Power operations and Lambda-structure-like lifts of Frobenius in $E_\infty$-geometry? 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 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 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, the $E_\infty$-analog of "this" are the power operations 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? 
 A: Although I don't have much to say here, perhaps the following example is worth pointing out. Let $R$ be a $K(1)$-local $E_\infty$-ring under ($p$-adic) complex $K$-theory $KU$. Then there exists a power operation $\theta: \pi_0 R \to \pi_0 R$ such that: 


*

*$\psi(x) \stackrel{\mathrm{def}}{=} x^p + p \theta(x)$ defines a ring homomorphism from $\pi_0 R \to \pi_0 R$.

*$\theta$ satisfies all the identities needed to make $\psi$ a ring-homomorphism after "division by p." For instance, $\psi(x+y) = \psi(x) + \psi(y)$ implies that $$\theta(x+y) = \theta(x) + \theta(y) + \frac{x^p - y^p - (x+y)^p}{p},$$
where the last term is an integral polynomial in $x,y$ and is interpreted as such. 


$\theta$ is the basic power operation for $K(1)$-local $E_\infty$-ring spectra, as explained in these notes of Hopkins, and the algebraic structure it gives is called a "$\theta$-algebra." (These also seem to be called $p$-derivations by algebraists.)
Note in particular that $\psi$ is a lift of the Frobenius. There are generalizations of $\psi, \theta$ at higher chromatic levels, too, and there (as I understand) a modular interpretation of the resulting algebraic structure in this paper of Rezk.
