Orientation Preserving Homeomorphism of the Circle $S^1$

Question

Ergodic Theory. Cornfeld, Fomin, Sinai. Page 73.

Consider a homeomorphism $T$ preserving the orientation of the circle $S^1$.

what does "preserving the orientation" mean?

Answer 1

Rougly, it means that $f$ preserves the direction of a "dot" walking on $S^1$. That is to say, if you have a "dot" walking on $S^1$ clockwise, then the $f$-image of the "dot" also walks clockwise.

Mathematically, this can be formalized as follows:

Let's parameterise $S^1$ with $[0,1]$, with $0$ identified with $1$. Let $\pi:\mathbb R\to S^1$ be the universal covering projection (or, if you prever, identify $S^1$ with the quotient $\mathbb R/\mathbb Z$ and take $\pi$ be the quotient map).

Any map $f:S^1\to S^1$ has a lift $\tilde f:\mathbb R\to \mathbb R$ so that the obviuos diagram commute, in formulae $$f\circ\pi=\pi\circ\tilde f$$ Moreover, such lift is unique once we choose $\tilde f(0)\in\pi^{-1}(f(0))$.

There three cases:

$\tilde f(1)=\tilde f(0)$, but in this case you can see that $f$ is not an homeo,

$\tilde f(1) >\tilde f(0)$, in this case $f$ is orientation preserving,

$\tilde f(1)< \tilde f(0)$, in this case $f$ is orientation inversing.