  We consider the second Ostrogradsky expansion from the number theory,
probability theory, dynamical systems and fractal geometry points of view, and
establish several new phenomena connected with this expansion.
  First of all we prove the singularity of the random second Ostrogradsky
expansion.
  Secondly we study properties of the symbolic dynamical system generated by
the natural one-sided shift-transformation $T$ on the second Ostrogradsky
expansion. It is shown, in particular, that there are no probability measures
which are simultaneously invariant and ergodic (w.r.t. $T$) and absolutely
continuous (w.r.t. Lebesgue measure). So, the classical ergodic approach to the
development of the metric theory is not applicable for the second Ostrogradsky
expansion.
  We develop instead the metric and dimensional theories for this expansion
using probabilistic methods. We show, in particular, that for Lebesgue almost
all real numbers any digit $i$ from the alphabet $A= \mathbb{N} $ appears only
finitely many times in the difference-version of the second Ostrogradsky
expansion, and the set of all reals with bounded digits of this expansion is of
zero Hausdorff dimension.
  Finally, we compare metric, probabilistic and dimensional theories for the
second Ostrogradsky expansions with the corresponding theories for the
Ostrogradsky-Pierce expansion and for the continued fractions expansion.
