Local rigidity of abelian actions of parabolic toral automorphisms with (at least) one step-2 generator

Two famous manifestations of local rigidity are KAM rigidity of Diophantine torus translations and smooth rigidity of hyperbolic or partially hyperbolic higher rank actions.
Damjanovic and Katok proved local rigidity for partially hyperbolic higher rank affine actions on tori. To complete the study of local rigidity of affine $\mathbb Z^k$ actions on the torus one needs to address the case of actions with parabolic generators.
We say that a linear map $A\in \mathrm{SL}(d,\mathbb Z)$ is (parabolic) of step $n$ if $ (A-\mathrm{Id})^n=0, $ and $ (A-\mathrm{Id})^{n-1}\ne0. $ An affine map $a(\cdot )=A(\cdot) +\alpha$ is said to be of step $n$ if $A$ is of step $n$.
We say that a $\mathbb Z^2$ affine action by parabolic elements is of step $n$ if all of its elements are of step at most $n$.
We say that an affine $\mathbb Z^2$-action $(a,b)$ is KAM-rigid under $\mu$-preserving perturbations, if there exists $\sigma \in \mathbb N$ and $r_0 \in \mathbb N$ and $\varepsilon>0$ that satisfy the following:
If $r\geq r_0$ and $(F,G)=(a+f,b+g)$ is a smooth $\mu$-preserving $\mathbb Z^2$ action such that
\begin{equation}\label{eq_Commut} (a+f)\circ (b+g)=(b+g)\circ (a+f), \end{equation}

$$ \|f\|_r\leq \varepsilon, \quad \|g\|_r\leq \varepsilon, \quad \widehat f:= \int_{\mathbb T^d} f d\lambda=0, \quad \widehat g:=\int_{\mathbb T^d} g d\lambda=0, $$
then there exists $H=\mathrm{Id} +h \in \text{Diff}^\infty_\mu (\mathbb T^d)$ such that $\|h\|_{r-\sigma}\leq \varepsilon$ and
\begin{equation}\label{eq_System} H \circ (a+f) \circ H^{-1} = a, \quad H \circ (b+g) \circ H^{-1} = b. \end{equation}

We denote by $\mathcal T(A,B)$ the set of possible translations $(\alpha,\beta)$ for affine actions $(A+\alpha,B+\beta)$, that is
$$\mathcal T(A,B):=\{\alpha, \beta\in \mathbb R^d : (A-\mathrm{Id})\beta=(B-\mathrm{Id})\alpha\}.$$

Theorem. Given a commuting pair $(A,B)$ of parabolic matrices where $A$ is step-2 ($(A-\mathrm{Id})^2=0$), we have the following dichotomy

• For any choice of $(\alpha,\beta) \in \mathcal T(A,B)$, the action of $(a,b)$ has a rank one factor that is not a (nonzero) translation and is thus not locally rigid.
• For almost every choice of $(\alpha,\beta)\in \mathcal T(A,B)$, the action of $(a,b)$ is ergodic and KAM-rigid under volume preserving perturbations.

This is a joint work with Danijela Damjanovic and Maria Saprykina.