  A mechanism of inflation from higher dimensions compactification is studied.
An Early Universe capable of providing exponential growth for some dimensions
and exponential contraction for others, giving therefore an explanation for the
big size of the observed four dimensional Universe as well as the required
smallness of the extra dimensions is obtained. The mechanism is formulated in
the context of dynamical space time theory which produces a unified picture of
dark energy, dark matter and can also provides a bounce for the volume of the
universe. A negative vacuum energy puts an upper bound on the maximum volume
and the bounce imposes a lower bound. So that in the early universe the volume
oscillates, but in each oscillation the extra dimensions contract
exponentially, and the ordinary dimension expand exponentially. The dynamical
space time theory provides a natural way to exit from the inflation
compactification epoch since the scalar field that drives the vacuum energy can
smoothly climb into small positive values of vacuum energy, which is the end of
the inflation compactification. A semi analytic solution for a step function
potential is also studied, where all of these effects are shown. Especially the
jump of the vacuum energy affect only on the derivative of dynamical space time
vector field, and not the volume or it's derivatives, which match smoothly.
