Gibbs and equilibrium measures for elliptic functions
- First Online:
- 37 Downloads
Because of its double periodicity, each elliptic function canonically induces a holomorphic dynamical system on a punctured torus. We introduce on this torus a class of summable potentials. With each such potential associated is the corresponding transfer (Perron-Frobenius-Ruelle) operator. The existence and uniquenss of “Gibbs states” and equilibrium states of these potentials are proved. This is done by a careful analysis of the transfer operator which requires a good control of all inverse branches. As an application a version of Bowen’s formula for expanding elliptic maps is obtained.
Mathematics Subject Classification (2000):30D05
Unable to display preview. Download preview PDF.
- 1.Walters, P.: Invariant measures and equilibrium states for some mappings which expand distances. Transactions of the AMS, 236, 121–153 (1978)Google Scholar
- 2.Denker, M., Mauldin, R.D., Nitecki, Z., Urbański, M.: Conformal measures for rational functions revisited. Fundamenta Math. 157, 161–173 (1998)Google Scholar
- 3.Denker, M., Urbański, M.: On the existence of conformal measures. Transactions of the AMS 328(2), 563–587 (1991)Google Scholar
- 4.Denker, M., Urbański, M.: Ergodic theory of equilibrium states for rational maps. Nonlinearity 4, 103–134 (1991)Google Scholar
- 5.Kotus, J., Urbański, M.: Hausdorff dimension and Hausdorff measures of Julia sets of elliptic functions. Bull. London Math. Soc. 35, 269–275 (2003)Google Scholar
- 7.McMullen, C.T.: Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps. Comm. Math. Helv. 75, 535–593 (2000)Google Scholar
- 8.McMullen, C.T.: Self-similarity of Siegel disks and Hausdorff dimension of Julia sets. Acta Math. 180, 247–292 (1998)Google Scholar
- 9.Parry, W.: Entropy and Generators in Ergodic Theory. Benjamin 1969Google Scholar
- 10.Przytycki, F., Urbański, M.: Fractals in the Plane - Ergodic Theory Methods. To appear Cambridge Univ. PressGoogle Scholar