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Gaétan Leclerc

Publications and source records attributed to Gaétan Leclerc.

10 recordsLinked to original sources
arXiv2026

Fourier Decay of SRB Measures for Skew Products

We study Fourier decay of natural invariant measures for real analytic skew products $T(x,y)=(f(x),g(x,y))$ on $\mathbb{S}^1 \times [-1,1]$, where $f$ is expanding and $0<|\partial_y g(x,y)|<1$. Let $\mathcal S$ denote the associated solenoidal attractor. We prove a dichotomy: either $\mathcal S$ is a single horizontal graph, or both the SRB measure and the measure of maximal entropy have polynomial Fourier decay. No transversality assumption is imposed.

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arXiv2026

Arbitrarily Fast Quantum Dispersion in Long-Range Crystals

We construct the first examples of long-range crystals exhibiting arbitrarily fast polynomial quantum dispersion. The Floquet functions of our Hamiltonians are highly oscillatory Weierstrass functions, whose rough autosimilar structure drives the fast dispersion. The proof develops a new Fourier decay theory for $C^α$ images of Lebesgue measure, based on a Dolgopyat-inspired transfer operator method, and yields a van der Corput lemma for Weierstrass functions. As a consequence, the local time of classical Weierstrass functions of sufficiently large lacunarity exists and is $C^k$, answering a question raised by Geman and Horowitz in 1980.

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arXiv2026

Fourier decay of equilibrium states and the Fibonacci Hamiltonian

We show power Fourier decay for equilibrium states of nonlinear, area preserving, smooth Axiom-A diffeomorphisms on surfaces. This implies positivity of the lower Fourier dimension for self-conformal measures under $C^{1+}$ iterated function systems that are factors of hyperbolic diffeomorphisms, which is the first result of this kind in this low-regularity setting. To do so, we use the sum-product phenomenon to reduce Fourier decay to the study of a temporal distance function for a well chosen suspension flow, behaving like a 3-dimensional Axiom A flow, whose mixing properties reflects the nonlinearity of our base dynamics. We then generalize in an Axiom A setting the methods of Tsujii-Zhang, dealing with exponential mixing of three-dimensional Anosov flows arXiv:2006.04293. The nonlinearity condition is generic and can be checked in concrete contexts. To illustrate the applications, we prove two corollaries. We first establish a spectral gap, proving exponential mixing for generic circle extensions over hyperbolic maps on surfaces. As a second application, we prove power Fourier decay for the density of states measure of the Fibonacci Hamiltonian. This implies phase-averaged escape-of-mass estimates, which is the first result of this type in a quasicrystal.

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arXiv2026

Fourier decay in parabolic $C^{1+α}$ systems with overlaps

We establish power Fourier decay for equilibrium states of parabolic $C^{1+α}$ iterated function systems with overlaps satisfying a multiscale nonlinearity condition. This class includes the Lyons conductance measures $ν_t$, $0<t<1$, associated to Galton-Watson trees with equal weights yielding advance towards a conjecture of Lyons on the absolute continuity of $ν_t$ for small $t$. Further applications include Patterson-Sullivan measures for cusped hyperbolic surfaces, extending the work of Bourgain and Dyatlov to parabolic settings, conformal measures for Manneville-Pommeau and Lorenz-type maps, and the construction of the first genuinely $C^{1+α}$ IFSs whose attractors have positive Fourier dimension but are not $C^1$-conjugate to linear IFSs. The proof combines the Bourgain-Dyatlov sum-product strategy with a multiscale induction approach that bypasses the use of spectral gaps for twisted transfer operators needed in several other works in the area.

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arXiv2024

Nonlinearity, Fractals, Fourier decay -- Harmonic analysis of equilibrium states for hyperbolic dynamical systems

This is my (reviewed) PhD manuscript. It contains 6 Chapters, which contains mostly already published work, except for Chapter 5 which is new. Chapter 1 introduce basic notions on fractal geometry: the Fourier dimension, the thermodynamical formalism and additive combinatorics. Chapter 2 is a generalized version of arXiv:2211.08088. Chapter 3 is a slightly upgraded version of arXiv:2112.00701. Chapter 4 is a generalized version of arXiv:2301.10623. Chapter 5 and Chapter 4 together contains the first proof of the positivity of the Fourier dimension for basic sets of nonlinear, area-preserving, smooth Axiom A diffeomorphisms on surfaces. This is possible by adapting some ideas that can be found in Tsujii-Zhang's work arXiv:2006.04293. Some future work still needs to be done to prove that the nonlinearity conditions are generic in this setting: this should become a proper research article in the future. Chapter 6 is a slighty upgraded version of arXiv:2307.10755.

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arXiv2023

Fourier decay of equilibrium states on hyperbolic surfaces

Let $Γ$ be a (convex-)cocompact group of isometries of the hyperbolic space $\mathbb{H}^d$, let $M := \mathbb{H}^d/Γ$ be the associated hyperbolic manifold, and consider a real valued potential $F$ on its unit tangent bundle $T^1 M$. Under a natural regularity condition on $F$, we prove that the associated $(Γ,F)$-Patterson-Sullivan densities are stationary measures with exponential moment for some random walk on $Γ$. As a consequence, when $M$ is a surface, the associated equilibrium state for the geodesic flow on $T^1 M$ exhibit "Fourier decay", in the sense that a large class of oscillatory integrals involving it satisfies power decay. It follows that the non-wandering set of the geodesic flow on convex-cocompact hyperbolic surfaces has positive Fourier dimension, in a sense made precise in the appendix.

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arXiv2023

Fourier decay of equilibrium states for bunched attractors

Let $M$ be a closed manifold, and let $f:M \rightarrow M$ be a $C^{2+α}$ Axiom A diffeomorphism. Suppose that $f$ has an attractor $Ω$ with codimension 1 stable lamination. Under a generic nonlinearity condition and a suitable bunching condition, we prove polynomial Fourier decay in the unstable direction for a large class of invariant measures on $Ω$. Our result applies in particular for the measure of maximal entropy. We construct in the appendix an explicit solenoid that satisfies the nonlinearity and bunching assumption.

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arXiv2022

On oscillatory integrals with Hölder phases

We exhibit a family of autosimilar Hölder maps that satisfies a fractal version of the Van Der Corput Lemma, despite not being absolutely continuous. The result is a direct consequence of a recent work of Sahlsten and Steven arXiv:2009.01703, which is based on a powerful theorem of Bourgain known as a sum-product phenomenon estimate. We give a substantially simpler proof of this fact in our particular context, using an elementary method inspired from arXiv:1704.02909 to check the non-concentration estimates that are needed to apply the sum-product phenomenon. This method allows us to gain additional control over the decay rate.

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arXiv2022

Julia sets of hyperbolic rational maps have positive Fourier dimension

Let $f:\widehat{\mathbb{C}}\rightarrow \widehat{\mathbb{C}}$ be a hyperbolic rational map of degree $d \geq 2$, and let $J \subset \mathbb{C}$ be its Julia set. We prove that $J$ always has positive Fourier dimension. The case where $J$ is included in a circle follows from a recent work of Sahlsten and Stevens, see arXiv:2009.01703. In the case where $J$ is not included in a circle, we prove that a large family of probability measures supported on $J$ exhibit polynomial Fourier decay: our result applies in particular to the measure of maximal entropy and to the conformal measure.

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