The Ahlfors-regular conformal dimension, ARCdim(X), of a compact metric space X is the infimal Hausdorff dimension in the Ahlfors-regularly quasi-symmetric class of X. As a fractal embedded in the Riemann sphere, the Julia set J_f of a hyperbolic rational map f has Ahlfors-regular conformal dimension between 1 and 2. We have ARCdim(J_f)=2 iff J_f is the entire Riemann sphere. The other extreme case ARCdim(J_f)=1, however, contains a variety of Julia sets, including Julia sets of critically finite polynomials and Newton maps. In this talk, we show that for a critically finite hyperbolic rational map f, ARCdim(J_f)=1 if and only if there exists an f-invariant graph G containing all the critical points such that the topological entropy of the induced dynamics on G is zero. We also show that for a (possibly non-hyperbolic) critically finite rational map f, ARCdim(X)=1 is attained as the minimal Hausdorff dimension if and only if f is conjugate to the monomial map z^{\pm} or the Chebyshev polynomial. This talk is partially based on joint work with Angela Wu.

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