Invariant measure and Lyapunov exponents for birational maps of P2
J Diller - Commentarii Mathematici Helvetici, 2001 - Springer
Commentarii Mathematici Helvetici, 2001•Springer
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.
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.
Abstract
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.
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