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

DOI:
https://doi.org/10.1090/S0002-9947-1991-1049612-4

MathSciNet review:
1049612

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a diffeomorphism which leaves a compact set invariant. Let be such that can map out of . Assume that has a hyperbolic fixed point in . Let be a space of smooth curves in . We define a normalized Frobenius-Perron operator on the vector bundle of Lipschitz continuous functions labelled by the curves in , and use it to prove the existence of a unique, smooth conditionally invariant measure on a segment of the unstable manifold of . A formula for the computation of , the density of , is derived, and is shown to be equal to the reciprocal of the maximal modulus eigenvalue of the Jacobian of at .

**[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)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
58F11,
28D05

Retrieve articles in all journals with MSC: 58F11, 28D05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1991-1049612-4

Article copyright:
© Copyright 1991
American Mathematical Society