[Paper Review] Polynomial entropy of Brouwer homeomorphisms

Louis Hauseux, Frédéric Le Roux|arXiv (Cornell University)|Dec 5, 2017

Mathematical Dynamics and Fractals2 references3 citations

TL;DR

This paper introduces a novel approach to computing polynomial entropy for Brouwer homeomorphisms—fixed-point-free, orientation-preserving homeomorphisms of the plane—by analyzing the wandering part of the system via a quotient construction. It establishes that polynomial entropy can take any real value in [2, ∞], providing the first natural examples of systems with non-integer polynomial entropy and zero topological entropy but infinite polynomial entropy.

ABSTRACT

We study the polynomial entropy of the wandering part of any invertible dynamical system on a compact metric space. As an application we compute the polynomial entropy of Brouwer homeomorphisms (fixed point free orientation preserving homeomorphisms of the plane), and show in particular that it takes every real value greater or equal to 2.

Motivation & Objective

• To develop a general theory of polynomial entropy for the wandering part of dynamical systems on compact metric spaces.
• To apply this theory to compute the polynomial entropy of Brouwer homeomorphisms, which are fixed-point-free, orientation-preserving homeomorphisms of the plane.
• To demonstrate that polynomial entropy serves as a complete conjugacy invariant for Brouwer homeomorphisms, distinguishing uncountably many conjugacy classes.
• To construct explicit examples of Brouwer homeomorphisms with arbitrary polynomial entropy in [2, ∞), including non-integer values.
• To explore connections between polynomial entropy and other invariants, such as the oscillating set, suggesting that high polynomial entropy may imply a non-empty oscillating set.

Proposed method

• Define the wandering polynomial entropy as the polynomial entropy of the quotient space where all non-wandering points are collapsed to a single point ∞.
• Use a finite family of compact subsets {U₁,…,Uₗ} in X\{∞} to code orbits via sequences of sets visited, with ∞ representing time intervals between visits.
• Establish lower and upper bounds on the number of distinct codings of length n using combinatorial counting and interval analysis of transition maps φ_{k,k+1}.
• Apply asymptotic comparison with integrals to estimate the growth rate of coding sets, leading to the polynomial entropy via limsup_n→∞ log S(n,ε)/log n.
• Use Nakayama’s technique and smooth flows with Reeb components to construct explicit examples of Brouwer homeomorphisms with controlled dynamics.
• Prove that the number of codings grows like n^α for α ≥ 2, leading to polynomial entropy h_pol(f) = α.

Experimental results

Research questions

• RQ1 What is the range of possible polynomial entropy values for Brouwer homeomorphisms?
• RQ2 Can polynomial entropy distinguish between non-conjugate Brouwer homeomorphisms, especially when topological entropy vanishes?
• RQ3 Is it possible to construct Brouwer homeomorphisms with non-integer polynomial entropy?
• RQ4 How does the structure of the wandering set and transition maps influence the growth rate of distinguishable orbits?
• RQ5 What is the relationship between polynomial entropy and other conjugacy invariants such as the oscillating set?

Key findings

• The polynomial entropy of a Brouwer homeomorphism is 1 if and only if it is conjugate to a translation.
• There are no Brouwer homeomorphisms with polynomial entropy in the interval (1,2), indicating a gap in the possible values.
• For every real number α ∈ [2, ∞), there exists a Brouwer homeomorphism f_α with polynomial entropy exactly α.
• The constructed examples are time-one maps of C^∞ flows with a single Reeb component and finitely many boundary components.
• The paper presents the first known natural examples of dynamical systems with non-integer polynomial entropy and zero topological entropy but infinite polynomial entropy.
• The proof technique relies on constructing orbit codings using interval dynamics and transition maps φ_{k,k+1}, showing that the number of distinct codings grows asymptotically as n^α, yielding h_pol(f) = α.