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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The Frobenius-Perron operator on spaces of curves

Authors: P. Góra and A. Boyarsky
Journal: Trans. Amer. Math. Soc. 324 (1991), 731-746
MSC: Primary 58F11; Secondary 28D05
MathSciNet review: 1049612
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Abstract: Let $ \tau :{R^2} \to {R^2}$ be a diffeomorphism which leaves a compact set $ A$ invariant. Let $ B \subset A$ be such that $ \tau $ can map out of $ B$. Assume that $ \tau $ has a hyperbolic fixed point $ p$ in $ B$. Let $ \mathcal{C}$ be a space of smooth curves in $ B$. We define a normalized Frobenius-Perron operator on the vector bundle of Lipschitz continuous functions labelled by the curves in $ \mathcal{C}$, and use it to prove the existence of a unique, smooth conditionally invariant measure $ \mu $ on a segment $ {V^u}$ of the unstable manifold of $ p$. A formula for the computation of $ {f^{\ast}}$, the density of $ \mu $, is derived, and $ \mu ({\tau ^{ - 1}}{V^u})$ is shown to be equal to the reciprocal of the maximal modulus eigenvalue of the Jacobian of $ \tau $ at $ p$.

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Article copyright: © Copyright 1991 American Mathematical Society