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- Publisher
- Wiley
- Copyright
- © London Mathematical Society
- 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. Manning and H. McCluskey, ‘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. Mihailescu and M. Urbanski, ‘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 ω(·) on Λ, then the stable dimension at every point is greater than or equal to the zero of the pressure function t → P(tΦs−log ω(·)). 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 td of the pressure function t → P(t Φs − log d), then the number of preimages in Λ of any point y is equal to d, and the stable dimension is td 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 Society – Wiley
Published: Feb 1, 2010