I'm interested in finding a continuous and ideally smooth family of real-valued monotonic functions $H_t(x)$ that interpolate between iterations of exponentials, and also iterations of logarithms. So the ideal such family would have the properties
- $H_0(x) = x$
- $H_{t+1}(x) = \exp\left(H_t(x)\right)$
- $H_{t-1}(x) = \log\left( H_t(x) \right)$
for all $t$. Do such families exist? If there are more than one, is there a categorization of the collection, or any particular such family considered canonical?
It's clear that we only need really define $H_t$ for $t \in [0, 1]$ in such a way so as to ensure smoothness, so it seems like this should reduce to the question of looking for a family of compositional $n^\textrm{th}$ roots for $\exp$, and seeing if the analytic completion of that family is smooth.
I'm aware of some similar results, e.g. there being many compositional square roots of the exponential function in $\mathbb{R}$. But from the wikipedia article, it isn't immediately clear to me if this process yields $n^\textrm{th}$ roots of $\exp$, or if so if the set of all $n^\textrm{th}$ roots are unique, or if the induced family $$\{H_{q}(x) \; : \; q \in \mathbb{Q} \cap [0, 1] \}$$
can be completed to yield a smooth family.
EDIT: Actually I think we want the following slightly stronger set of properties.
- $H_0(x) = x$
- $H_1(x) = \exp(x)$
- $H_\alpha(x) \circ H_\beta(x) = H_{\alpha + \beta}(x)$
If you have Tet(x), then $$H_n(x)=\text{Tet}(\text{Tet}^{−1}(x)+n)$$ You could visit https://math.eretrandre.org/tetrationforum/index for questions and details. Also, math.eretrandre.org/tetrationforum/showthread.php?tid=1017 for an implmentation of $\text{Tet}^{−1}(x)$ and $\text{Tet}(x)$.
Kneser's 1949 paper used exactly that equation to generate the half iterate of the exponential which the Op would call $H_{0.5}(x)$ from analytic tetration which Kneser showed could be generated from a Riemann mapping.
$$\phi(x)=\exp^{[0.5]}(x);\;\;\;\phi(\phi(x))=\exp(x)$$.