AI 中文总结
本文研究具有Lipschitz极限集的射影Anosov表示的Falconer泛函的临界指数，在稠密性和不可约性假设下证明其等于极限集的Hausdorff维数，还构造相关反例并得到独立意义的结果。
AI 中文摘要
我们研究具有Lipschitz极限集的射影Anosov表示的Falconer泛函的临界指数及其与极限集的Hausdorff维数的关系。在该表示满足稠密性和不可约性假设的条件下，我们扩展了Pozzetti、Sambarino和Wienhard的结果，证明该临界指数等于极限集的Hausdorff维数。我们还证明了这些假设对于他们及本文所采用的策略是必要的。为此，我们研究了$\boldsymbol{\text{SO}}(p,p)$在$\boldsymbol{\text{R}}^{p,p}$的极大迷向子空间空间上的作用，并提供了具有虚拟上同调维数$p$的格的$\boldsymbol{\text{H}}^{p,q}$-凸余紧表示的例子。这些例子不仅对$\boldsymbol{\text{SO}}(p,q+1)$中关于不可约表示的看似无害的断言给出了反例，还具有独立意义：它们允许黎曼对称空间的等变类空嵌入到$\boldsymbol{\text{H}}^{p,q}$中，而均匀格在该空间上余紧作用。
英文摘要
We study the critical exponent of the Falconer functional for projective Anosov representations with Lipschitz limit sets and its relationship to the Hausdorff dimension of the limit set. Extending a result of Pozzetti, Sambarino, and Wienhard, we show, under density and irreducibility assumptions on the representation, that this exponent equals the Hausdorff dimension of the limit set. We also show that these assumptions are necessary for the strategy used there and in the present paper. To this end, we study the action of $\mathrm{SO}(p,p)$ on the space of maximal isotropic subspaces of $\mathbb R^{p,p}$ and provide examples of $\mathbf H^{p,q}$-convex-cocompact representations of lattices with virtual cohomological dimension $p$. Besides yielding counterexamples to seemingly innocent claims about irreducible representations in $\mathrm{SO}(p,q+1)$, these examples are of independent interest: they admit equivariant spacelike embeddings of Riemannian symmetric spaces into $\mathbf H^{p,q}$ on which uniform lattices act cocompactly.
