtibos keltirish

APASattarov, Iskandar (2026). Ergodicity of isometries mapping a sphere into itself in the field of p-adic numbers. Ўзбекистон Миллий университети хабарлари, 2(2.3), 202-220. https://journals.nuu.uz/index.php/actanuuz/article/view/15289
GOST (OAK)Sattarov, Iskandar Ergodicity of isometries mapping a sphere into itself in the field of p-adic numbers // Ўзбекистон Миллий университети хабарлари. – 2026. – T. 2, № 2.3. – P. 202-220.
MLASattarov, Iskandar. “Ergodicity of isometries mapping a sphere into itself in the field of p-adic numbers.” Ўзбекистон Миллий университети хабарлари, 2(2.3), 202-220, 2026. https://journals.nuu.uz/index.php/actanuuz/article/view/15289
ChicagoSattarov, Iskandar. “Ergodicity of isometries mapping a sphere into itself in the field of p-adic numbers.” Ўзбекистон Миллий университети хабарлари, 2(2.3), 202-220 (2026). https://journals.nuu.uz/index.php/actanuuz/article/view/15289
HarvardSattarov, Iskandar (2026) ‘Ergodicity of isometries mapping a sphere into itself in the field of p-adic numbers’, Ўзбекистон Миллий университети хабарлари, 2(2.3), 202-220. https://journals.nuu.uz/index.php/actanuuz/article/view/15289
BibTeX
@article{sattaroviskandar2026,
  title = {Ergodicity of isometries mapping a sphere into itself in the field of p-adic numbers},
  author = {Sattarov, Iskandar},
  journal = {Ўзбекистон Миллий университети хабарлари},
  year = {2026},
  volume = {2},
  number = {2.3},
  pages = {202-220},
  url = {https://journals.nuu.uz/index.php/actanuuz/article/view/15289}
}

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