Given a sequence (M k , Q k ) k≥1 of independent, identically distributed random vectors with nonnegative components, we consider the recursive Markov chain (X n ) n≥0 , defined by the random difference equation X n = M n X n−1 + Q n for n ≥ 1, where X 0 is independent of (M k , Q k ) k≥1 . Criteria for the null recurrence/transience are provided in the situation where (X n ) n≥0 is contractive in the sense that M 1 · . . . · M n → 0 a.s., yet occasional large values of the Q n overcompensate the contractive behavior so that positive recurrence fails to hold. We also investigate the attractor set of (X n ) n≥0 under the sole assumption that this chain is locally contractive and recurrent.