Let us consider a sequence of continuous functions $g\_{q}:ℝ^2\to ℝ^2$. Let $(A\_{q})\_{q\geq 1}$ be a sequence of compact sets in $ℝ^2$. Assuming that each function $g\_{q}$ is topologically mixing in $A\_{q}$ for all $q\geq 1$, i.e., for every open subsets $U,V$ of $ℝ^2$ such that $U\cap A\_{q}$ and $V\cap A\_{q}$ are non-empty, there exists $k\_0=k\_0(q,U,V)$ such that for every $k\geq k\_0$ the set $g\_{q}^{k}(U)\cap V\cap A\_{q}$ is non-empty.


My **problem** is about finding a common set $A$ in which all the functions $g\_{q}$ are topologically mixing in $A$. My approach based on several techniques (symbolic dynamics) leads to this diophantine equation:


$$16(n+1)^2q^8+16(n+1)^2q^6+1=m^2$$


and the **main problem** is equivalent to the fact that the above diophantine equation has an infinite number of positive integers solutions $q,n,m$. So, the **question** is how one can prove that the above diophantine equation has an infinite number of positive integers solutions $q,n,m$.



Diophantine equation that has an infinite number of positive integers solutions
It is just Pell's equation $m^2-Ny^2=1$, for $N=16q^6(q^2+1) $. Thus even for fixed $q$ it has infinitely many solutions, that was proved by Lagrange and you may find the proof in many textbooks.


