  A one-dimensional confined Nonlinear Random Walk is a tuple of $N$
diffeomorphisms of the unit interval driven by a probabilistic Markov chain.
For generic such walks, we obtain a geometric characterization of their ergodic
stationary measures and prove that all of them have negative Lyapunov
exponents.
  These measures appear to be probabilistic manifestations of physical measures
for certain deterministic dynamical systems. These systems are step skew
products over transitive subshifts of finite type (topological Markov chains)
with the unit interval fiber.
  For such skew products, we show there exist only finite collection of
alternating attractors and repellers; we also give a sharp upper bound for
their number. Each of them is a graph of a continuous map from the base to the
fiber defined almost everywhere w.r.t. any ergodic Markov measure in the base.
The orbits starting between the adjacent attractor and repeller tend to the
attractor as $t \to +\infty$, and to the repeller as $t \to -\infty$. The
attractors support ergodic hyperbolic physical measures.
