Lower bounds for the dynamically defined measures

Abstract

The dynamically defined measure (DDM) $Φ$ arising from a finite measure $ϕ_0$ on an initial $σ$-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 $Λ$ such that $Λ\llϕ_0$. In particular, DDMs arising from the Hellinger integral $\mathcal{J}_α(Λ,ϕ_0)\geq\mathcal{H}^{α,0}(Λ,ϕ_0)\geq\mathcal{H}_α(Λ,ϕ_0)$ are constructed with $\mathcal{H}_{0}\left(Λ,ϕ_0\right)(Q) = Φ(Q)$, $\mathcal{H}_{1}\left(Λ,ϕ_0\right)(Q) = Λ(Q)$, and \[Φ(Q)^{1-α}Λ(Q)^α\geq\mathcal{J}_α\left(Λ,ϕ_0\right)(Q)\] for all measurable $Q$ and $α\in[0,1]$, and further computable lower bounds for them are obtained and analyzed. The function $(0,γ]\ownsα\longmapsto\mathcal{H}_α(Λ,ϕ_0)$ is computed explicitly for $γ\geq 1$ such that $\int(dΛ/dϕ_0)^{γ-1}dΛ<\infty$ in the case of a discrete ergodic decomposition of $Λ$, and the other two functions are computed under the additional condition of the equivalence of $ϕ_0$ and $Λ$. In particular, if $Λ$ is ergodic, it is shown that the first function is completely determined by the $Λ$-essential supremum (infimum) of $dΛ/dϕ_0$ for all $0<α<1$ ($1<α\leqγ$), and, if it is continuous at $0$, the above inequalities become equalities. The computation of it enables an explicit computation of some DDMs arising as outer measure approximations with respect to it, which demonstrates that this technique allows to obtain new measures, and that such measures can have phase transitions with respect to the DDM specifying the covering sets.