About the transformation $(x,y)\mapsto (ax,\frac{y}{a^2})$

Question

Suppose $f(x)=\frac{1}{x^2}$ and $g_k(x)=kx$ for parameter $k\in\Bbb R_{>0}.$

$g_k(x)$ maps onto itself as $k$ increases. The transformation maps a point on $f(x)$ to another point on $f(x).$

How would you write down this transformation more mathematically? Is it related to another, more well-known transformation?

I can write the transformation as $(x,y)\mapsto (ax,\frac{y}{a^2})$ for parameter $a$. I noticed that this is very similar to the squeeze mapping. The squeeze mapping is defined as: $(x,y)\mapsto (ax,\frac{y}{a}).$

Squeeze mappings are a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is not a rotation or shear mapping.

Does the transformation $(x,y)\mapsto (ax,\frac{y}{a^2})$ have any important uses? Does this transformation preserve anything?

I know that squeeze mappings are pretty important. They find applications in special relativity with Lorentz boosts.

Answer 1

Since it can be written as $$\begin{cases} x' = ax \\ y' = \dfrac{y}{a^2} \end{cases}$$ we see, for example, that $$(x')^2y' = x^2y = \text{const}$$ So, the curves $x^2y = c$ are invariant under this transformation for each constant $c$.