On the ergodicity of anti-symmetric skew products with singularities and its applications
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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909036657508352 |
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| author | Berk, Przemysław Frączek, Krzysztof Trujillo, Frank |
| author_facet | Berk, Przemysław Frączek, Krzysztof Trujillo, Frank |
| contents | We introduce a novel method for proving ergodicity for skew products of interval exchange transformations (IETs) with piecewise smooth cocycles having singularities at the ends of exchanged intervals. This approach is inspired by Borel-Cantelli-type arguments from Fayad and Lemańczyk (2006). The key innovation of our method lies in its applicability to singularities beyond the logarithmic type, whereas previous techniques were restricted to logarithmic singularities. Our approach is particularly effective for proving the ergodicity of skew products for symmetric IETs and antisymmetric cocycles. Moreover, its most significant advantage is its ability to study the equidistribution of error terms in the spectral decomposition of Birkhoff integrals for locally Hamiltonian flows on compact surfaces, applicable not only when all saddles are perfect (harmonic) but also in the case of some non-perfect saddles. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_21067 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the ergodicity of anti-symmetric skew products with singularities and its applications Berk, Przemysław Frączek, Krzysztof Trujillo, Frank Dynamical Systems 37E35, 37A10, 37C40, 37C83, 37J12 We introduce a novel method for proving ergodicity for skew products of interval exchange transformations (IETs) with piecewise smooth cocycles having singularities at the ends of exchanged intervals. This approach is inspired by Borel-Cantelli-type arguments from Fayad and Lemańczyk (2006). The key innovation of our method lies in its applicability to singularities beyond the logarithmic type, whereas previous techniques were restricted to logarithmic singularities. Our approach is particularly effective for proving the ergodicity of skew products for symmetric IETs and antisymmetric cocycles. Moreover, its most significant advantage is its ability to study the equidistribution of error terms in the spectral decomposition of Birkhoff integrals for locally Hamiltonian flows on compact surfaces, applicable not only when all saddles are perfect (harmonic) but also in the case of some non-perfect saddles. |
| title | On the ergodicity of anti-symmetric skew products with singularities and its applications |
| topic | Dynamical Systems 37E35, 37A10, 37C40, 37C83, 37J12 |
| url | https://arxiv.org/abs/2412.21067 |