This paper studies quasiconformal non-equivalence of Julia sets and limit sets. We proved that any Julia set is quasiconformally different from the Apollonian gasket. We also proved that any Julia set of a quadratic rational map is quasiconformally different from the gasket limit set of a geometrically finite Kleinian group.

Let $\Gamma =\langle I_{1}, I_{2}, I_{3}\rangle $ be the complex hyperbolic $(4,4,\infty )$ triangle group with $I_1I_3I_2I_3$ being unipotent. We show that the limit set of $\Gamma $ is connected and the closure of a countable union of $\mathbb {R}$-circles.

Given an integer $n\,\ge \,3$ , a metrizable compact topological $n$ -manifold $X$ with boundary, and a finite positive Borel measure $\mu$ on $X$ , we prove that for the typical homeomorphism $f:\,X\,\to \,X$ , it is true that for $\mu$ -almost every point $x$ in $X$ the restriction of $f$ (respectively of ${{f}^{-1}}$ ) to the omega limit set $\omega \left( f,\,x \right)$ (respectively to the alpha limit set $\alpha \left( f,\,x \right)$ ) is topologically conjugate to the universal odometer.

Given an integer $n\,\ge \,3$ , a metrizable compact topological $n$ -manifold $X$ with boundary, and a finite positive Borel measure $\mu $ on $X$ , we prove that for the typical homeomorphism $f\,:\,X\,\to \,X$ , it is true that for $\mu $ -almost every point $x$ in $X$ the limit set $\omega (f,\,x)$ is a Cantor set of Hausdorff dimension zero, each point of $\omega (f,\,x)$ has a dense orbit in $\omega (f,\,x)$ , $f$ is non-sensitive at each point of $\omega (f,\,x)$ , and the function $a\,\to \,\omega (f,\,a)$ is continuous at $x$ .