A good reading! If you are working in the smooth ($C^\infty$) regularity class, then the Reeb foliations are in infinite number, up to diffeomorphism. Fix a small enough compact arc $A=[a,b]$ transverse to the foliation and with one endpoint $a$ on the boundary. Then, for one of the two orientations of the boundary, the first return map $r:A\to A$ is contracting, in the sense that it is a diffeomorphism of $[a,b]$ onto $[a, r(b)]$ fixing $a$ and such that $r(x)<x$ for every $x\in(a,b]$. Conversely, every germ at $a$ of such a contraction can be realized by a Reeb foliation of the Möbius strip. Beware that, although every leaf meets $A$, the space of leaves is NOT the quotient of $A$ by $r$: because we are on the Möbius strip rather than the usual strip, there is moreover, in the holonomy of the foliation on $A$, a diffeomorphism $s:(a,b]\to(a,s(b)]$, such that $s\circ s=r$. You can realize $s$ by going across the strip along the leaves, rather than staying close to the boundary. You will see $s$ if you make a paper model and draw a Reeb component on it. The space of leaves is the quotient topological space $A/R$ where $x R y$ iff there is an $n\in Z$ such that $y=s^n(x)$: this is a circle plus one point adherent to all points of the circle.

A good reading! If you are working in the smooth ($C^\infty$) regularity class, then the Reeb foliations $F$ on the Möbius strip $M$ are in infinite number, up to diffeomorphism. Fix a small enough compact arc $A=[a,b]$ transverse to $F$ and with one endpoint $a$ on the boundary. Then, for one of the two orientations of the boundary, the first return map $r:A\to A$ is contracting, in the sense that it is a diffeomorphism of $[a,b]$ onto $[a, r(b)]$, with $a<r(b)<b$, fixing $a$ and such that $r(x)<x$ for every $x\in(a,b]$. Conversely, every germ at $a$ of such a contraction can be realized by a Reeb foliation of the Möbius strip. Beware that, although every leaf of $F$ meets $A$, the space of leaves is NOT the quotient of $A$ by $r$: because we are on the Möbius strip rather than the usual strip, there is moreover, in the holonomy of the foliation on $A$, a diffeomorphism $s:(a,b]\to(a,s(b)]$, with $a<s(b)<b$, such that $s\circ s=r$. You can realize $s$ by going across the strip along the leaves, rather than staying close to the boundary. You will see $s$ if you make a paper model and draw a Reeb component on it. The space of leaves is the quotient topological space $A/R$ where $x R y$ iff there is an $n\in Z$ such that $y=s^n(x)$: this is a circle plus one point adherent to all points of the circle. You can also see this by considering the circle $C$ which is in the middle of $M$: it meets transversely at one point every leaf, but the boundary leaf.

A good reading! If you are working in the smooth ($C^\infty$) regularity class, then the Reeb foliations are in infinite number, up to diffeomorphism. Fix a small enough compact arc $A=[a,b]$ transverse to the foliation and with one endpoint $a$ on the boundary. Then, for one of the two orientations of the boundary, the first return map $r:A\to A$ is contracting, in the sense that it is a diffeomorphism of $[a,b]$ onto $[a, r(b)]$ fixing $a$ and such that $r(x)<x$ for every $x\in(a,b]$. Conversely, every germ at $a$ of such a contraction can be realized by a Reeb foliation of the Möbius strip. Beware that, although every leaf meets $A$, the space of leaves is NOT the quotient of $A$ by $r$: because we are on the Möbius strip rather than the usual strip, there is moreover, in the holonomy of the foliation on $A$, a diffeomorphism $s:(a,b)\to(a,b')$ such that $s\circ s=r$. You can realize $s$ by going across the strip along the leaves, rather than staying close to the boundary. You will see it if you make a paper model and draw a Reeb component on it.

A good reading! If you are working in the smooth ($C^\infty$) regularity class, then the Reeb foliations are in infinite number, up to diffeomorphism. Fix a small enough compact arc $A=[a,b]$ transverse to the foliation and with one endpoint $a$ on the boundary. Then, for one of the two orientations of the boundary, the first return map $r:A\to A$ is contracting, in the sense that it is a diffeomorphism of $[a,b]$ onto $[a, r(b)]$ fixing $a$ and such that $r(x)<x$ for every $x\in(a,b]$. Conversely, every germ at $a$ of such a contraction can be realized by a Reeb foliation of the Möbius strip. Beware that, although every leaf meets $A$, the space of leaves is NOT the quotient of $A$ by $r$: because we are on the Möbius strip rather than the usual strip, there is moreover, in the holonomy of the foliation on $A$, a diffeomorphism $s:(a,b]\to(a,s(b)]$, such that $s\circ s=r$. You can realize $s$ by going across the strip along the leaves, rather than staying close to the boundary. You will see $s$ if you make a paper model and draw a Reeb component on it. The space of leaves is the quotient topological space $A/R$ where $x R y$ iff there is an $n\in Z$ such that $y=s^n(x)$: this is a circle plus one point adherent to all points of the circle.