I am a brand-new comer in dynamical system. I find it interesting that when defining ergodicity of classical dynamical system $(X,\sigma)$, they use $\mu(\sigma^{-1}(E))$ there. Since $\sigma$ is a homeomorphism, why not use $\sigma$ directly?
Moreover, from a classical dynamical system $(X,\sigma)$, we can induce a $C^*$-dynmaical system $(C(X),\mathbb{Z},\sigma)$ where $\sigma_n(f):=f\circ\sigma^{-n}$. Here, they use inverse homeomorphism again. Is it a convention in history or with theoretical background? It would be nice if somebody could explain a little bit. Thank you in advance!
If you have a measurable function $\sigma\colon X \to Y$ and a measure $\mu$ on $X$ you could define a measure $\sigma_\# \mu$ (push forward) as $$ \sigma_\# \mu(E) = \mu(\sigma^{-1}(E)). $$ So, measures go forward and use the inverse of the mapping. Obviously if $\sigma$ is an homeomorphism you could also go backward but this is not so natural...
When applied to sets the inverse of a map has best properties than the direct map itself. For example: $$ \sigma^{-1}(A \cap B) = \sigma^{-1}(A) \cap \sigma^{-1}(B) $$ (the same is true for the union). This explains why continuous and measurable functions can by defined by properties of the inverse map.