Smooth rigidity for higher-dimensional contact Anosov flows
UDC 515.12 We apply the technique of matching functions in the setting of contact Anosov flows satisfying a bunching assumption. This allows us to generalize the 3-dimensional rigidity result of Feldman and Ornstein [Ergodic Theory Dynam. Syst., 7, No. 1, 49–72 (1987)]. Nam...
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Institute of Mathematics, NAS of Ukraine
2023
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512635202568192 |
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| author | Gogolev, Andrey Hertz, Federico Rodriguez Gogolev, Andrey Hertz, Federico Rodriguez |
| author_facet | Gogolev, Andrey Hertz, Federico Rodriguez Gogolev, Andrey Hertz, Federico Rodriguez |
| author_sort | Gogolev, Andrey |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2024-06-19T00:34:40Z |
| description | UDC 515.12
We apply the technique of matching functions in the setting of contact Anosov flows satisfying a bunching assumption. This allows us to generalize the 3-dimensional rigidity result of Feldman and Ornstein [Ergodic Theory Dynam. Syst., 7, No. 1, 49–72 (1987)]. Namely, we show that if two Anosov flow of this kind are $C^0$ conjugate, then they are $C^{r}$ conjugate for some $r\in[1,2)$ or even $C^\infty$ conjugate under certain additional assumptions. This, for example, applies to geodesic flows on compact Riemannian manifolds of $1/4$-pinched negative sectional curvature. We can also use our result to recover Hamendstўаdt's marked length spectrum rigidity result for real hyperbolic manifolds. |
| doi_str_mv | 10.3842/umzh.v75i9.7253 |
| first_indexed | 2026-03-24T03:31:55Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-7253 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:31:55Z |
| publishDate | 2023 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-72532024-06-19T00:34:40Z Smooth rigidity for higher-dimensional contact Anosov flows Smooth rigidity for higher-dimensional contact Anosov flows Gogolev, Andrey Hertz, Federico Rodriguez Gogolev, Andrey Hertz, Federico Rodriguez Anosov flow Contact flow Rigidity Marked length spectrum Smooth conjugacy UDC 515.12 We apply the technique of matching functions in the setting of contact Anosov flows satisfying a bunching assumption. This allows us to generalize the 3-dimensional rigidity result of Feldman and Ornstein [Ergodic Theory Dynam. Syst., 7, No. 1, 49–72 (1987)]. Namely, we show that if two Anosov flow of this kind are $C^0$ conjugate, then they are $C^{r}$ conjugate for some $r\in[1,2)$ or even $C^\infty$ conjugate under certain additional assumptions. This, for example, applies to geodesic flows on compact Riemannian manifolds of $1/4$-pinched negative sectional curvature. We can also use our result to recover Hamendstўаdt's marked length spectrum rigidity result for real hyperbolic manifolds. УДК 515.12 Гладка жорсткість для контактних потоків Аносова вищої розмірності  Tехніку узгоджених функцій застосовано до контактних потоків Аносова, що задовольняють умови угруповання.  Це дозволяє узагальнити результат про 3-вимірну жорсткість Фельдмана та Орнштейна [Ergodic Theory Dynam. Syst., 7, № 1, 49-72 (1987)].  А саме, показано, що якщо два таких потоки Аносова є $C^0$ спряженими, то вони є $C^{r}$ спряженими для деякого $r\in[1,2),$ або навіть $C^\infty$ спряженими за деяких додаткових припущень. Це, наприклад, стосується геодезичних потоків на компактних ріманових многовидах $1/4$-стисненої негативної секційної кривини. Наш результат можна також використати, щоб отримати результат Хамендстадт про жорсткість зі спектру маркованих довжин для дійсних гіперболічних многовидів. Institute of Mathematics, NAS of Ukraine 2023-09-26 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7253 10.3842/umzh.v75i9.7253 Ukrains’kyi Matematychnyi Zhurnal; Vol. 75 No. 9 (2023); 1195 - 1203 Український математичний журнал; Том 75 № 9 (2023); 1195 - 1203 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7253/9746 Copyright (c) 2023 Andrey Gogolev |
| spellingShingle | Gogolev, Andrey Hertz, Federico Rodriguez Gogolev, Andrey Hertz, Federico Rodriguez Smooth rigidity for higher-dimensional contact Anosov flows |
| title | Smooth rigidity for higher-dimensional contact Anosov flows |
| title_alt | Smooth rigidity for higher-dimensional contact Anosov flows |
| title_full | Smooth rigidity for higher-dimensional contact Anosov flows |
| title_fullStr | Smooth rigidity for higher-dimensional contact Anosov flows |
| title_full_unstemmed | Smooth rigidity for higher-dimensional contact Anosov flows |
| title_short | Smooth rigidity for higher-dimensional contact Anosov flows |
| title_sort | smooth rigidity for higher-dimensional contact anosov flows |
| topic_facet | Anosov flow Contact flow Rigidity Marked length spectrum Smooth conjugacy |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7253 |
| work_keys_str_mv | AT gogolevandrey smoothrigidityforhigherdimensionalcontactanosovflows AT hertzfedericorodriguez smoothrigidityforhigherdimensionalcontactanosovflows AT gogolevandrey smoothrigidityforhigherdimensionalcontactanosovflows AT hertzfedericorodriguez smoothrigidityforhigherdimensionalcontactanosovflows |