Define: $\phi(x)=\exp(1/\ln(x)).$
Q: Is the following sufficient notation to express a homotopy between $\phi$ and some arbitrary power of it, $\phi^t?$
$$\phi,\phi^t:\phi \to \phi^t$$
I want to prove that this is a homotopic map between the two functions, meaning that $\phi$ can be continuously deformed into $\phi^t, t\in\Bbb R^+.$
I'd like for $t$ to be the amount of time it takes for the mapping to occur. For example, $\phi$ continuously deformed into itself would take $t=0$ units of time. And $\phi$ mapped to $\phi^t$ would take $t$ units of time. Hopefully that makes sense.
Q: Does a homotopic map encode anything about a "speed" at which the functions are mapped to each other?
I wouldn't talk about "speed", since of course you can define the same homotopy by making $t$ vary over an interval of any lenght, using a reparametrization. Moreover, it's very important to know which are the sets between your functions are defined since, for example, if codomain is a convex vector space (or more generally star-shaped) every two functions are homotopic.