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arxiv: 2605.06962 · v1 · submitted 2026-05-07 · 🧮 math.DS

Recognition: 2 theorem links

· Lean Theorem

Expanding Maps on Flowers, Interval Exchange Transformations, and Ergodic Optimization

Margaret Brown

Pith reviewed 2026-05-11 01:12 UTC · model grok-4.3

classification 🧮 math.DS
keywords flowersexpanding mapsinterval exchange transformationsergodic optimizationlinear complexitySturmian systemstrigonometric polynomialsmaximizing measures
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The pith

Any set contained in a flower has at most linear complexity, and flowers relate to interval exchange transformations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that flowers, special invariant sets for expanding maps, ensure any contained set has word complexity that grows at most linearly. It establishes a direct relationship between flowers and a special class of interval exchange transformations. This extends earlier embeddings of Sturmian systems and supplies numerical evidence that trigonometric polynomials are maximized on flowers in the setting of ergodic optimization.

Core claim

Flowers are invariant sets under expanding maps such that any subset has at most linear complexity and such sets correspond to a special class of interval exchange transformations. The work extends the result that any Sturmian system embeds into the circle as a doubling-invariant subset contained in a half-circle. In ergodic optimization, where flowers were introduced as candidate supports for maximizing measures, numerical computations support the conjecture that trigonometric polynomials attain their maxima on flowers.

What carries the argument

The flower, defined as an invariant set for an expanding map that restricts subset complexity to linear growth and corresponds to interval exchange transformations.

Load-bearing premise

The definitions of flowers from ergodic optimization match those used to prove the linear complexity bound and the interval exchange relationship, and the numerical evidence for trigonometric polynomials generalizes.

What would settle it

Construct a concrete flower that contains a subset whose symbolic complexity grows faster than linearly, or exhibit a trigonometric polynomial whose maximizing measure is supported outside every flower.

Figures

Figures reproduced from arXiv: 2605.06962 by Margaret Brown.

Figure 1
Figure 1. Figure 1: Examples of preimage selectors and corresponding flowers [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A 3-IET with length data (0.5, 0.2, 0.3) and permutation 1 7→ 2, 2 7→ 1, 3 7→ 3 Definition 3.3. An IET is called minimal if every orbit is dense. Definition 3.4. An IET is said to satisfy Keane’s property if every end point of a starting interval, 0, l1, l1 + l2, ..., 1 has a distinct infinite orbit. It was proved in [Kea75] that irreducibility plus rational independence of the lengths l1, ..., lm imply Ke… view at source ↗
Figure 3
Figure 3. Figure 3: A 3-flower and its image under D Note that because we take P ∗ i and because there are no atoms of µ at any end points pi,j , the definitions of the Ai , Bi give left closed/right open intervals as desired in the definition of deck-shuffler IETs. We will verify that Tµ defined by these intervals makes diagram 5 commute. The petals of the flower satisfy the antipodal condition: P ∗ m − 1 2 < P∗ 1 < P∗ m+1− … view at source ↗
Figure 4
Figure 4. Figure 4: Graph of Hl for l = ( 2 5 , 1 5 , 1 5 , 1 5 ) and a flower containing the image The image under Hl is exactly the orbit { 3 31 , 6 31 , 12 31 , 24 31 , 17 31 } [PITH_FULL_IMAGE:figures/full_fig_p031_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Graph of Hl for l = ( 3 10 , 2 10 , 2 10 , 3 10 ) and a flower containing the image a b+1/4 b 1/4 b a 1/4 b+1/4 [0,1): T([0,1)) [PITH_FULL_IMAGE:figures/full_fig_p033_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The first iterate of Tl for example 3 33 [PITH_FULL_IMAGE:figures/full_fig_p033_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Graph of Hl for l = (a, b + 1 4 , b, 1 4 ) where b /∈ Q and a 3−flower containing the image. where S : X → X is a topological dynamical system, MS is the set of S invariant measures on X, and f : X → R is a real valued potential function. A classical example in ergodic optimization was proved by Bousch ([Bou00]) for the dynamics of the doubling map on the circle, and the potential functions of degree one t… view at source ↗
Figure 8
Figure 8. Figure 8: Visualization of interlacing number 36 [PITH_FULL_IMAGE:figures/full_fig_p036_8.png] view at source ↗
read the original abstract

In this paper, we discuss expanding maps on a class of invariant sets called flowers. We show that any set contained in a flower has at most linear complexity, and we present a relationship between flowers and a special class of interval exchange transformations. This extends work of Bullett and Sentenac, who showed that any Sturmian system may be embedded into the circle as a doubling-invariant subset that is contained in a half circle. Flowers were first introduced in the context of ergodic optimization, as candidate sets for supporting maximizing measures. We discuss the relationship to ergodic optimization, and present numerical results that support the conjecture that trigonometric polynomials are maximized on flowers.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper studies expanding maps on invariant sets called flowers. It proves that any subset contained in a flower has at most linear complexity and establishes an explicit correspondence between flowers and a special class of interval exchange transformations, extending the embedding result of Bullett and Sentenac for Sturmian systems. In the ergodic optimization setting, flowers are discussed as candidate supports for maximizing measures, and numerical experiments are presented as supporting evidence for the conjecture that trigonometric polynomials attain their maxima on flowers.

Significance. If the claims hold, the work supplies a combinatorial framework linking linear complexity of invariant sets under expanding maps to interval exchange transformations and ergodic optimization. The direct combinatorial argument for linear complexity and the explicit coding that preserves the invariant set and expansion (matching the Bullett-Sentenac construction) are strengths. The numerical results are framed strictly as supporting evidence with discretization details supplied, without overclaiming generality.

minor comments (2)
  1. [§6] §6 (numerical experiments): the discretization and optimization procedures are described, but adding a short table summarizing the tested trigonometric polynomials, grid sizes, and observed maxima would improve reproducibility and clarity of the supporting evidence.
  2. [Introduction] Introduction: the extension beyond Bullett-Sentenac is stated clearly, yet a one-sentence comparison table or diagram contrasting the Sturmian half-circle case with the general flower case would aid readers unfamiliar with the prior embedding.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive assessment. We are pleased that the referee recommends acceptance and that the summary accurately reflects the paper's contributions on linear complexity, the correspondence with interval exchange transformations, and the numerical support for the ergodic optimization conjecture.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives linear complexity for subsets of flowers directly from the combinatorial structure of the expanding map and flower partition, without reducing to fitted parameters or self-definitions. The explicit coding establishing the correspondence to a subclass of interval exchange transformations is constructed to preserve the invariant set and expansion, extending the Bullett-Sentenac embedding for Sturmian systems via independent combinatorial arguments rather than self-citation load-bearing. Numerical results are explicitly framed as supporting evidence for the ergodic-optimization conjecture on trigonometric polynomials, with discretization details provided, and do not serve as load-bearing predictions. No ansatz smuggling, renaming of known results, or uniqueness theorems imported from the authors' prior work appear in the chain.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the prior definition of flowers from ergodic optimization literature and the embedding results of Bullett and Sentenac; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Flowers are invariant sets under expanding maps that serve as candidate supports for maximizing measures
    Invoked as the foundation for both the complexity claims and the optimization conjecture.

pith-pipeline@v0.9.0 · 5396 in / 1180 out tokens · 35395 ms · 2026-05-11T01:12:01.628353+00:00 · methodology

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Reference graph

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