We show that a class of higher-dimensional hyperbolic endomorphisms admit absolutely continuous invariant probabilities whose densities are regular and vary differentiably with respect to the dynamical system. The maps we consider are skew-products given by $T(x,y) = (E (x), C(x,y))$ , where E is an expanding map of $\mathbb {T}^u$ and C is a contracting map on each fiber. If $\inf |\!\det DT| \inf \| (D_yC)^{-1}\| ^{-2s}>1$ for some ${s<r-(({u+d})/{2}+1)}$ , $r \geq 2$ , and T satisfies a transversality condition between overlaps of iterates of T (a condition which we prove to be $C^r$ -generic under mild assumptions), then the SRB measure $\mu _T$ of T is absolutely continuous and its density $h_T$ belongs to the Sobolev space $H^s({\mathbb {T}}^u\times {\mathbb {R}}^d)$ . When $s> {u}/{2}$ , it is also valid that the density $h_T$ is differentiable with respect to T. Similar results are proved for thermodynamical quantities for potentials close to the geometric potential.

中文翻译：

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螺线管吸引器SRB测量的规律性和线性响应公式

我们证明了一类高维双曲自同态允许绝对连续的不变概率，其密度是规则的并且相对于动力系统可微地变化。我们考虑的地图是由下式给出的偏斜积 $T(x,y) = (E(x), C(x,y))$ ， 在哪里乙是一张展开的地图 $\mathbb {T}^u$ 和C是每根纤维上的收缩图。如果 $\inf |\!\det DT| \inf\| (D_yC)^{-1}\| ^{-2s}>1$ 对于一些 ${s<r-(({u+d})/{2}+1)}$ , $r \geq 2$ ， 和时间满足迭代重叠之间的横向条件时间（我们证明的条件是 $C^r$ - 温和假设下的通用），然后是 SRB 测量 $\亩_T$ 的时间是绝对连续的并且它的密度 $h_T$ 属于索博列夫空间 $H^s({\mathbb {T}}^u\times {\mathbb {R}}^d)$ 。什么时候 $s> {u}/{2}$ ，密度也是有效的 $h_T$ 相对于可微分时间。对于接近几何势的势的热力学量，也证明了类似的结果。