  In this paper we study the ergodicity and the related semigroup property for
a class of symmetric Markov jump processes associated with time changed
symmetric $\alpha$-stable processes. For this purpose, explicit and sharp
criteria for Poincar\'{e} type inequalities (including Poincar\'{e}, super
Poincar\'{e} and weak Poincar\'{e} inequalities) of the corresponding non-local
Dirichlet forms are derived. Moreover, our main results, when applied to a
class of one-dimensional stochastic differential equations driven by symmetric
$\alpha$-stable processes, yield sharp criteria for their various ergodic
properties and corresponding functional inequalities.
