Critical Exponent Rigidity for $\mathrm{\Theta }-$positive Representations

We prove for a $\mathrm{\Theta }-$positive representation from a discrete subgroup $\mathrm{\Gamma }\subset \mathsf{P}\mathsf{S}\mathsf{L}(2,\mathbb{R})$, the critical exponent for any $\alpha \in \mathrm{\Theta }$ is not greater than one. When $\mathrm{\Gamma }$ is geometrically finite, the equality holds if and only if $\mathrm{\Gamma }$ is a lattice.