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Unmarked spectral rigidity of expanding circle maps

Kostiantyn Drach, Vadim Kaloshin

TL;DR

The paper investigates whether the unmarked length spectrum of a smooth expanding circle map determines its smooth conjugacy class. It proves a local rigidity result: if a map $g$ is close to the linear model $L_d$ and its length spectrum is sufficiently sparse in a quantified $(\beta,\gamma)$-sense, then any nearby map $f$ with the same unmarked spectrum is $C^r$-conjugate to $g$, with the conjugacy gaining $C^r$ regularity. The approach combines a sharp Whitney extension scheme, a quantitative Livšic-type theorem, and an iterative distortion-control framework, carefully preserving a Lebesgue measure normalization along the way. The paper also demonstrates that general unmarked rigidity fails by constructing near-linear counterexamples with identical spectra but non-smooth markings. Together, these results illuminate how spectral data robustly determines smooth structure in one dimension under sparsity, while also highlighting the limits of unmarked spectral rigidity.

Abstract

For a smooth expanding map $f$ of the circle, its (unmarked) length spectrum is defined as the set of logarithms of multipliers of periodic orbits of $f$. This spectrum is analogous to the set of lengths of all closed geodesics on negatively curved surfaces -- the classical length spectrum. In the paper, we prove a length spectral rigidity result for expanding circle maps. Namely, we show that a smooth expanding circle map $f$ of degree $d \ge 2$, under certain assumptions on the sparsity of its length spectrum, cannot be perturbed with an arbitrarily small perturbation (depending on $f$) so that its length spectrum stays the same. The proof uses the Whitney extension theorem, a quantitative Livsic-type theorem, and a novel iterative scheme.

Unmarked spectral rigidity of expanding circle maps

TL;DR

The paper investigates whether the unmarked length spectrum of a smooth expanding circle map determines its smooth conjugacy class. It proves a local rigidity result: if a map is close to the linear model and its length spectrum is sufficiently sparse in a quantified -sense, then any nearby map with the same unmarked spectrum is -conjugate to , with the conjugacy gaining regularity. The approach combines a sharp Whitney extension scheme, a quantitative Livšic-type theorem, and an iterative distortion-control framework, carefully preserving a Lebesgue measure normalization along the way. The paper also demonstrates that general unmarked rigidity fails by constructing near-linear counterexamples with identical spectra but non-smooth markings. Together, these results illuminate how spectral data robustly determines smooth structure in one dimension under sparsity, while also highlighting the limits of unmarked spectral rigidity.

Abstract

For a smooth expanding map of the circle, its (unmarked) length spectrum is defined as the set of logarithms of multipliers of periodic orbits of . This spectrum is analogous to the set of lengths of all closed geodesics on negatively curved surfaces -- the classical length spectrum. In the paper, we prove a length spectral rigidity result for expanding circle maps. Namely, we show that a smooth expanding circle map of degree , under certain assumptions on the sparsity of its length spectrum, cannot be perturbed with an arbitrarily small perturbation (depending on ) so that its length spectrum stays the same. The proof uses the Whitney extension theorem, a quantitative Livsic-type theorem, and a novel iterative scheme.

Paper Structure

This paper contains 24 sections, 25 theorems, 199 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Setup and the statements of the main results
  3. Some history of the question
  4. Strategy of the proof
  5. The structure of the paper
  6. Preliminaries from analysis and on extension theorems
  7. Some preliminaries from calculus
  8. Rolle's Theorem and its application
  9. The Whitney extension theorem
  10. Preliminaries on expanding circle maps
  11. Distortion estimates for expanding circle endomorphisms
  12. Combinatorics of periodic points
  13. Density and sparsity of $\mathbb P_k^g$
  14. Length spectrum
  15. Marked length spectrum and recovering of markings
  16. ...and 9 more sections

Key Result

Lemma 2.1

Let $f, g \colon \mathbb S^1 \to \mathbb S^1$ be, respectively, a $C^{s,1}$-smooth and $C^{s+1,1}$-smooth maps of the circle (not necessarily bijective), $s \geqslant 1$, and let $h \colon \mathbb S^1 \to \mathbb S^1$ be a $C^{s,1}$-smooth diffeomorphism. Assume that for some $M > 0$. Then $h \circ f \circ h^{-1}$ is $C^{s,1}$-smooth, and there is a constant $T = T(s, M)$ such that

Figures (4)

  • Figure 1: The structure of a $(\beta,\gamma)$-sparse length spectrum at the given level $n$. The subset of $\mathop{\mathrm{\mathscr{L}}}\nolimits_n(f)$ is shown in green. A similar structure holds for $\mathop{\mathrm{\mathscr{L}}}\nolimits(f)$ with overlap.
  • Figure 2: A $(p,p)$-messenger connecting $0$ to $x_i \in \mathbb P^g_n$ is shown in blue, while a $(p,p)$-messenger connecting $0$ to $x_j = g^t(x_i)$ is shown in green. A hybrid messenger (in red) connects $0$, $x_i$ and $x_j = g^t(x_i)$ by shadowing halves of the orbits of blue and green messengers and the orbit $x_i, g(x_i), \ldots, g^t(x_i) = x_j$ (in black).
  • Figure 3: Local inductive adjustments. Note that all for all $k \geqslant \kappa_0$, we have $\widetilde{f}_{k-1} = \widetilde{f}_k$ by the uniqueness of a Lebesgue measure preserving map in the smooth conjugacy class of $f$.
  • Figure 4: The example of two length iso-spectral expanding circle maps that are not smoothly conjugate via an orientation preserving diffeomorphism. The graph of $f(x) = 2x+0.1 \cdot \sin^2(\pi x) \colon [0,1] \to [0,1]$ is shown in orange, and the graph of $g(x) = - f(-x) = 2x-0.1 \cdot \sin^2(\pi x)$ is shown in green. The maps $f$ and $g$ are not smoothly conjugate: the multiplier at the points corresponding to the symbolic sequence $\overline{001}$ of period $3$ for $f$ is $\approx 2.31$ (the orange points), while for $g$ it is $\approx 1.9$ (the green points).

Theorems & Definitions (54)

  • Remark 1.1: On the choice of the sparsity parameters $\beta$ and $\gamma$
  • Remark 1.2: On the final smoothness of $\varphi$
  • Remark 1.3: Beyond expanding circle maps
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3: Improvement by density
  • proof
  • Lemma 2.4: À la Rolle
  • proof
  • ...and 44 more