Multiple mixing and multiple fractional cohomological equation: semisimple setting
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913149587816448 |
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| author | Wang, Zhenqi Jenny |
| author_facet | Wang, Zhenqi Jenny |
| contents | The purpose of this paper is to develop a new effective approach to higher-order mixing in the semisimple setting. We prove effective exponential mixing of all orders for partially hyperbolic algebraic actions, under a strong spectral-gap assumption. The decay rates are explicit in the Lyapunov and spectral-gap data, and the required Sobolev orders are explicit. Already at order two, our estimates require only partial Sobolev/Hölder regularity along weak stable and unstable subgroup directions, with no transverse derivatives. For representations admitting better-than-tempered decay, the resulting order-two estimate attains the optimal matrix-coefficient exponent.
The proof introduces a new fractional-cohomological method in the semisimple setting. The central analytic input is a solvability theory for multiple fractional cohomological equations of Type~$II$ (sum-of-product type). These equations are solvable in a cohomology-free range governed by the spectral behavior near the edge \(0\), and the solutions satisfy estimates in partial Sobolev norms. This mechanism converts fractional solvability into order-two decay of correlations under partial regularity, and then into effective higher-order mixing, yielding a quantitative form of Rokhlin's multiple-mixing problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_21173 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multiple mixing and multiple fractional cohomological equation: semisimple setting Wang, Zhenqi Jenny Dynamical Systems The purpose of this paper is to develop a new effective approach to higher-order mixing in the semisimple setting. We prove effective exponential mixing of all orders for partially hyperbolic algebraic actions, under a strong spectral-gap assumption. The decay rates are explicit in the Lyapunov and spectral-gap data, and the required Sobolev orders are explicit. Already at order two, our estimates require only partial Sobolev/Hölder regularity along weak stable and unstable subgroup directions, with no transverse derivatives. For representations admitting better-than-tempered decay, the resulting order-two estimate attains the optimal matrix-coefficient exponent. The proof introduces a new fractional-cohomological method in the semisimple setting. The central analytic input is a solvability theory for multiple fractional cohomological equations of Type~$II$ (sum-of-product type). These equations are solvable in a cohomology-free range governed by the spectral behavior near the edge \(0\), and the solutions satisfy estimates in partial Sobolev norms. This mechanism converts fractional solvability into order-two decay of correlations under partial regularity, and then into effective higher-order mixing, yielding a quantitative form of Rokhlin's multiple-mixing problem. |
| title | Multiple mixing and multiple fractional cohomological equation: semisimple setting |
| topic | Dynamical Systems |
| url | https://arxiv.org/abs/2605.21173 |