### 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) |
---|---|

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 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### Cite this

*Ergodic Theory and Dynamical Systems*,

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

}

*Ergodic Theory and Dynamical Systems*, vol. 39, no. 3, pp. 764-794. https://doi.org/10.1017/etds.2017.35

**Thermodynamics of the Katok map.** / Pesin, Y.; Senti, S.; Zhang, K.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Thermodynamics of the Katok map

AU - Pesin, Y.

AU - Senti, S.

AU - Zhang, K.

PY - 2019/3/1

Y1 - 2019/3/1

N2 - 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 φt =-t log |df|Eu(x)| for any t ∞ (t0, ∞), t ≠ 1 where Eu(x) denotes the unstable direction. We show that t0 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 φt for all values of t ∞ (t0, 1).

AB - 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 φt =-t log |df|Eu(x)| for any t ∞ (t0, ∞), t ≠ 1 where Eu(x) denotes the unstable direction. We show that t0 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 φt for all values of t ∞ (t0, 1).

UR - http://www.scopus.com/inward/record.url?scp=85021410885&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85021410885&partnerID=8YFLogxK

U2 - 10.1017/etds.2017.35

DO - 10.1017/etds.2017.35

M3 - Article

AN - SCOPUS:85021410885

VL - 39

SP - 764

EP - 794

JO - Ergodic Theory and Dynamical Systems

JF - Ergodic Theory and Dynamical Systems

SN - 0143-3857

IS - 3

ER -