Commentarii Mathematici Helvetici


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Volume 76, Issue 4, 2001, pp. 754–780
DOI: 10.1007/s00014-001-8327-6

Invariant measure and Lyapunov exponents for birational maps of P^2

Jeffrey Diller[1]

(1) Department of Mathematics, University of Notre Dame, 255 Hurley Hall, IN 46556-4618, NOTRE DAME, UNITED STATES

In this paper we construct and study a natural invariant measure for a birational self-map of the complex projective plane. Our main hypothesis - that the birational map be "separating" - is a condition on the indeterminacy set of the map. We prove that the measure is mixing and that it has distinct Lyapunov exponents. Under a further hypothesis on the indeterminacy set we show that the measure is hyperbolic in the sense of Pesin theory. In this case, we also prove that saddle periodic points are dense in the support of the measure.

Keywords: Holomorphic dynamics, birational maps, Lyapunov exponents, invariant measures

Diller Jeffrey: Invariant measure and Lyapunov exponents for birational maps of P^2. Comment. Math. Helv. 76 (2001), 754-780. doi: 10.1007/s00014-001-8327-6