Considering the Julia sets of a family of rational maps concerning two-dimensional diamond hierarchical Potts models in statistical mechanics, we show the continuity of their Hausdorff dimension. 1. Introduction The continuity of Hausdorff dimension of Julia sets is an important and interesting problem for rational maps with degree . In general, this problem adheres to the continuity of Julia sets which is response to the stability of system. It is well known that both the Julia set and its Hausdorff dimension of a rational map vary continuously in the parameter space if is hyperbolic [1, 2]. However, as we know, there are no direct relationship between them when is not hyperbolic though there are many works devoted to the two problems [1, 3, 4]. In this paper, we discuss a family of rational maps for ; here with two parameters and . is a renormalization transformation of -state Potts models on the two-dimensional diamond-like hierarchical lattice with bifurcation number in statistical mechanics [5]. In turn, the zeros of the partition function for the model with bifurcation number condense to the Julia sets of [6]. It has been shown that there exists some relationship between the critical temperatures, the critical amplitudes, and the structures of the Julia sets [7]. Therefore, much interest has been devoted to these physical models, since they exhibit a connection between statistical mechanics and complex dynamics [6, 8–15]. We have known that, for any given , the Julia set of is continuous in the Hausdorff distance for any except two points [11]. Whether the Hausdorff dimension of is also continuous for any except two points? From the proof of the main result in [10, 11], for even integer , it is easy to see that is hyperbolic in the real axis except countable points. Except at most three points from those countable points, is subhyperbolic but not hyperbolic; though the dynamical property of is simple, it is difficult to compute all the iteration number of critical points which are eventually equal to the repelling fixed points in the iteration of . Therefore, we cannot give a quantitative analysis for the corresponding critical points when the parameter is close to the above points. For any odd integer , there exist at least two real numbers such that and are Feigenbaum-like maps [15]. As we have seen, for the simplest Feigenbaum quadratic polynomials, the continuity of Hausdorff dimension of its Julia sets is unknown. Based on the above reason, we just consider the case for . We define the following constants: We have the following result.
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