- 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.