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Ergodic and Spectral Theory of Area-Preserving Flows on Surfaces

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Ergodic Theory

Part of the book series: Encyclopedia of Complexity and Systems Science Series ((ECSSS))

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  • Originally published in
  • R. A. Meyers (ed.), Encyclopedia of Complexity and Systems Science, © Springer Science+Business Media LLC 2021

Glossary

Borel flow is a family φℝ = (φt)t∈ℝ of Borel maps on a topological space X such that (t, x) ↦ φtx is also Borel, φ0 = IdX and \( {\varphi}_{t_1+{t}_2}={\varphi}_{t_1}\circ {\varphi}_{t_2} \) for all t1, t2 ∈ ℝ. The flow preserves a Borel probability measure μ on X if μ(φt(A)) = μ(A) for any Borel set A and any t ∈ ℝ. Then we say that φℝ is a measure-preserving flow of (X, μ).

If S is compact smooth manifold and the map (t, x) ↦ φtx is smooth, then φℝ is called a smooth flow. If additionally S is a surface (dimension two manifold) and an invariant measure μ is smooth and positive (has a smooth positive density), then φℝ is called an area-preserving smooth flow. Such flows are also called multi-valued or locally Hamiltonian.

We say that two flows φℝ on (X, μ) and ϕℝ on (Y, v) and are isomorphic as measure-preserving flows if there exists an isomorphism Φ : X → Y, i.e., a bimeasurable map which transports the measure μ to v (i.e., μ(Φ−1(A)) = ν(A) for any Borel set A ⊂ Y) and...

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Acknowledgments

K. F. acknowledges the supporf of the Narodowe Centrum Nauki Grant number 2017/27/B/ST1/00078. C. U. is part of SwissMAP (The Mathematics of Physics National Centre for Compentence in Research) and is currently supported by a SNSF (Swiss National Science Foundation) Grant number 200021_188617/1. Both are acknowledged for their support.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland

    Krzysztof Frączek

  2. Institut für Mathematik, Universität Zürich, Zürich, Switzerland

    Corinna Ulcigrai

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  1. Krzysztof Frączek
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  2. Corinna Ulcigrai
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Correspondence to Corinna Ulcigrai .

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  1. Mathematics, Williams College, Williamstown, MA, USA

    Cesar E. Silva

  2. B.Verkin Institute for Low Temperature Physics and Engineering of the NAS of Ukraine, Kharkiv, Ukraine

    Alexandre I. Danilenko

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Frączek, K., Ulcigrai, C. (2023). Ergodic and Spectral Theory of Area-Preserving Flows on Surfaces. In: Silva, C.E., Danilenko, A.I. (eds) Ergodic Theory. Encyclopedia of Complexity and Systems Science Series. Springer, New York, NY. https://doi.org/10.1007/978-1-0716-2388-6_775

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