In the field of random matrix products, it seems a lot of theorems which give nice statistical properties (central limit theorem, large deviation, etc.) assume—among other things—a Zariski density property.
The problem is, I have no clue how to prove Zariski density at all!
Here is a toy example in two dimensions: Let $$A_1 = \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array} \right), \qquad A_2 = \left( \begin{array}{cc} 1 & 0 \\ 1 & 1 \\ \end{array} \right). $$
Question:
Is the semigroup generated by these two matrices,
$$ \left\langle A_1, A_2 \right\rangle = \bigcup_{n=1}^∞ \big\{A_{i_1}A_{i_2}\cdots A_{i_n}\;:\; (i_1,i_2,\ldots,i_n)\in \{1,2\}^n\big\},$$
Zariski dense in $\mathrm{SL}_2(\mathbb R)$? In other words, must every polynomial in 4 variables which vanishes on this semigroup vanish on $\mathrm{SL}_2(\mathbb R)$?