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<p>Jonsson and Reschke [JR] showed that birational selfmaps on a projective surface defined over a number field satisfy the energy condition of Bedford and Diller [BD] so their ergodic properties are very well understood. Under suitable hypotheses on the indeterminacy loci, we extend that result to birational maps ℙ<sup><i>k</i></sup> ⇢ ℙ<sup><i>k</i></sup>, <i>k</i> ≥ 2, defined over a number field, showing that they satisfy a similar energy condition introduced by De Thélin and the second author [DTV]. As a consequence, we can construct for such maps their Green measure and deduce several important ergodic consequences.</p><p>Under a mild additional hypothesis, we show that generic sequences of a Galois invariant subset of periodic points equidistribute toward the Green measure.</p>

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Complex dynamics of birational maps

  • 期刊论文

  • 发表日期：

    2025年10月27日

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  • Thomas Gauthier,
  • Gabriel Vigny

摘要

Jonsson and Reschke [JR] showed that birational selfmaps on a projective surface defined over a number field satisfy the energy condition of Bedford and Diller [BD] so their ergodic properties are very well understood. Under suitable hypotheses on the indeterminacy loci, we extend that result to birational maps ℙk ⇢ ℙk, k ≥ 2, defined over a number field, showing that they satisfy a similar energy condition introduced by De Thélin and the second author [DTV]. As a consequence, we can construct for such maps their Green measure and deduce several important ergodic consequences.

Under a mild additional hypothesis, we show that generic sequences of a Galois invariant subset of periodic points equidistribute toward the Green measure.