Lower bounds for the dynamically defined measures

The dynamically defined measure (DDM) $\Phi$ arising from a finite measure $\phi_0$ on an initial $\sigma$-algebra on a set and an invertible map acting on the latter is considered. Several lower bounds for it are obtained and sufficient conditions for its positivity are deduced under the general assumption that there exists an invariant measure $\Lambda$ such that $\Lambda\ll\phi_0$. In particular, DDMs arising from the Hellinger integral $\mathcal{J}_\alpha(\Lambda,\phi_0)\geq\mathcal{H}^{\alpha,0}(\Lambda,\phi_0)\geq\mathcal{H}_\alpha(\Lambda,\phi_0)$ are constructed with $\mathcal{H}_{0}\left(\Lambda,\phi_0\right)(Q) = \Phi(Q)$, $\mathcal{H}_{1}\left(\Lambda,\phi_0\right)(Q) = \Lambda(Q)$, and \[\Phi(Q)^{1-\alpha}\Lambda(Q)^{\alpha}\geq\mathcal{J}_{\alpha}\left(\Lambda,\phi_0\right)(Q)\] for all measurable $Q$ and $\alpha\in[0,1]$, and further computable lower bounds for them are obtained and analyzed. It is shown, in particular, that $(0,1)\owns\alpha\longmapsto\mathcal{H}_{\alpha}(\Lambda,\phi_0)(Q)$ is completely determined by the $\Lambda$-essential supremum of $d\Lambda/d\phi_0$ for all $0<\alpha<1$ if $\Lambda$ is ergodic, and if also a condition for the continuity at $0$ is satisfied, the above inequalities become equalities. In general, for every measurable $Q$, it is shown that $[0,1]\owns\alpha\longmapsto\mathcal{J}_{\alpha}(\Lambda,\phi_0)(Q)$ is log-convex, all one-sided derivatives of $(0,1)\owns\alpha\longmapsto\mathcal{H}^{\alpha,0}(\Lambda,\phi_0)(Q)$ and $(0,1)\owns\alpha\longmapsto\mathcal{J}_{\alpha}(\Lambda,\phi_0)(Q)$ are obtained, and some lower bounds for the functions by means of the derivatives are given. Some sufficient conditions for the continuity and a one-sided differentiability of $(0,1)\owns\alpha\longmapsto\mathcal{H}_{\alpha}(\Lambda,\phi_0)(Q)$ are provided.