Towards Brin’s conjecture on frame flow ergodicity:  new progress and perspectives

Cekić, Mihajlo; Lefeuvre, Thibault; Moroianu, Andrei; Semmelmann, Uwe

doi:10.5802/mrr.11

Cited by 2 publications

(2 citation statements)

References 28 publications

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“…In [BG80], Brin-Gromov verified the case when n is odd and n = 7. Recently, Cekić-Lefeuvre-Moroianu-Semmelmann [CLMS21] made progress on the case when n is even or n = 7. For the quantitative theory, Dolgopyat [Dol02] treated the mixing properties of compact group extensions of hyperbolic diffeomorphisms which are discrete-time versions of frame flows.…”

Section: Introductionmentioning

confidence: 99%

Exponential mixing of frame flows for geometrically finite hyperbolic manifolds

Li¹,

Pan²,

Sarkar³

2023

Preprint

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As a final work to establish that frame flows for geometrically finite hyperbolic manifolds of arbitrary dimensions are exponentially mixing with respect to the Bowen-Margulis-Sullivan measure, this paper focuses on the case with cusps. To prove this, we utilize the countably infinite symbolic coding of the geodesic flow of Li-Pan and perform a frame flow version of Dolgopyat's method à la Sarkar-Winter and Tsujii-Zhang. This requires the local non-integrability condition and the non-concentration property but the challenge in the presence of cusps is that the latter holds only on a large proper subset of the limit set. To overcome this, we use a large deviation property for symbolic recurrence to the large subset. It is proved by studying the combinatorics of cusp excursions and using an effective renewal theorem as in the work of Li; the latter uses the exponential decay of the transfer operators for the geodesic flow of Li-Pan.

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Section: Introductionmentioning

confidence: 99%

Exponential mixing of frame flows for geometrically finite hyperbolic manifolds

Li¹,

Pan²,

Sarkar³

2023

Preprint

View full text Add to dashboard Cite

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“…Nevertheless, the present article is not a mere adaptation of [CLMS21] as we had to develop new techniques in order to take into account the specificities of the Kähler setting, see Theorem 3.1, §3.2 or §5.3 for instance. Although it is meant to be self-contained, we encourage the reader to consult [Lef21,CLMS21,CLMS22] as we build here on the framework developed in these articles.…”

mentioning

confidence: 99%

On the ergodicity of unitary frame flows on Kähler manifolds

Cekić¹,

Lefeuvre²,

Moroianu³

et al. 2023

Preprint

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Let (M, g, J) be a closed Kähler manifold with negative sectional curvature and complex dimension m := dim C M ≥ 2. In this article, we study the unitary frame flow, that is, the restriction of the frame flow to the principal U(m)-bundle F C M of unitary frames. We show that if m ≥ 6 is even, and m = 28, there exists λ(m) ∈ (0, 1) such that if (M, g) has negative λ(m)-pinched holomorphic sectional curvature, then the unitary frame flow is ergodic and mixing. The constants λ(m) satisfy λ(6) = 0.9330..., lim m→+∞ λ(m) = 11 12 = 0.9166..., and m → λ(m) is decreasing. This extends to the even-dimensional case the results of Brin-Gromov [BG80] who proved ergodicity of the unitary frame flow on negatively-curved compact Kähler manifolds of odd complex dimension.

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Towards Brin’s conjecture on frame flow ergodicity: new progress and perspectives

Abstract: We report on some recent progress on the ergodicity of the frame flow of negatively-curved Riemannian manifolds. We explain the new ideas leading to ergodicity for nearly 0.25-pinched manifolds and give perspectives for future work.

Cited by 2 publications

References 28 publications

Exponential mixing of frame flows for geometrically finite hyperbolic manifolds

Exponential mixing of frame flows for geometrically finite hyperbolic manifolds

On the ergodicity of unitary frame flows on Kähler manifolds

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