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Relations between stable dimension and the preimage counting function on basic sets with overlaps

Relations between stable dimension and the preimage counting function on basic sets with overlaps In this paper we study non-invertible hyperbolic maps f and the relation between the stable dimension (that is, the Hausdorff dimension of the intersection between local stable manifolds of f and a given basic set Λ) and the preimage counting function of the map f restricted to the fractal set Λ. The case of diffeomorphisms on surfaces was considered in ( A. M anning and H. M c C luskey , ‘Hausdorff dimension for horseshoes’, Ergodic Theory Dynam. Systems 3 (1983) 251–260), where thermodynamic formalism was used to study the stable/unstable dimensions. In the case of endomorphisms, the non-invertibility generates new phenomena and new difficulties due to the overlappings coming from the different preimages of points, and also due to the variations of the number of preimages belonging to Λ (when compared with (E. M ihailescu and M. U rbanski , ‘Estimates for the stable dimension for holomorphic maps’, Houston J. Math. 31 (2005) 367–389)). We show that, if the number of preimages belonging to Λ of any point is less than or equal to a continuous function ३(·) on Λ, then the stable dimension at every point is greater than or equal to the zero of the pressure function t → P ( t Φ s −log ३(·)). As a consequence we obtain that, if d is the maximum value of the preimage counting function on Λ and if there exists x ∈ Λ with the stable dimension at x equal to the zero t d of the pressure function t → P ( t Φ s − log d ), then the number of preimages in Λ of any point y is equal to d , and the stable dimension is t d everywhere on Λ. This has further consequences to estimating the stable dimension for non-invertible skew products with overlaps in fibers. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Oxford University Press

Relations between stable dimension and the preimage counting function on basic sets with overlaps

Relations between stable dimension and the preimage counting function on basic sets with overlaps

Bulletin of the London Mathematical Society , Volume 42 (1) – Feb 1, 2010

Abstract

In this paper we study non-invertible hyperbolic maps f and the relation between the stable dimension (that is, the Hausdorff dimension of the intersection between local stable manifolds of f and a given basic set Λ) and the preimage counting function of the map f restricted to the fractal set Λ. The case of diffeomorphisms on surfaces was considered in ( A. M anning and H. M c C luskey , ‘Hausdorff dimension for horseshoes’, Ergodic Theory Dynam. Systems 3 (1983) 251–260), where thermodynamic formalism was used to study the stable/unstable dimensions. In the case of endomorphisms, the non-invertibility generates new phenomena and new difficulties due to the overlappings coming from the different preimages of points, and also due to the variations of the number of preimages belonging to Λ (when compared with (E. M ihailescu and M. U rbanski , ‘Estimates for the stable dimension for holomorphic maps’, Houston J. Math. 31 (2005) 367–389)). We show that, if the number of preimages belonging to Λ of any point is less than or equal to a continuous function ३(·) on Λ, then the stable dimension at every point is greater than or equal to the zero of the pressure function t → P ( t Φ s −log ३(·)). As a consequence we obtain that, if d is the maximum value of the preimage counting function on Λ and if there exists x ∈ Λ with the stable dimension at x equal to the zero t d of the pressure function t → P ( t Φ s − log d ), then the number of preimages in Λ of any point y is equal to d , and the stable dimension is t d everywhere on Λ. This has further consequences to estimating the stable dimension for non-invertible skew products with overlaps in fibers.

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References (27)

Publisher
Oxford University Press
Copyright
© 2009 London Mathematical Society
Subject
PAPERS
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/bdp092
Publisher site
See Article on Publisher Site

Abstract

In this paper we study non-invertible hyperbolic maps f and the relation between the stable dimension (that is, the Hausdorff dimension of the intersection between local stable manifolds of f and a given basic set Λ) and the preimage counting function of the map f restricted to the fractal set Λ. The case of diffeomorphisms on surfaces was considered in ( A. M anning and H. M c C luskey , ‘Hausdorff dimension for horseshoes’, Ergodic Theory Dynam. Systems 3 (1983) 251–260), where thermodynamic formalism was used to study the stable/unstable dimensions. In the case of endomorphisms, the non-invertibility generates new phenomena and new difficulties due to the overlappings coming from the different preimages of points, and also due to the variations of the number of preimages belonging to Λ (when compared with (E. M ihailescu and M. U rbanski , ‘Estimates for the stable dimension for holomorphic maps’, Houston J. Math. 31 (2005) 367–389)). We show that, if the number of preimages belonging to Λ of any point is less than or equal to a continuous function ३(·) on Λ, then the stable dimension at every point is greater than or equal to the zero of the pressure function t → P ( t Φ s −log ३(·)). As a consequence we obtain that, if d is the maximum value of the preimage counting function on Λ and if there exists x ∈ Λ with the stable dimension at x equal to the zero t d of the pressure function t → P ( t Φ s − log d ), then the number of preimages in Λ of any point y is equal to d , and the stable dimension is t d everywhere on Λ. This has further consequences to estimating the stable dimension for non-invertible skew products with overlaps in fibers.

Journal

Bulletin of the London Mathematical SocietyOxford University Press

Published: Feb 1, 2010

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