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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The Frobenius-Perron operator on spaces of curves
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by P. Góra and A. Boyarsky PDF
Trans. Amer. Math. Soc. 324 (1991), 731-746 Request permission

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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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 324 (1991), 731-746
  • MSC: Primary 58F11; Secondary 28D05
  • DOI: https://doi.org/10.1090/S0002-9947-1991-1049612-4
  • MathSciNet review: 1049612