Lower bounds for the dynamically defined measures

TL;DR

This paper investigates lower bounds and positivity conditions for dynamically defined measures (DDMs) derived from a base measure and an invertible transformation, providing explicit formulas and conditions for their computation and properties.

Contribution

It introduces new lower bounds for DDMs, establishes conditions for their positivity, and computes explicit formulas for these measures under ergodic and invariance assumptions.

Findings

Derived explicit formulas for DDMs in ergodic cases.

Established conditions for positivity of DDMs.

Demonstrated phase transitions in measures based on covering sets.

Abstract

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. The function $(0,\gamma]\owns\alpha\longmapsto\mathcal{H}_{\alpha}(\Lambda,\phi_0)$ is computed explicitly for $\gamma\geq 1$ such that $\int(d\Lambda/d\phi_0)^{\gamma-1}d\Lambda<\infty$ in the case of a discrete ergodic decomposition of $\Lambda$, and the other two functions are computed under the additional condition of the equivalence of $\phi_0$ and $\Lambda$. In particular, if $\Lambda$ is ergodic, it is shown that the first function is completely determined by the $\Lambda$-essential supremum (infimum) of $d\Lambda/d\phi_0$ for all $0<\alpha<1$ ($1<\alpha\leq\gamma$), 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.