  We study the scalar sector of the Two Measures Field Theory (TMT) model in
the context of cosmological dynamics. The scalar sector includes the inflaton
\phi and the Higgs \upsilon fields. The model possesses gauge and scale
invariance. The latter is spontaneously broken due to intrinsic features of the
TMT dynamics. In the model with the inflaton \phi alone, in different regions
of the parameter space the following different effects can take place without
fine tuning of the parameters and initial conditions: a) Possibility of
resolution of the old cosmological constant problem: this is done in a
consistent way hinted by S. Weinberg in his comment concerning the question of
how one can avoid his no-go theorem. b) The power law inflation without any
fine tuning may end with damped oscillations of $\phi$ around the state with
zero cosmological constant. c) There are regions of the parameters where the
equation-of-state w=p/\rho in the late time universe is w<-1 and w
asymptotically (as t\to\infty) approaches -1 from below. This effect is
achieved without any exotic term in the action. In a model with both \phi and
\upsilon fields, a scenario which resembles the hybrid inflation is realized
but there are essential differences, for example: the Higgs field undergos
transition to a gauge symmetry broken phase <\upsilon>\neq 0 soon after the end
of a power law inflation; there are two oscillatory regimes of \upsilon, one
around \upsilon =0 at 50 e-folding before the end of inflation, another -
during transition to a gauge symmetry broken phase where the scalar dark energy
density approaches zero without fine tuning; the gauge symmetry breakdown is
achieved without tachyonic mass term in the action.
