We discovered the following operator which acts on the space of polynomials (or series) of the type $\sum a_{ij}x^iy^j$ where $a_{ij}$ belong to a field $k$ of characteristic two.

For a point $p\in (k^*)^2$ and $f$ as above, we define $$S_p f(z) = f(z) + \sqrt{f(z^2/p)f(p)}.$$ Here we multiply and divide points coordinatewise, i.e. $z^2=(z_1,z_2)^2=(z_1^2,z_2^2)$.

Note that $\mathrm{char}(k) = 2$ facilitates taking the square root, so $S_p f$ is again a polynomial (series).

The properties are as follows: $S_pS_pf=S_pf$ and $(S_pf)(p)=0$.

We can take several points $p_1,p_2,\dots\in (k^*)^2$ and look on the dynamic generated by $S_{p_i}$.

The question is as follows: have you seen something like that? Could you advise anything on studying of this dynamic?

In a sense $S_p$ changes the curve $f(z_1,z_2)=0$ to a curve $(S_pf)(z_1,z_2)=0$ which passes through $p$. So the limit of applications of $S_{p_i}$ will be a curve which passes through all $p_i$.

Motivation: this operator appeared as a detropicalization of a certain operator in sandpiles. See https://arxiv.org/abs/1509.02303 for details about the problem. The idea is that in the limit, sandpile pictures become tropical curves. And adding a grain to a point corresponds in the limit to the following operator $G_p$ -- we take the face of the tropical curve where $p$ belongs and contract this face until this new tropical curve passes through $p$. Trying to interpret $G_p$ algebraically we ended up with the above operator $S_p$. We can not reproduce the same behavior for other characteristics of the field.

Added: another function like that is $S_p f(z) = f(z)+\frac{f(\sqrt{zp})^2}{f(p)}$ in a field of characteristic two.

We discovered the following operator which acts on the space of polynomials (or series) of the type $\sum a_{ij}x^iy^j$ where $a_{ij}$ belong to a field $k$ of characteristic two.

For a point $p\in (k^*)^2$ and $f$ as above, we define $$S_p f(z) = f(z) + \sqrt{f(z^2/p)f(p)}.$$ Here we multiply and divide points coordinatewise, i.e. $z^2=(z_1,z_2)^2=(z_1^2,z_2^2)$.

Note that $\mathrm{char}(k) = 2$ facilitates taking the square root, so $S_p f$ is again a polynomial (series).

The properties are as follows: $S_pS_pf=S_pf$ and $(S_pf)(p)=0$.

We can take several points $p_1,p_2,\dots\in (k^*)^2$ and look on the dynamic generated by $S_{p_i}$.

The question is as follows: have you seen something like that? Could you advise anything on studying of this dynamic?

In a sense $S_p$ changes the curve $f(z_1,z_2)=0$ to a curve $(S_pf)(z_1,z_2)=0$ which passes through $p$. So the limit of applications of $S_{p_i}$ will be a curve which passes through all $p_i$.

Motivation: this operator appeared as a detropicalization of a certain operator in sandpiles. See https://arxiv.org/abs/1509.02303 for details about the problem. The idea is that in the limit, sandpile pictures become tropical curves. And adding a grain to a point corresponds in the limit to the following operator $G_p$ -- we take the face of the tropical curve where $p$ belongs and contract this face until this new tropical curve passes through $p$. Trying to interpret $G_p$ algebraically we ended up with the above operator $S_p$. We can not reproduce the same behavior for other characteristics of the field.