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On Dirac Physical Measures for Transitive Flows

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Abstract

We discuss some examples of smooth transitive flows with physical measures supported at fixed points. We give some conditions under which stopping a flow at a point will create a Dirac physical measure at that indifferent fixed point. Using the Anosov-Katok method, we construct transitive flows on surfaces with the only ergodic invariant probabilities being Dirac measures at hyperbolic fixed points. When there is only one such point, the corresponding Dirac measure is necessarily the only physical measure with full basin of attraction. Using an example due to Hu and Young, we also construct a transitive flow on a three-dimensional compact manifold without boundary, with the only physical measure the average of two Dirac measures at two hyperbolic fixed points.

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Authors and Affiliations

  1. Departamento de Matematica, IME-USP, Sao Paulo, Brasil

    Radu Saghin & Edson Vargas

  2. School of Mathematical Sciences, PKU, Beijing, China

    Wenxiang Sun

Authors
  1. Radu Saghin
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  2. Wenxiang Sun
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  3. Edson Vargas
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Correspondence to Radu Saghin.

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Communicated by G. Gallavotti

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Saghin, R., Sun, W. & Vargas, E. On Dirac Physical Measures for Transitive Flows. Commun. Math. Phys. 298, 741–756 (2010). https://doi.org/10.1007/s00220-010-1077-9

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  • DOI: https://doi.org/10.1007/s00220-010-1077-9

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