Let the dynamical system $(\mathbb{T}^2,\mathcal{B}_{\mathbb{T}^2},m_{\mathbb{T^2}},T)$ where $m_{\Bbb{T}^2}=m_{\Bbb{T}} \times m_{\Bbb{T}}$ the product measure of the Lebesgue measure $m$ on the circle $\Bbb{T}$ and $$T(x,y)=(x+a,y+2x+b) \text{mod1}$$
It is not difficult to prove using Fubini's theorem that $T$ is measure preserving
For which $a,b \in \Bbb{R}$ is $T$ ergodic?
I'm new to ergodic theory and i would appreciate any help solving this problem.
My thoughts are to use the theorem which states that:
If $f:\Bbb{T^2} \to \Bbb{T}^2$ such that $f \in L^2(\Bbb{T}^2)$ and $f(T(x,y))=f(x,y) \Rightarrow$ $f$ is constant almost everywhere,then $T$ is ergodic.
To use this i took the Fourier expansion of $f$ in $\Bbb{T^2}$ (which converges to $f$ with respect to the $L^2$ norm) to find the form of the values $a,b$ but i could not solve it.
Can someone help me?
Thank you in advance.