Abstract

The investigation of the non-conformal repellers are the important topic in the research on the hyperbolic dynamical systems. Unlike the conformal case, the scenario of the non-conformal one is quite different, e.g., the dimension formula is poorly known and it’s hard to determine the coincidence of the Hausdorff dimension and box dimension. Thus, it becomes more and more important to consider the continuity of the dimension of the non-conformal repeller near the conformal one. In this seminar we focus on such topic. First, we review some previous results on the cases of sub-additive pressures for self-affine sets (Falconer and Sloan, Real Analysis Exchange 2009) and continuity of pressures for matrix cocycles (Feng and Shmerkin, Geom. Funct. Anal. 2014). Some recent results and unsolved problems will also be discussed at this seminar and hope to approach these problems.