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I assume your map is surjective, thus an increasing $C^k$ diffeo $f:(0,1)\to(0,1)$ (say $1\le k\le\infty).$ The latter, as a discrete dynamical system, turns out to be $C^k$ conjugated with the shift by translation $t\mapsto t+1$ on $\mathbb{R}$. Now if $h:(0,1)\to\mathbb{R}$ is such an (increasing) conjugation, define a field $X$ by taking for all $t\in (0,1)$ $$X(t)=\frac{1}{h^'(t)}.$$ Then $f$ is the flow of $X$ at time $1$, just because $u:=h^{-1}:\mathbb{R}\to(0,1)$ solves the autonomous ODE $u'=X(u).$ So the solution at time $1$ corresponding to the initial value $t$ at time $0$ is actually $u\left(h(t)+1\right)=f(t).$

Rmk. The general fact behind is: for a diffeo on a manifold, being a time-one map of a flow, is a property invariant by smooth conjugation -the flow transforms by conjugation, and its generator is the pull back of $X$. And the shift $x\mapsto x+1$ is, of course, the map at time 1 of a flow.

Construction of the conjugation. Note that here a conjugation is an increasing diffeo $h:(0,1)\to\mathbb{R}$ solving the linear functional equation $h\left(f(x)\right) = h(x)+1.$ A solution of class $C^k$ can be constructed fixing any smooth diffeo $h_0:\left[\frac{1}{2},f(\frac{1}{2})\right]\to[0,1]$ with the convenient $k$-jets at the end-points of its domain, and extending it (uniquely) to a diffeo $h:(0,1)\to\mathbb{R}$ by means of the functional equation -the condition is that the $k$-jet of $h_0\left(f(x)\right)$ at $x=\frac{1}{2}$ and the $k$-jet of $h_0(x)+1$ at $x=f(\frac{1}{2})$ should coincide, in order that the glueing be $C^k$.

A maybe more clear way of achieving that, is: first fix any $C^k$-germ $H$ of local diffeo with $H(1/2)=0$ and $H'(0)>0$. Then take $h_0$ as a smooth diffeo from a nbd of $J:=\left[\frac{1}{2},f(\frac{1}{2})\right]$ to a nbd of $[0,1]$, whose germs at $\frac{1}{2}$, and at $f(\frac{1}{2})$, are respectively $H$ and $H\circ f^{-1} + 1$. The extension of $h_0$ from the two germs is of course immediate by using some cut-off function. Then we may proceed as said before, from ${h_0}_ { |J} $.

(Clearly, these naive constructions immediately fall, in the case of $C^\omega$ conjugation of $C^\omega$ diffeo's on open intervals with no fixed points.)

I assume your map is surjective, thus an increasing $C^k$ diffeo $f:(0,1)\to(0,1)$ (say $1\le k\le\infty).$ The latter, as a discrete dynamical system, turns out to be $C^k$ conjugated with the shift by translation $t\mapsto t+1$ on $\mathbb{R}$. Now if $h:(0,1)\to\mathbb{R}$ is such an (increasing) conjugation, define a field $X$ by taking for all $t\in (0,1)$ $$X(t)=\frac{1}{h^\prime(t)}.$$ Then $f$ is the flow of $X$ at time $1$, just because $u:=h^{-1}:\mathbb{R}\to(0,1)$ solves the autonomous ODE $u'=X(u).$ So the solution at time $1$ corresponding to the initial value $t$ at time $0$ is actually $u\left(h(t)+1\right)=f(t).$

Rmk. The general fact behind is: for a diffeo on a manifold, being a time-one map of a flow, is a property invariant by smooth conjugation -the flow transforms by conjugation, and its generator is the pull back of $X$. And the shift $x\mapsto x+1$ is, of course, the map at time 1 of a flow.

Construction of the conjugation. Note that here a conjugation is an increasing diffeo $h:(0,1)\to\mathbb{R}$ solving the linear functional equation $h\left(f(x)\right) = h(x)+1.$ A solution of class $C^k$ can be constructed fixing any smooth diffeo $h_0:\left[\frac{1}{2},f(\frac{1}{2})\right]\to[0,1]$ with the convenient $k$-jets at the end-points of its domain, and extending it (uniquely) to a diffeo $h:(0,1)\to\mathbb{R}$ by means of the functional equation -the condition is that the $k$-jet of $h_0\left(f(x)\right)$ at $x=\frac{1}{2}$ and the $k$-jet of $h_0(x)+1$ at $x=f(\frac{1}{2})$ should coincide, in order that the glueing be $C^k$.

A maybe more clear way of achieving that, is: first fix any $C^k$-germ $H$ of local diffeo with $H(1/2)=0$ and $H'(0)>0$. Then take $h_0$ as a smooth diffeo from a nbd of $J:=\left[\frac{1}{2},f(\frac{1}{2})\right]$ to a nbd of $[0,1]$, whose germs at $\frac{1}{2}$, and at $f(\frac{1}{2})$, are respectively $H$ and $H\circ f^{-1} + 1$. The extension of $h_0$ from the two germs is of course immediate by using some cut-off function. Then we may proceed as said before, from ${h_0}_ { |J} $.

(Clearly, these naive constructions immediately fall, in the case of $C^\omega$ conjugation of $C^\omega$ diffeo's on open intervals with no fixed points.)

I assume your map is surjective, thus an increasing $C^k$ diffeo $f:(0,1)\to(0,1)$ (say $1\le k\le\infty).$ The latter, as a discrete dynamical system, turns out to be $C^k$ conjugated with the shift by translation $t\mapsto t+1$ on $\mathbb{R}$. Now if $h:(0,1)\to\mathbb{R}$ is such an (increasing) conjugation, define a field $X$ by taking for all $t\in (0,1)$ $$X(t)=\frac{1}{h^'(t)}.$$ Then $f$ is the flow of $X$ at time $1$, just because $u:=h^{-1}:\mathbb{R}\to(0,1)$ solves the autonomous ODE $u'=X(u).$ So the solution at time $1$ corresponding to the initial value $t$ at time $0$ is actually $u\left(h(t)+1\right)=f(t).$

Rmk. The general fact behind is: for a diffeo on a manifold, being a time-one map of a flow, that is, being isotopic to the identity, is a property invariant by smooth conjugation -the flow transforms by conjugation, and its generator is the pull back of $X$. And the shift $x\mapsto x+1$ is, of course, the map at time 1 of a flow.

Construction of the conjugation. Note that here a conjugation is an increasing diffeo $h:(0,1)\to\mathbb{R}$ solving the linear functional equation $h\left(f(x)\right) = h(x)+1.$ A solution of class $C^k$ can be constructed fixing any smooth diffeo $h_0:\left[\frac{1}{2},f(\frac{1}{2})\right]\to[0,1]$ with the convenient $k$-jets at the end-points of its domain, and extending it (uniquely) to a diffeo $h:(0,1)\to\mathbb{R}$ by means of the functional equation -the condition is that the $k$-jet of $h_0\left(f(x)\right)$ at $x=\frac{1}{2}$ and the $k$-jet of $h_0(x)+1$ at $x=f(\frac{1}{2})$ should coincide, in order that the glueing be $C^k$.

A maybe more clear way of achieving that, is: first fix any $C^k$-germ $H$ of local diffeo with $H(1/2)=0$ and $H'(0)>0$. Then take $h_0$ as a smooth diffeo from a nbd of $J:=\left[\frac{1}{2},f(\frac{1}{2})\right]$ to a nbd of $[0,1]$, whose germs at $\frac{1}{2}$, and at $f(\frac{1}{2})$, are respectively $H$ and $H\circ f^{-1} + 1$. The extension of $h_0$ from the two germs is of course immediate by using some cut-off function. Then we may proceed as said before, from ${h_0}_ { |J} $.

(Clearly, these naive constructions immediately fall, in the case of $C^\omega$ conjugation of $C^\omega$ diffeo's on open intervals with no fixed points.)

I assume your map is surjective, thus an increasing $C^k$ diffeo $f:(0,1)\to(0,1)$ (say $1\le k\le\infty).$ The latter, as a discrete dynamical system, turns out to be $C^k$ conjugated with the shift by translation $t\mapsto t+1$ on $\mathbb{R}$. Now if $h:(0,1)\to\mathbb{R}$ is such an (increasing) conjugation, define a field $X$ by taking for all $t\in (0,1)$ $$X(t)=\frac{1}{h^'(t)}.$$ Then $f$ is the flow of $X$ at time $1$, just because $u:=h^{-1}:\mathbb{R}\to(0,1)$ solves the autonomous ODE $u'=X(u).$ So the solution at time $1$ corresponding to the initial value $t$ at time $0$ is actually $u\left(h(t)+1\right)=f(t).$

Rmk. The general fact behind is: for a diffeo on a manifold, being a time-one map of a flow, is a property invariant by smooth conjugation -the flow transforms by conjugation, and its generator is the pull back of $X$. And the shift $x\mapsto x+1$ is, of course, the map at time 1 of a flow.

Construction of the conjugation. Note that here a conjugation is an increasing diffeo $h:(0,1)\to\mathbb{R}$ solving the linear functional equation $h\left(f(x)\right) = h(x)+1.$ A solution of class $C^k$ can be constructed fixing any smooth diffeo $h_0:\left[\frac{1}{2},f(\frac{1}{2})\right]\to[0,1]$ with the convenient $k$-jets at the end-points of its domain, and extending it (uniquely) to a diffeo $h:(0,1)\to\mathbb{R}$ by means of the functional equation -the condition is that the $k$-jet of $h_0\left(f(x)\right)$ at $x=\frac{1}{2}$ and the $k$-jet of $h_0(x)+1$ at $x=f(\frac{1}{2})$ should coincide, in order that the glueing be $C^k$.

A maybe more clear way of achieving that, is: first fix any $C^k$-germ $H$ of local diffeo with $H(1/2)=0$ and $H'(0)>0$. Then take $h_0$ as a smooth diffeo from a nbd of $J:=\left[\frac{1}{2},f(\frac{1}{2})\right]$ to a nbd of $[0,1]$, whose germs at $\frac{1}{2}$, and at $f(\frac{1}{2})$, are respectively $H$ and $H\circ f^{-1} + 1$. The extension of $h_0$ from the two germs is of course immediate by using some cut-off function. Then we may proceed as said before, from ${h_0}_ { |J} $.

(Clearly, these naive constructions immediately fall, in the case of $C^\omega$ conjugation of $C^\omega$ diffeo's on open intervals with no fixed points.)

I assume your map is surjective, thus an increasing $C^k$ diffeo $f:(0,1)\to(0,1)$ (say $1\le k\le\infty).$ The latter, as a discrete dynamical system, turns out to be $C^k$ conjugated with the shift by translation $t\mapsto t+1$ on $\mathbb{R}$. Now if $h:(0,1)\to\mathbb{R}$ is such an (increasing) conjugation, define a field $X$ by taking for all $t\in (0,1)$ $$X(t)=\frac{1}{h^'(t)}.$$ Then $f$ is the flow of $X$ at time $1$, just because $u:=h^{-1}:\mathbb{R}\to(0,1)$ solves the autonomous ODE $u'=X(u).$ So the solution at time $1$ corresponding to the initial value $t$ at time $0$ is actually $u\left(h(t)+1\right)=f(t).$

Construction of the conjugation. Note that here a conjugation is an increasing diffeo $h:(0,1)\to\mathbb{R}$ solving the linear functional equation $h\left(f(x)\right) = h(x)+1.$ A solution of class $C^k$ can be constructed fixing any smooth diffeo $h_0:\left[\frac{1}{2},f(\frac{1}{2})\right]\to[0,1]$ with the convenient $k$-jets at the end-points of its domain, and extending it (uniquely) to a diffeo $h:(0,1)\to\mathbb{R}$ by means of the functional equation -the condition is that the $k$-jet of $h_0\left(f(x)\right)$ at $x=\frac{1}{2}$ and the $k$-jet of $h_0(x)+1$ at $x=f(\frac{1}{2})$ should coincide, in order that the glueing be $C^k$.

A maybe more clear way of achieving that, is: first fix any $C^k$-germ $H$ of local diffeo with $H(1/2)=0$ and $H'(0)>0$. Then take $h_0$ as a smooth diffeo from a nbd of $J:=\left[\frac{1}{2},f(\frac{1}{2})\right]$ to a nbd of $[0,1]$, whose germs at $\frac{1}{2}$, and at $f(\frac{1}{2})$, are respectively $H$ and $H\circ f^{-1} + 1$. The extension of $h_0$ from the two germs is of course immediate by using some cut-off function. Then we may proceed as said before, from ${h_0}_ { |J} $.

(Clearly, these naive constructions immediately fall, in the case of $C^\omega$ conjugation of $C^\omega$ diffeo's on open intervals with no fixed points.)

I assume your map is surjective, thus an increasing $C^k$ diffeo $f:(0,1)\to(0,1)$ (say $1\le k\le\infty).$ The latter, as a discrete dynamical system, turns out to be $C^k$ conjugated with the shift by translation $t\mapsto t+1$ on $\mathbb{R}$. Now if $h:(0,1)\to\mathbb{R}$ is such an (increasing) conjugation, define a field $X$ by taking for all $t\in (0,1)$ $$X(t)=\frac{1}{h^'(t)}.$$ Then $f$ is the flow of $X$ at time $1$, just because $u:=h^{-1}:\mathbb{R}\to(0,1)$ solves the autonomous ODE $u'=X(u).$ So the solution at time $1$ corresponding to the initial value $t$ at time $0$ is actually $u\left(h(t)+1\right)=f(t).$

Rmk. The general fact behind is: for a diffeo on a manifold, being a time-one map of a flow, that is, being isotopic to the identity, is a property invariant by smooth conjugation -the flow transforms by conjugation, and its generator is the pull back of $X$. And the shift $x\mapsto x+1$ is, of course, the map at time 1 of a flow.

Construction of the conjugation. Note that here a conjugation is an increasing diffeo $h:(0,1)\to\mathbb{R}$ solving the linear functional equation $h\left(f(x)\right) = h(x)+1.$ A solution of class $C^k$ can be constructed fixing any smooth diffeo $h_0:\left[\frac{1}{2},f(\frac{1}{2})\right]\to[0,1]$ with the convenient $k$-jets at the end-points of its domain, and extending it (uniquely) to a diffeo $h:(0,1)\to\mathbb{R}$ by means of the functional equation -the condition is that the $k$-jet of $h_0\left(f(x)\right)$ at $x=\frac{1}{2}$ and the $k$-jet of $h_0(x)+1$ at $x=f(\frac{1}{2})$ should coincide, in order that the glueing be $C^k$.

A maybe more clear way of achieving that, is: first fix any $C^k$-germ $H$ of local diffeo with $H(1/2)=0$ and $H'(0)>0$. Then take $h_0$ as a smooth diffeo from a nbd of $J:=\left[\frac{1}{2},f(\frac{1}{2})\right]$ to a nbd of $[0,1]$, whose germs at $\frac{1}{2}$, and at $f(\frac{1}{2})$, are respectively $H$ and $H\circ f^{-1} + 1$. The extension of $h_0$ from the two germs is of course immediate by using some cut-off function. Then we may proceed as said before, from ${h_0}_ { |J} $.

(Clearly, these naive constructions immediately fall, in the case of $C^\omega$ conjugation of $C^\omega$ diffeo's on open intervals with no fixed points.)