We establish that for a $C^r$-generic surface repeller ($1<r \leq \infty$), its Hausdorff and box dimensions are exactly the unique zero of the sub-additive topological pressure function. As an application of our framework, we show that the attractor of a uniformly non-conformal and weakly irreducible planar non-linear iterated function system (IFS) satisfying the strong separation condition (SSC) attains its expected Hausdorff and box dimensions. Furthermore, as a direct consequence of our generic repeller theorem, we deduce that this dimension formula also holds for a generic $C^r$ planar IFS satisfying the SSC. Using this approach, we also extend the dichotomy result for graphs of Weierstrass-type functions of Ren and Shen (2021) by weakening their real-analytic requirement to arbitrary $C^r$ regularity for $r>1$.
We prove the repelling-density conjecture for cubic rational maps over $\mathbf C_p$: for every prime $p$, the classical Julia set is the closure of the classical repelling periodic points. By B\'ezivin's theorem, it suffices to prove that a nonempty classical Julia set contains a classical repelling periodic point. Th...
We prove that the upper box dimension of an inhomogeneous attractor of a $C^1$ iterated function system in $\mathbb{R}^d$ is bounded above by the maximum of the singularity dimension of the system and the upper box dimension of the condensation set. This extends the upper bound of Burrell and Fraser (2020) for inhomoge...
We prove a sheeting theorem for stable solutions to the Allen--Cahn equation in ambient dimensions $2$ through $10$. We show that weak convergence of the measures $\varepsilon|Du_\varepsilon|^2\,dX$ to a hyperplane with integer multiplicity implies that the zero sets are disjoint smooth graphs. As a consequence, stable...
Yong Liu, Tian-Ci Luo, Ke-Lei Wang et al.· 0 citations
We study the Yamabe--Escobar equation prescribing positive constant scalar curvature and minimal boundary. The compactness conjecture asks whether being conformally equivalent to the round hemisphere is the only obstruction for its set of positive solutions to be compact. In this paper we show that this is true for dim...
We study the Hausdorff dimension of attracting sets of three-dimensional flows whose one-dimensional quotient dynamics is non-uniformly expanding. Our main assumption is that the quotient map admits a full-branch uniformly expanding Gibbs--Markov induced map. We first prove that every such attractor contains a hyperbol...
We establish characterizations of planar quasiconformal mappings in weighted Bergman spaces $A_\alpha^p(\mathbb D)$. For $p>0$ and $\alpha>-1$, our characterization involves the convergence of certain integrals involving superlevel set areas, the Hausdorff measure of boundary superlevel sets, and the averaged derivativ...
P. Koskela, Cheng-Xi Li, A. Pandey· 0 citations
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