We effect the thermodynamical formalism for the non-uniformly hyperbolic $C^{\infty }$ 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 $\unicode[STIX]{x1D711}_{t}=-t\log \mid df|_{E^{u}(x)}|$ for any $t\in (t_{0},\infty )$ , $t\neq 1$ , where $E^{u}(x)$ denotes the unstable direction. We show that $t_{0}$ tends to $-\infty$ 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 $\unicode[STIX]{x1D711}_{t}$ for all values of $t\in (t_{0},1)$ .

我们对二维环面的非均匀双曲 $ C ^ {\ infty} $ 映射（称为Katok映射[Katok。伯努利表面上的亚同形。安 数学。（2）110（3）1979，529–547]。它是原点附近线性Anosov贴图的减慢，并且是局部（但不小）扰动。我们证明了任何连续势函数的平衡测度的存在，并获得了与几何 $ t $- 电位 $ \ unicode [STIX] {x1D711} _ {t} =-t \ log \ mid df | _相关的平衡测度的唯一性。{E ^ {u}（x）} | $ 对于任何 $ t \ in（t_ {0}，\ infty）$ ， $ t \ neq 1 $ ，其中 $ E ^ {u}（x）$ 表示不稳定的方向。我们表明，随着扰动的域缩小到零， $ t_ {0} $ 趋向于 $-\ infty $ 。最后，对于 $ t \ in（t_ {0}，1的所有值，我们建立与 $ \ unicode [STIX] {x1D711} _ {t} $ 相关的平衡测度的相关性的指数衰减以及中心极限定理。）$ 。