Given a non-conformal repeller \(\Lambda \) of a \(C^{1+\gamma }\) map, we study the Hausdorff dimension of the repeller and continuity of the sub-additive topological pressure for the sub-additive singular valued potentials. Such a potential always possesses an equilibrium state. We then use a substantially modified version of Katok’s approximating argument, to construct a compact invariant set on which the corresponding dynamical quantities (such as Lyapunov exponents and metric entropy) are close to that of the equilibrium measure. This allows us to establish continuity of the sub-additive topological pressure and obtain a sharp lower bound of the Hausdorff dimension of the repeller. The latter is given by the zero of the super-additive topological pressure.

中文翻译：

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非保形驱避器的尺寸估计和次加拓扑压力的连续性

给定一个非共形排斥\（\ LAMBDA \）的\（C ^ {1+ \伽马} \）地图，我们研究了推斥的用于子添加剂子添加剂拓扑压的Hausdorff尺寸和连续性奇异价值的潜力。这样的电位总是具有平衡状态。然后，我们使用Katok逼近论的实质修改版本，构造一个紧凑的不变集，在该集合上相应的动力学量（例如Lyapunov指数和度量熵）接近于平衡测度。这使我们能够建立次加性拓扑压力的连续性，并获得推斥极的Hausdorff尺寸的急剧下界。后者由超加拓扑压力的零给出。