Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Entropy of Hilbert metrics and length spectrum of Hitchin
  representations in $\mathrm{PSL}(3,\mathbb{R})$   We prove a sharp inequality between the Blaschke and Hilbert distance on a
proper convex domain: for any two points $x$ and $y$, \[d^B(x,y) < d^H(x,y)
+1.\] We obtain two interesting consequences: the first one is the volume
entropy rigidity for Hilbert geometries : for any proper convex domain of
$\mathbb{R}\mathbf{P}^n$, the volume of a ball of radius $R$ grows at most like
$e^{(n-1)R}$. The second consequence is the following fact: for any Hitchin
representation of a surface group into $\mathrm{PSL}(3,\mathbb{R})$, there
exists a Fuchsian representation $j$ in $\mathrm{PSL}(2,\mathbb{R})$ such that
the length spectrum of $j$ is uniformly smaller than the length spectrum of
$\rho$.

12
The categories of this paper are: nucl-th
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci
14
The categories of this paper are: math-ph
19
The categories of this paper are: eess
1
The categories of this paper are: cs
5
The categories of this paper are: gr-qc
11
The categories of this paper are: hep-lat
34
The categories of this paper are: chem-ph
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
27
The categories of this paper are: comp-gas
3
The categories of this paper are: physics