In the field of dynamical systems, a hyperbolic map $T: \mathbb{R}^n\rightarrow \mathbb{R}^n$ can be defined in a few equivalent ways. One of them is: T is hyperbolic iff there exists eigenvalues of T with modulus not equal to 1, and there are eigenvalues $\lambda$ such that $|\lambda_1|>1$ and $|\lambda_2|<1$. Furthermore, mathworld wolfram suggests that this means that we can split such a map into a direct sum of generalized eigenspaces.
But the reals are not algebraically closed. This means that any map with a rotational component could possibly be hyperbolic as long as it has a direction in which in expands and another in which it contracts, yet such a map doesn't decompose into generalized eigenspaces.
Should we not be working over the complex field instead for this theory to work out properly?
Certainly the standard definition is, that a linear map $T\colon \mathbb{R}^n\to \mathbb{R}^n$ is called hyperbolic if $T$ is invertible (sometimes this is omitted) and has no eigenvalue of absolute value $1$. For the decomposition into $A$-invariant subspaces over the real numbers see Theorem 1.1 here. The statement of the Theorem does not refer to (generalised) eigenspaces, and is correct. The remark afterwards with the eigenspaces is not correct.