  We consider Markov decision processes with synchronizing objectives, which
require that a probability mass of $1-\epsilon$ accumulates in a designated set
of target states, either once, always, infinitely often, or always from some
point on, where $\epsilon = 0$ for sure synchronizing, and $\epsilon \to 0$ for
almost-sure and limit-sure synchronizing.
  We introduce two new qualitative modes of synchronizing, where the
probability mass should be either positive, or bounded away from $0$. They can
be viewed as dual synchronizing objectives.
  We present algorithms and tight complexity results for the problem of
deciding if a Markov decision process is positive, or bounded synchronizing,
and we provide explicit bounds on $\epsilon$ in all synchronizing modes. In
particular, we show that deciding positive and bounded synchronizing always
from some point on, is coNP-complete.
