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Motivated by some questions in the dimension theory of self-affine sets, a colleague and I are interested in the freeness (or otherwise) of the subsemigroup of $SL_\pm(2,\mathbb{R})$ generated by the matrices of determinant $\pm1$ which are proportional to $$A_1:=\left(\begin{array}{cc}0.85&0.04\\-0.04&0.85\end{array}\right),\quad A_2:=\left(\begin{array}{cc}0.20&-0.26\\0.23&0.22\end{array}\right),\quad A_3:=\left(\begin{array}{cc}-0.15&0.28\\0.26&0.24\end{array}\right).$$ In fact, our question is slightly more specific: we would like to know that there exist no nontrivial equations between these three matrices which (up to dividing out the determinant) have the form $A_{i_1}\cdots A_{i_n}=A_{j_1}\cdots A_{j_n}$, that is, the words on each side of the equation have the same length.

(If you are curious about why we choose these very specific matrices, they are the linear parts of the affinities which define the classical Barnsley fern. I would also be interested to know of very nearby triples of matrices which generate a free semigroup, if their entries can be explicitly written down. ( I am aware that freeness is satisfied Lebesgue a.e.))

I am aware that it is in general a very difficult problem to determine when a set of matrices generates a free semigroup, and that the problem is known to be computationally undecidable in dimensions higher than two. However, difficult general problems can still have easy special cases. Can anyone on this forum suggest a method for ruling out the possibility of nontrivial equations between normalised products of these matrices?

Thanks in advance!

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    $\begingroup$ It is a little jarring to have floating point numbers. If you really mean that the entries are approximate, then with probability $1$ your semigroup is free. $\endgroup$
    – Igor Rivin
    Commented Nov 16, 2015 at 22:29
  • $\begingroup$ Good point. Unfortunately almost sure results are two-a-penny in dimension theory, so I am interested in matrix triples whose entries can be written down explicitly, i.e. matrices with rational or algebraic entries. Close explicit approximations to the above matrices having the freeness property would be very interesting. I've edited the question to better reflect this. $\endgroup$
    – Ian Morris
    Commented Nov 16, 2015 at 22:39
  • $\begingroup$ Number $e/10$ can be effectively approximated. Are such entries allowed? $\endgroup$
    – markvs
    Commented Dec 10, 2020 at 23:25

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