  We study the probability that an AR(1) Markov chain $X_{n+1}=aX_n+\xi_{n+1}$,
where $a\in(0,1)$ is a constant, stays non-negative for a long time. We find
the exact asymptotics of this probability and the weak limit of $X_n$
conditioned to stay non-negative, assuming that the i.i.d.\ innovations $\xi_n$
take only two values $\pm1$ and $a \le \frac23$. This limiting distribution is
quasi-stationary. It has no atoms and is singular with respect to the Lebesgue
measure when $\frac12< a \le \frac23$, except for the case $a=\frac23$ and
$\mathbf{P}(\xi_n=1)=\frac12$, where this distribution is uniform on the
interval $[0,3]$. This is similar to the properties of Bernoulli convolutions.
  It turns out that for the $\pm1 $ innovations there is a close connection
between $X_n$ killed at exiting $[0, \infty)$ and the classical dynamical
system defined by the piecewise linear mapping $x \mapsto \frac1a x + \frac12
\pmod 1$. Namely, the trajectory of this system started at $X_n$
deterministically recovers the values of the killed chain in reversed time! We
use this property to construct a suitable Banach space, where the transition
operator of the killed chain has the compactness properties that let us apply a
conventional argument of Perron--Frobenius type. The difficulty in finding such
space stems from discreteness of the innovations.
