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A smooth Gaussian-Kronecker diffeomorphism

Mostapha Benhenda

TL;DR

This work answers Katok's question by constructing a smooth Gaussian-Kronecker diffeomorphism. It combines De La Rue's planar Brownian-motion approach with an infinite-dimensional Anosov–Katok scheme on the Hilbert cube $M=\mathbb{T}\times[0,1]^{\mathbb{N}}$, producing a $C^\infty$-regular, measure-preserving $T$ whose spectrum is Kronecker. The proof builds a sequence of finite-approximation diffeomorphisms $T_n=B_n^{-1}S_{p_n/q_n}B_n$ and shows convergence to a limit $T$ that is Gaussian-Kronecker with simple $L^p$-spectrum and the weak-closure property. The construction relies on a carefully engineered hierarchy of partitions $\zeta_n^\infty$, monotone refinements, and smooth conjugacies that respect the independent rotation structure, ensuring the Kronecker spectrum in the limit. As a result, the paper provides a concrete, smooth realization of a Gaussian-Kronecker system, expanding the class of known smooth dynamics with prescribed spectral characteristics and offering new tools for spectral control in high- and infinite-dimensional settings.

Abstract

We build a smooth Gaussian-Kronecker diffeomorphism, answering a question raised by Anatole Katok in his list of ''Five Most Resistant Problems in Dynamics''.

A smooth Gaussian-Kronecker diffeomorphism

TL;DR

This work answers Katok's question by constructing a smooth Gaussian-Kronecker diffeomorphism. It combines De La Rue's planar Brownian-motion approach with an infinite-dimensional Anosov–Katok scheme on the Hilbert cube , producing a -regular, measure-preserving whose spectrum is Kronecker. The proof builds a sequence of finite-approximation diffeomorphisms and shows convergence to a limit that is Gaussian-Kronecker with simple -spectrum and the weak-closure property. The construction relies on a carefully engineered hierarchy of partitions , monotone refinements, and smooth conjugacies that respect the independent rotation structure, ensuring the Kronecker spectrum in the limit. As a result, the paper provides a concrete, smooth realization of a Gaussian-Kronecker system, expanding the class of known smooth dynamics with prescribed spectral characteristics and offering new tools for spectral control in high- and infinite-dimensional settings.

Abstract

We build a smooth Gaussian-Kronecker diffeomorphism, answering a question raised by Anatole Katok in his list of ''Five Most Resistant Problems in Dynamics''.

Paper Structure

This paper contains 15 sections, 40 theorems, 334 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions
  3. Transformations of the planar Brownian motion
  4. Basic steps of the proof
  5. Construction of suitable sequences of integers: proof of lemma \ref{['lemme1point3rotisom']}
  6. Proof that $T$ is Kronecker
  7. Partitions of $\Omega= C_0^0([0,1], \varmathbb{C})$
  8. The metric isomorphism
  9. The sequence of conjugacies
  10. Re-writing and approximation of the partition $\eta_n^{n+1}$ by $\eta_n^{'n+1}$
  11. Definitions and basic properties
  12. Construction of the sequence of conjugacies
  13. Generation of $\xi_n$ and convergence of the sequence of diffeomorphisms
  14. Generation of $\xi_n$
  15. Convergence

Key Result

Theorem 1.1

There exists $T \in$ Diff$^\infty(\varmathbb{T} \times [0,1]^{\varmathbb{N}},Leb)$ that is Gaussian-Kronecker.

Figures (1)

  • Figure :

Theorems & Definitions (81)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 1.3
  • Lemma 1.4
  • Lemma 1.5
  • Lemma 1.6
  • proof
  • Lemma 1.7
  • proof
  • Corollary 1.8
  • ...and 71 more