Let the Markov chain \(\{X_ n, n \geq 0\}\) be defined by \[ X_{n + 1} = f(X_ n) + \sigma(X_ n) \varepsilon_{n + 1},\quad n \geq 0, \] where \(f\), \(\sigma\) are measurable, \(f\) is bounded on compacts, \(0 < \sigma_ 1 \leq \sigma(x) \leq \sigma_ 2 < \infty\) for all \(x\), \(\{\varepsilon_ n, n\geq 1\}\) is an i.i.d. sequence whose common distribution has a nonzero absolutely continuous component with a positive density, and \(X_ 0\) is independent of \(\{\varepsilon_ n, n\geq 1\}\). By making use of R. L. Tweedie's criteria for ergodicity and geometric ergodicity of Markov chains, the author provides some sufficient conditions for ergodicity and geometric ergodicity of \(\{X_ n\}\) and a necessary condition for recurrence of \(\{X_ n\}\), imposed on the asymptotic behaviours of \(f\) as \(x\to \infty\) and \(x\to -\infty\). For example, in case \(f(x)/x\) has limits \(\alpha\) and \(\beta\) as \(x \to \infty\), and \(x \to -\infty\) respectively, ``\(\alpha < 1\), \(\beta < 1,\alpha \beta < 1\)'' is sufficient for geometric ergodicity, and ``\(\alpha \leq 1\), \(\beta \leq 1\), \(\alpha\beta \leq 1\)'' is necessary for recurrence.