### Abstract

We effect the thermodynamical formalism for the non-uniformly hyperbolic C map of the two-dimensional torus known as the Katok map [Katok. Bernoulli diffeomorphisms on surfaces. Ann. of Math. (2) 110(3) 1979, 529-547]. It is a slow-down of a linear Anosov map near the origin and it is a local (but not small) perturbation. We prove the existence of equilibrium measures for any continuous potential function and obtain uniqueness of equilibrium measures associated to the geometric t-potential φ_{t} =-t log |df|E^{u}(x)| for any t ∞ (t_{0}, ∞), t ≠ 1 where E^{u}(x) denotes the unstable direction. We show that t_{0} tends to-∞ as the domain of the perturbation shrinks to zero. Finally, we establish exponential decay of correlations as well as the central limit theorem for the equilibrium measures associated to φ_{t} for all values of t ∞ (t_{0}, 1).

Original language | English (US) |
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Pages (from-to) | 764-794 |

Number of pages | 31 |

Journal | Ergodic Theory and Dynamical Systems |

Volume | 39 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1 2019 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

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## Cite this

*Ergodic Theory and Dynamical Systems*,

*39*(3), 764-794. https://doi.org/10.1017/etds.2017.35