Tent property of the growth indicator functions and applications

Let $\Gamma$ be a Zariski dense discrete subgroup of a connected semisimple real algebraic group $G$. Let $k=\operatorname{rank} G$. Let $\psi_\Gamma:\mathfrak{a} \to \mathbb{R}\cup \{-\infty\}$ be the growth indicator function of $\Gamma$, first introduced by Quint. In this paper, we obtain the following pointwise bound of $\psi_\Gamma$: for all $v\in \mathfrak{a}$, $$\psi_\Gamma(v) \le \min_{1\le i\le k} \delta_{\alpha_i} \alpha_i(v)$$ where $\Delta=\{\alpha_1, \cdots, \alpha_k\}$ is the set of all simple roots of $(\mathfrak{g},\mathfrak{a})$ and $0<\delta_{\alpha_i}\le \infty$ is the critical exponent of $\Gamma$ associated to $\alpha_i$. When $\Gamma$ is $\Delta$-Anosov, there are precisely $k$-number of directions where the equality is achieved, and the following strict inequality holds for $k\ge 2$: for all $v\in \mathfrak{a}-\{0\}$, $$\psi_\Gamma(v) <\frac{1}{k}\sum_{i=1}^k \delta_{\alpha_i} \alpha_i (v).$$ We discuss applications for self-joinings of convex cocompact subgroups in $\prod_{i=1}^k \operatorname{SO}(n_i,1)$ and Hitchin subgroups of $\operatorname{PSL}(d, \mathbb{R})$. In particular, for a Zariski dense Hitchin subgroup $\Gamma<\text{PSL}(d, \mathbb{R})$, we obtain that for any $v=\operatorname{diag}(t_1, \cdots, t_d)\in \mathfrak{a}^+$, $$\psi_\Gamma (v) \le \min_{1\le i\le d-1} (t_i -t_{i+1}).$$