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I am self-studying Rob Devaney's "An introduction to Chaotical Dynamical Systems".

"Decide whether each of the following functions are 1-1, onto, homemorphisms or diffeomorphisms on their domains of definition."

the fisrt one is;

$f(x) = x^{5/3}$

This function has domain $[0,\infty)$. It is continuous, one-one and onto.

Also its inverse;

$f^{-1}(x) = x^{3/5}$

satisfies the above criterion.

This means its a Homeomorphic function, i think. Is this correct?

Now, I have a few questions;

A function is of class $C^r$ on its domain if $f^{(c)}(x)$ exists and is continuous at all points on its domain.

what does if "$f^{(c)}(x)$ exists" mean?

so if $f(x) = x$, then $f$ is of class $C^1$? because its first derivative exists and is continuous.

What I don't understand is the definiton of a Diffeomorphism;

The function $f(x)$ is a $C^r-diffeomorphism$ if $f(x)$ is a $C^r-homeomorphism$ such that $f^{-1}(x)$ is also $C^r$.

so lets use, the example I chose earlier; $f(x) = x^{5/3}$

Is this a diffeomorphism? why/whynot?

I think so because both functions are infinitely differentiable...

Trajan
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1 Answers1

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The definition of $C^r$ is missing a little bit: a function is of class $C^r$ if all derivatives up to order $r$ exist and are continuous, i.e. $f'(x),f''(x),f^{(3)}(x),\ldots,f^{(r)}(x)$ all exist and are continuous. (It is sufficient to show all the derivatives exist and $f^{(r)}$ is continuous.) A $C^r$-diffeomorphism is a bijection $f$ such that both $f$ and $f^{-1}$ are of class $C^r$.

With your specific examples, $f(x)=x^{5/3}$ is indeed a homeomorphism. It is not a $C^r$-diffeomorphism for $r\geq1$ since $(f^{-1})'(x)=\frac35x^{-2/5}$ is not continuous at $x=0$. The function $f(x)=x$ is self-inverse and has continuous derivatives of all orders (most of them are identically zero, which is a continuous function), so $f$ is a $C^r$-diffeomorphism for any $r\geq1$.

Jason
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