A $\Lambda$-structure on a commutative ring $R$ is a ring endomorphism wich restricts to the $p$-Frobenius homomorphism after localizing at $(p)$. One may think of this as a "flow" $\Phi \colon Spec(R) \longrightarrow Spec(R)$ in arithmetic geometry, which lifts the [Fermat p-derivation](http://ncatlab.org/nlab/show/Fermat+quotient) on  the base $Spec(\mathbb{Z})$. If we allowed ourselves to denote derivations as endomorphisms, then with slight but very suggestive abuse of notation we have the picture

$$
  "\array{
    Spec(R) &\stackrel{\Phi + p \cdot \partial_p^{\Phi}}{\longrightarrow}& Spec(R)
    \\
    \downarrow && \downarrow
    \\
    Spec(\mathbb{Z}) &\stackrel{(-)^p = id + p\cdot \partial_p}{\longrightarrow}& Spec(\mathbb{Z})
  }
  "
$$

See on the $n$Lab at _[Borger's absolute geometry -- Motivation](http://nlab.mathforge.org/nlab/show/Borger's%20absolute%20geometry#Motivation)_ for more on what I have in mind here, following ideas famously promoted by James Borger and Alexadru Buium.

I would like to know if there is a sensible generalization of this from arithmetic geometry to $E_\infty$-arithmetic geometry, hence from commutative rings $R$ to $E_\infty$-rings.

Via discussion which is clearly articulated for instance starting from remark 2.2.9 in Jacob Lurie's DAGXIII _[Rational and p-adic homotopy theory](http://nlab.mathforge.org/nlab/show/Rational+and+p-adic+Homotopy+Theory)_, the $E_\infty$-analog of "this" are the [power operations](http://ncatlab.org/nlab/show/power+operation) in multiplicative cohomology theory. 

I am a little shaky on some details though. Therefore my question: what would be the good generalization of the concept of $\Lambda$-rings to $E_\infty$-algebra in the sense of Frobenius lifts and with an eye towards absolute geometry, as above? Can one say anything?