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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$.

References [Enhancements On Off] (What's this?)

  • [1] A. Boyarsky, A functional equation for a segment of the Henon map unstable manifold, Phys. D 21 (1986), 415-426. MR 862268 (88b:58083)
  • [2] J. Palis, Jr. and W. de Melo, Geometric theory of dynamical systems, Springer-Verlag, New York, 1982. MR 669541 (84a:58004)
  • [3] G. Pianigiani and J. Yorke, Expanding maps on sets which are almost invariant: decay and chaos, Trans. Amer. Math. Soc. 252 (1979), 351-366. MR 534126 (80i:58030)
  • [4] G. Pianigiani, Conditionally invariant measures and exponential decay, J. Math. Anal. 82 (1981), 75-88. MR 626742 (82j:58074)
  • [5] M. Misiurewicz and B. Szewc, Existence of a homoclinic point for the Henon map, Comm. Math. Phys. 75 (1980), 285-291. MR 581950 (82d:58046)
  • [6] A. Lasota and P. Rusek, Application of ergodic theory to the determining of cogged bit efficiency, Arch. Gornictwa 19 (1974), no. 3, 281.
  • [7] R. Bowen, Equilibrium states and ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math., vol. 470, Springer-Verlag, New York, 1975. MR 0442989 (56:1364)

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Article copyright: © Copyright 1991 American Mathematical Society

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