You're asking for a one-parameter family $\mathcal{C}$ of maps on a closed interval $I$ such that

1. the endpoints of $I$ are fixed by every map in $\mathcal{C}$;
2. the maps in $\mathcal{C}$ are strictly increasing;
3. $\mathcal{C}$ is closed under composition;
4. every positive real number appears exactly once as the ratio of the derivatives of $\mathcal{C}$ at the right and left endpoints of $I$.

The fourth condition means that $\mathcal{C}$ should be parametrised by $\mathbb{R}$, so let's write $\mathcal{C} = \{ \psi_c \mid c>0 \}$, where $c$ represents the square root of the ratio between the derivates at the endpoints, as in your question. The chain rule together with the third condition means that we should require $$\psi_{c_1 c_2} = \psi_{c_1} \circ \psi_{c_2}$$ for all $c_1,c_2>0$. Replacing $c$ with $t=\log c$ and writing $\phi_t = \psi_c$, this becomes $$ \phi_{t_1+t_2} = \phi_{t_1} \circ \phi_{t_2}.$$ In other words, the family $\mathcal{C}$ defines an action of $\mathbb{R}$ on $I$, or in the language of dynamical systems/ODEs, a flow on $I$. This flow comes from integrating a vector field $V(x)$, or if you prefer, solving the ODE $$(*) \qquad \qquad \frac d{dt} \phi_t(x) = V(\phi_t(x)).\qquad\qquad\qquad$$ Conditions 1 and 2 mean that $V$ should vanish at the endpoints of $I$ and should be positive everywhere else. But modulo some regularity concerns, that's all you need. If you let $V\colon I\to [0,\infty)$ be any Lipschitz continuous function that vanishes at the endpoints of $I$ and is positive on its interior, then solving $(*)$ will give you a family of smooth maps (the time-$t$ maps of the flow) that satisfy your conditions.

Incidentally, you can use $(*)$ to see how both the family $\mathcal{S}$ of FLTs and the family in Robert Israel's (second) answer fit into this scheme. If you write $$ \phi_t(x) = \frac{e^t x}{1+(1-e^t)x}$$ for the FLT satisfying your conditions (with $c=e^t$ and $t\in\mathbb{R}$ arbitrary), then an easy calculation shows that $\phi_{s+t} = \phi_s \circ \phi_t$, so this defines a flow, and we can compute $$ V(x) = \frac{d\phi_t(x)}{dt}|_{t=0} = x(1+x). $$ In other words, these FLTs are just the time-$t$ maps of the logistic flow on the interval. You can play a similar game with the arctan maps, but I haven't worked it out to see if the formula comes out cleanly.

Actually, the family of arctan maps in Robert Israel's answer illustrates a (superficially) different way of approaching the question, which is to fix a homeomorphism $h$ between the interior of $I$ and the real line (or the half-line) and then define the family of maps $\mathcal{C}$ to be those that are conjugated to translations (or homotheties) by $h$. So in his example, you conjugate to the negative half-line by $h(x) = \tan(\pi x/2)$ and then let $g_c = h^{-1} \circ (y\mapsto cy) \circ h$.

Of course, logs turn the half-line into the line and multiplication into addition, so this turns into translation on the line under another change of coordinates. And finding a change of coordinates that turns a map into translation on the line is just another way of saying that you find that map as a solution of an ODE, so it boils down to what I described above. Your choice... you can choose the vector field $V$, or you can choose the homeomorphism $h$. Either will give you quite a large number of examples.

You're asking for a one-parameter family $\mathcal{C}$ of maps on a closed interval $I$ such that

1. the endpoints of $I$ are fixed by every map in $\mathcal{C}$;
2. the maps in $\mathcal{C}$ are strictly increasing;
3. $\mathcal{C}$ is closed under composition;
4. every positive real number appears exactly once as the ratio of the derivatives of $\mathcal{C}$ at the right and left endpoints of $I$.

The fourth condition means that $\mathcal{C}$ should be parametrised by $\mathbb{R}^+$, so let's write $\mathcal{C} = \{ \psi_c \mid c>0 \}$, where $c$ represents the square root of the ratio between the derivates at the endpoints, as in your question. The chain rule together with the third condition means that we should require $$\psi_{c_1 c_2} = \psi_{c_1} \circ \psi_{c_2}$$ for all $c_1,c_2>0$. We can reparametrise $\mathcal{C}$ by putting $t=\log c$ (or $t=-\log c$) and writing $\phi_t = \psi_c$, this becomes $$ \phi_{t_1+t_2} = \phi_{t_1} \circ \phi_{t_2}.$$ In other words, the family $\mathcal{C}$ defines an action of $\mathbb{R}$ on $I$, or in the language of dynamical systems/ODEs, a flow on $I$. This flow comes from integrating a vector field $V(x)$, or if you prefer, solving the ODE $$(*) \qquad \qquad \frac d{dt} \phi_t(x) = V(\phi_t(x)).\qquad\qquad\qquad$$ Conditions 1 and 2 mean that $V$ should vanish at the endpoints of $I$ and should be positive everywhere else. But modulo some regularity concerns, that's all you need. If you let $V\colon I\to [0,\infty)$ be any Lipschitz continuous function that vanishes at the endpoints of $I$ and is positive on its interior, then solving $(*)$ will give you a family of smooth maps (the time-$t$ maps of the flow) that satisfy your conditions.

Incidentally, you can use $(*)$ to see how both the family $\mathcal{S}$ of FLTs and the family in Robert Israel's (second) answer fit into this scheme. If you write $$ \phi_t(x) = \frac{e^{-t} x}{1+(1-e^{-t})x}$$ for the FLT satisfying your conditions (with $c=e^{-t}$ and $t\in\mathbb{R}$ arbitrary), then an easy calculation shows that $\phi_{s+t} = \phi_s \circ \phi_t$, so this defines a flow, and we can compute $$ V(x) = \frac{d\phi_t(x)}{dt}|_{t=0} = -x(1+x). $$ In other words, these FLTs are just the time-$t$ maps of the logistic flow on the interval $[-1,0]$. You can play a similar game with the arctan maps, but I haven't worked it out to see if the formula comes out cleanly.

Actually, the family of arctan maps in Robert Israel's answer illustrates a (superficially) different way of approaching the question, which is to fix a homeomorphism $h$ between the interior of $I$ and the real line (or the half-line) and then define the family of maps $\mathcal{C}$ to be those that are conjugated to translations (or homotheties) by $h$. So in his example, you conjugate to the negative half-line by $h(x) = \tan(\pi x/2)$ and then let $g_c = h^{-1} \circ (y\mapsto cy) \circ h$.

Of course, logs turn the half-line into the line and multiplication into addition, so this turns into translation on the line under another change of coordinates. And finding a change of coordinates that turns a map into translation on the line is just another way of saying that you find that map as a solution of an ODE, so it boils down to what I described above. Your choice... you can choose the vector field $V$, or you can choose the homeomorphism $h$. Either will give you quite a large number of examples.

You're asking for a one-parameter family $\mathcal{C}$ of maps on a closed interval $I$ such that

1. the endpoints of $I$ are fixed by every map in $\mathcal{C}$;
2. the maps in $\mathcal{C}$ are strictly increasing;
3. $\mathcal{C}$ is closed under composition;
4. every positive real number appears exactly once as the ratio of the derivatives of $\mathcal{C}$ at the right and left endpoints of $I$.

The fourth condition means that $\mathcal{C}$ should be parametrised by $\mathbb{R}$, so let's write $\mathcal{C} = \{ \phi_t \mid t\in\mathbb{R} \}$, where $t$ represents the square root of the ratio in question. The chain rule together with the third condition means that we should require

• $\phi_{s+t} = \phi_s \circ \phi_t$

In other words, the family $\mathcal{C}$ defines an action of $\mathbb{R}$ on $I$, or in the language of dynamical systems/ODEs, a flow on $I$. This flow comes from integrating a vector field $V(x)$, or if you prefer, solving the ODE $$(*) \qquad \qquad \frac d{dt} \phi_t(x) = V(\phi_t(x)).\qquad\qquad\qquad$$ Conditions 1 and 2 mean that $V$ should vanish at the endpoints of $I$ and should be positive everywhere else. But modulo some regularity concerns, that's all you need. If you let $V\colon I\to [0,\infty)$ be any Lipschitz continuous function that vanishes at the endpoints of $I$ and is positive on its interior, then solving $(*)$ will give you a family of smooth maps (the time-$t$ maps of the flow) that satisfy your conditions.

Incidentally, you can use $(*)$ to see how both the family $\mathcal{S}$ of FLTs and the family in Robert Israel's (second) answer fit into this scheme. If you write $$ \phi_t(x) = \frac{tx}{1+(1-t)x}$$ for the FLT satisfying your conditions (with $t=c$), then an easy calculation shows that $\phi_{s+t} = \phi_s \circ \phi_t$, so this defines a flow, and we can compute $$ V(x) = \frac{d\phi_t(x)}{dt}|_{t=0} = x(1+x). $$ In other words, these FLTs are just the time-$t$ maps of the logistic flow on the interval. You can play a similar game with the arctan maps, but I haven't worked it out to see if the formula comes out cleanly.

Actually, the family of arctan maps in Robert Israel's answer illustrates a (superficially) different way of approaching the question, which is to fix a homeomorphism $h$ between the interior of $I$ and the real line (or the half-line) and then define the family of maps $\mathcal{C}$ to be those that are conjugated to translations (or homotheties) by $h$. So in his example, you conjugate to the negative half-line by $h(x) = \tan(\pi x/2)$ and then let $g_c = h^{-1} \circ (y\mapsto cy) \circ h$.

Of course, logs turn the half-line into the line and multiplication into addition, so this turns into translation on the line under another change of coordinates. And finding a change of coordinates that turns a map into translation on the line is just another way of saying that you find that map as a solution of an ODE, so it boils down to what I described above. Your choice... you can choose the vector field $V$, or you can choose the homeomorphism $h$. Either will give you quite a large number of examples.

You're asking for a one-parameter family $\mathcal{C}$ of maps on a closed interval $I$ such that

1. the endpoints of $I$ are fixed by every map in $\mathcal{C}$;
2. the maps in $\mathcal{C}$ are strictly increasing;
3. $\mathcal{C}$ is closed under composition;
4. every positive real number appears exactly once as the ratio of the derivatives of $\mathcal{C}$ at the right and left endpoints of $I$.

The fourth condition means that $\mathcal{C}$ should be parametrised by $\mathbb{R}$, so let's write $\mathcal{C} = \{ \psi_c \mid c>0 \}$, where $c$ represents the square root of the ratio between the derivates at the endpoints, as in your question. The chain rule together with the third condition means that we should require $$\psi_{c_1 c_2} = \psi_{c_1} \circ \psi_{c_2}$$ for all $c_1,c_2>0$. Replacing $c$ with $t=\log c$ and writing $\phi_t = \psi_c$, this becomes $$ \phi_{t_1+t_2} = \phi_{t_1} \circ \phi_{t_2}.$$ In other words, the family $\mathcal{C}$ defines an action of $\mathbb{R}$ on $I$, or in the language of dynamical systems/ODEs, a flow on $I$. This flow comes from integrating a vector field $V(x)$, or if you prefer, solving the ODE $$(*) \qquad \qquad \frac d{dt} \phi_t(x) = V(\phi_t(x)).\qquad\qquad\qquad$$ Conditions 1 and 2 mean that $V$ should vanish at the endpoints of $I$ and should be positive everywhere else. But modulo some regularity concerns, that's all you need. If you let $V\colon I\to [0,\infty)$ be any Lipschitz continuous function that vanishes at the endpoints of $I$ and is positive on its interior, then solving $(*)$ will give you a family of smooth maps (the time-$t$ maps of the flow) that satisfy your conditions.

Incidentally, you can use $(*)$ to see how both the family $\mathcal{S}$ of FLTs and the family in Robert Israel's (second) answer fit into this scheme. If you write $$ \phi_t(x) = \frac{e^t x}{1+(1-e^t)x}$$ for the FLT satisfying your conditions (with $c=e^t$ and $t\in\mathbb{R}$ arbitrary), then an easy calculation shows that $\phi_{s+t} = \phi_s \circ \phi_t$, so this defines a flow, and we can compute $$ V(x) = \frac{d\phi_t(x)}{dt}|_{t=0} = x(1+x). $$ In other words, these FLTs are just the time-$t$ maps of the logistic flow on the interval. You can play a similar game with the arctan maps, but I haven't worked it out to see if the formula comes out cleanly.

Actually, the family of arctan maps in Robert Israel's answer illustrates a (superficially) different way of approaching the question, which is to fix a homeomorphism $h$ between the interior of $I$ and the real line (or the half-line) and then define the family of maps $\mathcal{C}$ to be those that are conjugated to translations (or homotheties) by $h$. So in his example, you conjugate to the negative half-line by $h(x) = \tan(\pi x/2)$ and then let $g_c = h^{-1} \circ (y\mapsto cy) \circ h$.

Of course, logs turn the half-line into the line and multiplication into addition, so this turns into translation on the line under another change of coordinates. And finding a change of coordinates that turns a map into translation on the line is just another way of saying that you find that map as a solution of an ODE, so it boils down to what I described above. Your choice... you can choose the vector field $V$, or you can choose the homeomorphism $h$. Either will give you quite a large number of examples.