I am working on a problem relating to Lyapunov exponents of products of random matrices, and this has led me to the following question which I suspect is best approached using techniques outside my area of expertise:

> Let $\tau \colon SL_2^\pm(\mathbb{R})^k \to \mathbb{R}$ be a function which has the form
$$\tau(A_1,\ldots,A_k):=\mathrm{tr }\left(A_{i_n}A_{i_{n-1}}\cdots A_{i_2}A_{i_1}\right)$$
for some fixed indices $i_1,i_2,\ldots,i_n \in \{1,\ldots,k\}$. Can $\tau$ have a local maximum or a local minimum at one of its zeros?

My main area of mathematical expertise is analysis, so I apologise if this is actually very easy to solve using tools with which I am unfamiliar. Since I am not sure which aspects of the problem are most significant in its solution I have taken a somewhat scattershot approach to tagging: please feel free to adjust the tags if you feel that they are inappropriate.

Some remarks on the question:

 - Local maxima and minima are of course not strict, since if $B \in SL_2(\mathbb{R})$ is close to the identity then $(B^{-1}A_1B,\ldots,B^{-1}A_kB)$ is close to, and typically distinct from, $(A_1,\ldots,A_k)$ but is taken to the same value by $\tau$.
 - It is certainly possible for a function of this type to have a local minimum. For example, if $k=1$, $\tau(A_1):=\mathrm{tr }A_1^2$ then a local minimum is achieved when $A_1$ has trace zero and determinant $-1$. The value of $\tau$ at the minimum in this case is $2$.
 - A solution which treated only the case $k=2$, but in which $n$ and $i_1,\ldots,i_n \in \{1,2\}$ were allowed to be arbitrary, would still be interesting to me.