We construct an explicit family of arithmetic Teichmüller curves ${𝒞}_{2k}$, $k\in \mathbb{N}$, supporting $\mathrm{SL}(2,ℝ)$-invariant probabilities ${\mu }_{2k}$ such that the associated $\mathrm{SL}(2,ℝ)$-representation on $L^2({𝒞}_{2k},{\mu }_{2k})$ has complementary series for every $k\ge 3$. Actually, the size of the spectral gap along this family goes to zero. In particular, the Teichmüller geodesic flow restricted to these explicit arithmetic Teichmüller curves ${𝒞}_{2k}$ has arbitrarily slow rate of exponential mixing.

We introduce the notion of exponential limit shadowing and show that it is a persistent property near a hyperbolic set of a dynamical system. We show that Ω-stability implies the exponential limit shadowing property.

We study the dynamics of the Teichmüller flow in the moduli space of abelian differentials (and more generally, its restriction to any connected component of a stratum). We show that the (Masur-Veech) absolutely continuous invariant probability measure is exponentially mixing for the class of Hölder observables. A geometric consequence is that the $SL(2,ℝ)$ action in the moduli space has a spectral gap.