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Devaney's definition of chaos assumes the function $f: V \rightarrow V$. Is there a reason for restricting this to $V \rightarrow V $ and not two different sets $X\rightarrow Y$?. Specifically, my question is, does the definition still hold if $f: X \rightarrow Y$? Also, can anyone explain topological transitivity in layman terms?

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    You can't iterate $f$ if it's a function from a set to a different set. Chaos is about the sensitivity of _repeated iterations_ of $f$ to initial conditions.2012-04-26

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Qiaochu’s comment explains why you want $f$ to have the same domain and codomain.

Intuitively, the idea behind transitivity of $f$ is that you can get from anywhere to anywhere by applying $f$ enough times.

Suppose that $f:X\to X$. Each $x\in X$ has an orbit under $f$, i.e., $\{f^n(x):n\in\Bbb N\}$. Taken to its extreme, the intuitive idea would require that every point of $f$ have all of $X$ as its orbit. Since the orbit of $x$ is countable, and most spaces of interest in this context are not, this is plainly asking too much of $f$. Instead, we ask that for any $x,y\in X$, we can find points arbitrarily close to $x$ whose orbits come arbitrarily close to $y$. More precisely, for any $\epsilon>0$ there are a point $z$ and a non-negative integer $n$ such that d(x,z)<\epsilon and d\left(f^n(z),y\right)<\epsilon, where $d$ is the distance function in $X$. (The actual definition of a transitive operator is stated in terms of open sets in general and doesn’t require that the space be metric, but for an intuitive description I think that the metric version makes the most sense.)