arXiv subjects

Zhufeng Yao

Publications and source records attributed to Zhufeng Yao.

5 recordsLinked to original sources

arXiv2026

Non-Hitchin Borel Anosov representations from surface groups to $\mathrm{SL}_{3k}\mathbb{R}$

We use the Labourie--Wentworth's formula and the thermodynamic formalism to show that, along the slice constructed by Bronstein--Davalo, the logarithmic top-eigenvalue length spectrum has a uniformly positive second variation near the Barbot representation. As a major application, we show that every closed surface group admits a non-Hitchin Borel Anosov representation into $\mathrm{SL}_{3k}\mathbb{R}$ for every $k\geqslant 1$. In particular, we obtain the first such examples in the even dimensions $6k$. We also study the local behavior of related objects of this slice around the Barbot representation, including the Lyapunov exponent of the flat bundle, the Hausdorff dimension of the limit set, and the Hilbert entropy of the representation.

math.GT↗

arXiv2026

Geometric finiteness in paracomplex hyperbolic spaces

We develop a framework for studying discrete subgroups of $\mathsf{PGL}(d+1,\mathbb{R})$ via the paracomplex hyperbolic space $\mathbb{H}_τ^d$, a rank-$1$ pseudo-Riemannian symmetric space. We characterize projective transverse, relatively Anosov, and Anosov subgroups in terms of properly discontinuous, geometrically finite, and convex-cocompact actions respectively on their weak hulls, which are canonical flow spaces in the spacelike unit tangent bundle of $\mathbb{H}_τ^d$. A key ingredient is the construction of a Busemann-type horofunction on the spacelike unit tangent bundle with the properties needed to describe cuspidal geometry. We further prove for relatively Anosov subgroups that the geodesic flows on their weak hulls are uniformly hyperbolic, giving a relative analogue of the Axiom A property.

math.DG↗

arXiv2026

Hausdorff Dimension of Anosov Subgroups' Limit Sets with Special Self-Affine Complexity

Let $Γ\subset \mathsf{PGL}(d,\mathbb{R})$ be an irreducible projective Anosov subgroup and let $Λ^1(Γ)$ be its projective limit set. Viewing $Λ^1(Γ)$ as an analogue of a self-affine set, we investigate the Hausdorff dimension of $Λ^1(Γ)$ under specific assumptions regarding its affine complexity: 1. If $Λ^1(Γ)$ is of full Hausdorff dimension, then $d= 2$ and $Γ$ is a cocompact lattice. 2. If $d = 3$ and $Γ$ is the image of a closed surface group under an irreducible Anosov representation, then $Λ^1(Γ)$ never has Hausdorff dimension $1$ unless the representation is Hitchin. 3. If the limit set $Λ^1(Γ)$ exhibits a partial quasi-self-similarity (in the sense of Falconer~\cite{falconerselfsimilar1}) -- which can be implied by the ``regular distortion property'' of $Γ$ -- then the Hausdorff dimension of $Λ^1(Γ)$ equals the critical exponent of the first simple root. An application of this result is the computation of the Hausdorff dimension of the limit set for arbitrary $Θ$-positive representations of convex cocompact Fuchsian groups.

math.DG↗

arXiv2026

Critical Exponent Rigidity for $Θ-$positive Representations

We prove for a $Θ-$positive representation from a discrete subgroup $Γ\subset \mathsf{PSL}(2,\mathbb{R})$, the critical exponent for any $α\in Θ$ is not greater than one. When $Γ$ is geometrically finite, the equality holds if and only if $Γ$ is a lattice.

math.DG↗

arXiv2026

Entropy Rigidity for Maximal Representations

Let $Γ\subset \mathsf{PSL}(2,\mathbb{R})$ be a lattice and $ρ:Γ\to \mathsf{Sp}(2n,\mathbb{R})$ be a maximal representation. We show that $ρ$ satisfies a measurable $(1,1,2)-$hypertransversality condition. With this we define a measurable Gromov product and the Bowen-Margulis-Sullivan measure associated to the unstable Jacobian introduced by Pozzetti, Sambarino and Wienhard. As a main application, we prove a strong entropy rigidity result for $ρ$.

math.DG↗