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On a question of Kwakkel and Markovic on existence of wandering domains with bounded geometry

Sergei Merenkov

TL;DR

The paper proves that for any closed surface $M$ with genus $g(M)\ge1$, no $C^1$-diffeomorphism can permute a dense collection of domains with bounded geometry, extending Kwakkel–Markovi7's negative result beyond $C^{1+\alpha}$ to the $C^1$ setting. The authors develop an invariant conformal-structure reduction, introduce transboundary modulus with extremal mass distributions, and employ Nielsen–Thurston theory alongside holomorphic quadratic differentials to rule out all dynamical types (pseudo-Anosov, periodic, reducible). The combination of invariant Beltrami forms, transboundary modulus, and differential-geometric techniques leads to a contradiction, establishing nonexistence except on the sphere. The work clarifies the rigidity of wandering-domain-type phenomena on higher-genus surfaces and highlights the role of conformal/quasiconformal tools in complex dynamics on manifolds.

Abstract

A question of F. Kwakkel and V. Markovic on existence of C^1-diffeomorphisms of closed surfaces that permute a dense collection of domains with bounded geometry is answered in the negative. In fact, it is proved that for closed surfaces of genus at least one such diffeomorphisms do not exist regardless of whether they have positive or zero topological entropy.

On a question of Kwakkel and Markovic on existence of wandering domains with bounded geometry

TL;DR

The paper proves that for any closed surface with genus , no -diffeomorphism can permute a dense collection of domains with bounded geometry, extending Kwakkel–Markovi7's negative result beyond to the setting. The authors develop an invariant conformal-structure reduction, introduce transboundary modulus with extremal mass distributions, and employ Nielsen–Thurston theory alongside holomorphic quadratic differentials to rule out all dynamical types (pseudo-Anosov, periodic, reducible). The combination of invariant Beltrami forms, transboundary modulus, and differential-geometric techniques leads to a contradiction, establishing nonexistence except on the sphere. The work clarifies the rigidity of wandering-domain-type phenomena on higher-genus surfaces and highlights the role of conformal/quasiconformal tools in complex dynamics on manifolds.

Abstract

A question of F. Kwakkel and V. Markovic on existence of C^1-diffeomorphisms of closed surfaces that permute a dense collection of domains with bounded geometry is answered in the negative. In fact, it is proved that for closed surfaces of genus at least one such diffeomorphisms do not exist regardless of whether they have positive or zero topological entropy.
Paper Structure (9 sections, 20 theorems, 72 equations)

This paper contains 9 sections, 20 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Invariant Conformal Structure
  3. Transboundary modulus
  4. Auxiliary lemmas
  5. Proof of Proposition \ref{['P:Extremaldistr']}
  6. Nielsen--Thurston Classification
  7. Pseudo-Anosov diffeomorphisms
  8. Holomorphic quadratic differentials
  9. Periodic and reducible diffeomorphisms

Key Result

Theorem 1.4

If $M$ is a closed surface, other than the sphere, then there does not exist $f\in{\rm Diff}^1(M)$ that permutes a dense collection of domains with bounded geometry.

Theorems & Definitions (39)

  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Remark 2.2
  • proof : Proof of Lemma \ref{['L:UQC']}
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • Corollary 2.5
  • ...and 29 more