Fourier decay in parabolic $C^{1+α}$ systems with overlaps
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917300961017856 |
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| author | Leclerc, Gaétan Paukkonen, Sampo Sahlsten, Tuomas |
| author_facet | Leclerc, Gaétan Paukkonen, Sampo Sahlsten, Tuomas |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_15468 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fourier decay in parabolic $C^{1+α}$ systems with overlaps Leclerc, Gaétan Paukkonen, Sampo Sahlsten, Tuomas Dynamical Systems Classical Analysis and ODEs Spectral Theory 37A46 (Primary) 37D35, 37D45 (Secondary) 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. |
| title | Fourier decay in parabolic $C^{1+α}$ systems with overlaps |
| topic | Dynamical Systems Classical Analysis and ODEs Spectral Theory 37A46 (Primary) 37D35, 37D45 (Secondary) |
| url | https://arxiv.org/abs/2505.15468 |