This might be a basic question, but I think I'm completely missing the idea of this. The question comes from Devaney's An Introduction to Chaotic Dynamical Systems (p.g. 59, Ex. 11)

We define the notion of linear structural stability for linear maps by replacing the notion of topological conjugacy by that of linear conjugacy. Two maps $T_1,T_2:\mathbb{R}\to\mathbb{R}$ are linearly conjugate if there is a linear map $L$ such that $T_1\circ L=L\circ T_2$. $T_1(x)=ax$ is a linearly stable map if there is a neighborhood $N$ about $a$ such that for $b\in N$, then $T_2(x)=bx$ is linearly conjugate to $T_1$. Find all linearly stable maps and identify all elements of a linear conjugacy class.

Now, I think I'm reading this wrong because this seems somewhat trivial to me. All linear maps on $\mathbb{R}$ are given by $T(x)=ax$ for some $a\in\mathbb{R}$. We have that two linear maps are conjugate if there is a linear map $L$ such that $$T_1\circ L=L\circ T_2$$ Let $T_1(x)=ax, T_2(x)=bx, L(x)=rx$ for some $a,b,r\in\mathbb{R}$. Then the conjugacy condition requires that $a(rx)=r(bx)$. Since $r\neq 0$ (it must be an isomorphism), then $ax=bx$ for all $x$, or $a=b$.

This seems to suggest that the classes are just elements of $\mathbb{R}$, but I think I'm misreading this. Maybe the question is asking for all linearly stable continuous maps, in which case I could use some hints since this seems like a very long and difficult exercise.


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