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}^+$, $$\psi_\Gamma (v) \le \min_{ 1\le i\le d-1} (t_i -t_{ i+1}). $$
Categories: math.GT math.DS