  In this paper we study a class of \emph{self-consistent dynamical systems},
self-consistent in the sense that the discrete time dynamics is different in
each step depending on current statistics. The general framework admits popular
examples such as coupled map systems. Motivated by an example of M. Blank, we
concentrate on a special case where the dynamics in each step is a $\beta$-map
with some $\beta \geq 2$. Included in the definition of $\beta$ is a parameter
$\varepsilon > 0$ controlling the strength of self-consistency. We show such a
self-consistent system which has a unique absolutely continuous invariant
measure (acim) for $\varepsilon=0$, but at least two for any $\varepsilon > 0$.
With a slight modification, we transform this system into one which produces a
phase transition-like behavior: it has a unique acim for $0< \varepsilon <
\varepsilon^*$, and multiple for sufficiently large values of $\varepsilon$. We
discuss the stability of the invariant measures by the help of computer
simulations employing the numerical representation of the self-consistent
transfer operator.
