The stability of Hausdorff dimension for the level sets under the perturbation of conformal repellers

Let $M$ be a $C^\infty$ compact Riemann manifold. $f:M\to M$ is a $C^1$ map and $\Lambda_f \subset M$ is a conformal repeller of $f$. Suppose $\varphi:M\to\mathbb{R}$ is a continuous function and let $f_k$ be nonconformal perturbation of the map $f$. We consider the stability of Hausdorff dimension of level sets for Birkhorff average of potential function $\varphi$ with respect to $f_k$ and $f$.