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The classical result of Patterson and Sullivan says that for a non-elementaryconvex cocompact subgroup $\Gamma<\text{SO}^\circ (n,1)$, $n\ge 2$, theHausdorff dimension of the limit set of $\Gamma$ is equal to the criticalexponent of $\Gamma$. In this paper, we generalize this result forself-joinings of convex cocompact groups in two ways. Let $\Delta$ be afinitely generated group and $\rho_i:\Delta\to \text{SO}^\circ(n_i,1)$ be aconvex cocompact faithful representation of $\Delta$ for $1\le i\le k$.Associated to $\rho=(\rho_1, \cdots, \rho_k)$, we consider the followingself-joining subgroup of $\prod_{i=1}^k \text{SO}(n_i,1)$:$$\Gamma=\left(\prod_{i=1}^k\rho_i\right)(\Delta)=\{(\rho_1(g), \cdots,\rho_k(g)):g\in \Delta\} .$$ (1). Denoting by $\Lambda\subset \prod_{i=1}^k\mathbb{S}^{n_i-1}$ the limit set of $\Gamma$, we first prove that$$\text{dim}_H \Lambda=\max_{1\le i\le k} \delta_{\rho_i}$$ where$\delta_{\rho_i}$ is the critical exponent of the subgroup $\rho_{i}(\Delta)$.  (2). Denoting by $\Lambda_u\subset \Lambda$ the $u$-directional limit set foreach $u=(u_1, \cdots, u_k)$ in the interior of the limit cone of $\Gamma$, weobtain that for $k\le 3$, $$ \frac{\psi_\Gamma(u)}{\max_i u_i }\le \text{dim}_H\Lambda_u \le \frac{\psi_\Gamma(u)}{\min_i u_i }$$ where$\psi_\Gamma:\mathbb{R}^k\to \mathbb{R}\cup\{-\infty\}$ is the growth indicatorfunction of $\Gamma$.

Hausdorff dimension of directional limit sets for self-joinings of hyperbolic manifolds