Denote by $\Phi(x)$ the cumulative distribution function of $\Gamma$, the standard normal  distribution. More explicitly,
$$
\Phi(x)=\int_{-\infty}^x \Gamma[dt]:=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-t^2/2} dt. $$

It defines a homeomorphisms  $\Phi:\bR\to (0,1)$. Its inverse $Q=\Phi^{-1}$ is the so called *quantile function* of $\Gamma$. If $\mu$ is the uniform distribution on $[0,1]$, then $Q_\#\mu$,  the push forward of $\mu$  is the normal distribution.  The push forward  is defined  as in Iosif Pinelis' answer, $\newcommand{\bR}{\mathbb{R}}$

$$Q_\#\mu[S]= \mu\big[ Q^{-1}(S)\big],$$

for any Borel subset $S\subset \bR$.

Take an ergodic  transformation of $T$ of the $\big((0,1),\mu\big)$.  Then $Q\circ T \circ \Phi$ is an ergodic transformation of $(\bR,\Gamma)$.