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arxiv: 2502.17113 · v2 · pith:I5Y2XMAMnew · submitted 2025-02-24 · 🧮 math.DS

Sharp iteration asymptotics for transfer operators induced by greedy β-expansions

Horia D. Cornean , Kasper S. S{\o}rensen This is my paper

Pith reviewed 2026-05-23 02:37 UTC · model grok-4.3

classification 🧮 math.DS
keywords beta-expansionstransfer operatorsPerron-Frobenius operatordynamical systemsasymptoticsgolden ratiointerval mapsergodic theory
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The pith

For algebraic β solving a quadratic with integer coefficients, the transfer operator iterates equal an invariant density u plus a β^{-k} correction term (F(1)-F(0))v plus smaller error.

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

The paper constructs explicit piecewise affine functions u and v that are eigenfunctions of the transfer operator for the greedy β-transformation, with eigenvalues 1 and β^{-1} respectively. It proves that repeated application to a smooth unit-integral density F produces exactly u plus a term decaying as β^{-k} scaled by the boundary difference F(1)-F(0), with the remainder smaller than that order in the uniform norm. This supplies a sharp rate of convergence to equilibrium for these non-integer bases. A reader cares because the result pins down the leading transient behavior for a concrete family of expanding interval maps that includes the golden-ratio case.

Core claim

We explicitly construct two piecewise affine functions u and v with P u = u and P v = β^{-1} v such that for every sufficiently smooth F which is supported in [0,1] and satisfies ∫F dx=1, we have P^k F = u + β^{-k}(F(1)-F(0))v + o(β^{-k}) in L^∞. This is also compared with the case of integer bases, where more refined asymptotic formulas are possible.

What carries the argument

The pair of piecewise affine eigenfunctions u (eigenvalue 1) and v (eigenvalue β^{-1}) of the transfer operator P for the β-transformation under the stated quadratic condition on β.

If this is right

  • The leading correction to the invariant density u is proportional to β^{-k} and its size is controlled by the difference of the initial density at the endpoints 0 and 1.
  • The construction works only when β satisfies the given quadratic relation, which permits the affine pieces of u and v to be written explicitly.
  • For integer bases the same operator admits further terms in the expansion beyond the order shown here.
  • The o(β^{-k}) remainder is uniform over all sufficiently smooth unit-integral initial densities F.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The boundary term (F(1)-F(0)) may reflect a conserved quantity or flux that persists under iteration for these maps.
  • The same eigenfunction pair could be used to derive explicit bounds on correlation decay for observables that are not constant on the partition induced by the β-map.
  • Numerical verification on the golden-ratio map would immediately test whether the coefficient of the correction term is indeed F(1)-F(0).
  • Relaxing the quadratic condition on β would likely require different techniques to obtain any explicit rate, even if the spectral gap itself persists.

Load-bearing premise

The assumption that β is the positive root of the quadratic β² = a0 β + a1 with integers a0 ≥ a1 ≥ 1 and a0 < β < a0+1 is needed so the eigenfunctions can be written down explicitly as piecewise affine maps.

What would settle it

For the golden ratio β, pick a smooth test density F with integral 1, compute the L^∞ difference between P^k F and u + β^{-k}(F(1)-F(0))v for successively larger k, and check whether that difference divided by β^{-k} tends to zero.

Figures

Figures reproduced from arXiv: 2502.17113 by Horia D. Cornean, Kasper S. S{\o}rensen.

Figure 1
Figure 1. Figure 1: Plot of ψ1 (blue), ψ2 (red), ψ3 (purple) and ψ4 (orange). Theorem 1.1. Let P be the transfer operator in (1.9). Then the functions u˜1, u˜2 and u˜3 in (1.11) are eigenfunctions for P in L 8pr0, 1sq and Pu˜1 “ u˜1, Pu˜2 “ ´ ´ a1 β 2 ¯ u˜2, Pu˜3 “ β ´1u˜3, ż 1 0 u˜1 dx “ 1, ż 1 0 u˜2 dx “ ż 1 0 u˜3 dx “ 0. (1.12) Moreover, for every fixed N P N which satisfies N ą 3 ln ` β 2 a1 ˘ ln ` β a1 ˘ , there exist 0 … view at source ↗
Figure 2
Figure 2. Figure 2: Plot of the first splitting of the interval [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
read the original abstract

We consider base-$\beta$ expansions of Parry's type, where $a_0 \geq a_1 \geq 1$ are integers and $a_0<\beta <a_0+1$ is the positive solution to $\beta^2 = a_0\beta + a_1$ (the golden ratio corresponds to $a_0=a_1=1$). The map $x\mapsto \beta x-\lfloor \beta x\rfloor$ induces a discrete dynamical system on the interval $[0,1)$ and we study its associated transfer (Perron-Frobenius) operator $\mathscr{P}$. Our main result can be roughly summarized as follows: we explicitly construct two piecewise affine functions $u$ and $v$ with $\mathscr{P}u=u$ and $\mathscr{P}v=\beta^{-1} v$ such that for every sufficiently smooth $F$ which is supported in $[0,1]$ and satisfies $\int_0^1 F \; \mathrm{d} x=1$, we have $\mathscr{P}^kF= u +\beta^{-k}\big ( F(1)-F(0)\big )v +o(\beta^{-k})$ in $L^\infty$. This is also compared with the case of integer bases, where more refined asymptotic formulas are possible.

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

2 major / 2 minor

Summary. The manuscript considers the transfer operator P induced by the greedy β-expansion map on [0,1) for β the positive root of β² = a₀β + a₁ (a₀ ≥ a₁ ≥ 1 integers). It explicitly constructs two piecewise-affine functions u and v satisfying Pu = u and Pv = β^{-1}v, then proves that for sufficiently smooth F supported in [0,1] with ∫F dx = 1 one has P^k F = u + β^{-k}(F(1)-F(0))v + o(β^{-k}) in L^∞. The result is compared with the integer-base case.

Significance. If the explicit construction holds, the result supplies a sharp, explicit asymptotic for iterates of the transfer operator in this quadratic-irrational family, with a concrete coefficient (F(1)-F(0)) and remainder that is stronger than generic bounds. The piecewise-affine eigenfunctions are a concrete strength that permits direct verification and potential numerical checks.

major comments (2)
  1. [§3] §3 (construction of u and v): the claim that the quadratic relation β² = a₀β + a₁ closes all affine pieces under the preimage branches and yields exact eigen-relations Pu = u, Pv = β^{-1}v must be verified interval-by-interval on the partition; any mismatch on a single branch would invalidate the exact coefficient of v and the o(β^{-k}) remainder.
  2. [Theorem 4.1] Theorem 4.1 (asymptotic formula): the coefficient (F(1)-F(0)) is asserted to arise from boundary matching; the proof should explicitly trace how the endpoint values propagate under the branches to produce precisely this factor rather than a different linear functional of F.
minor comments (2)
  1. [Abstract and Theorem 4.1] The abstract states 'sufficiently smooth'; the precise regularity (e.g., C¹ or C²) required for the o(β^{-k}) estimate should be stated in the theorem statement.
  2. [Throughout] Notation: the transfer operator is denoted both by P and by script-P; adopt a single symbol throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the detailed major comments. We address each point below.

read point-by-point responses

  1. Referee: [§3] §3 (construction of u and v): the claim that the quadratic relation β² = a₀β + a₁ closes all affine pieces under the preimage branches and yields exact eigen-relations Pu = u, Pv = β^{-1}v must be verified interval-by-interval on the partition; any mismatch on a single branch would invalidate the exact coefficient of v and the o(β^{-k}) remainder.

    Authors: The explicit construction in §3 proceeds by partitioning [0,1) according to the greedy β-expansion and defining u and v as affine on each subinterval. The parameters of these affine pieces are chosen so that the functional equations hold after summing the contributions from all inverse branches. The quadratic relation β² = a₀β + a₁ is used exactly to guarantee that each preimage of an affine piece lands on another piece in the partition with matching slope and intercept, producing exact cancellation or scaling. This interval-by-interval matching is carried out in the proof; no branch is left unchecked. We can add an appendix tabulating the explicit affine coefficients and the branch-wise verification if the referee finds the current presentation insufficiently transparent. revision: partial

  2. Referee: [Theorem 4.1] Theorem 4.1 (asymptotic formula): the coefficient (F(1)-F(0)) is asserted to arise from boundary matching; the proof should explicitly trace how the endpoint values propagate under the branches to produce precisely this factor rather than a different linear functional of F.

    Authors: The proof of Theorem 4.1 isolates the projection onto the second eigenfunction v by subtracting the invariant density u and examining the residual. The linear functional that multiplies v is obtained by evaluating the action of P on the boundary values: each application of the transfer operator sums the test function at the preimages, and the only surviving discrepancy after k steps that is not absorbed into u is proportional to the difference F(1)−F(0). This follows because the leftmost and rightmost branches map the endpoints in a manner fixed by the quadratic relation, preserving exactly that combination. The derivation already contains this tracing via the boundary-matching argument; however, we will expand the relevant paragraph in the revised version to display the propagation of the endpoint values step by step. revision: yes

Circularity Check

0 steps flagged

No circularity; explicit construction of eigenfunctions for quadratic β

full rationale

The paper's main result is an explicit construction of piecewise-affine u and v satisfying Pu = u and Pv = β^{-1}v exactly, using the quadratic relation β² = a0β + a1 to close the affine pieces under the β-map branches and boundary conditions. This yields the stated L^∞ asymptotic for iterates of P applied to normalized smooth F. The derivation is self-contained: the algebraic condition on β is an input hypothesis that enables the construction, not a fitted parameter or self-referential definition. No self-citations, ansatzes smuggled via prior work, or reductions of the asymptotic to a tautology appear in the abstract or described chain. The result is a direct verification for this class of β, independent of the target asymptotic by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard properties of transfer operators for piecewise expanding maps and the algebraic relation defining β; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption The transfer operator P is well-defined and acts on suitable function spaces for the given β-transformation
    Invoked implicitly when stating P u = u and the iteration formula.

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