# Link between the Hartman-Grobman and Hadamard Perron Theorem

Hartman-Grobman Theorem: If $\phi$ is a hyperbolic map, and we write $\phi(x) = Ax +\bar\phi(x)$, where $A$ is the linearization of $\phi$, then for a sufficiently small neighbourhood about the origin we can find a homeomorphism $h$ such that $A\circ h = h\circ\phi$.

Hadamard-Perron Theorem: If $\phi$ is a hyperbolic map of class $C^k$, then for a neighbourhood $Q$ about the origin we can write $W^+(Q):=\{x\in Q|\lim_{j\rightarrow\infty}\phi^j(x)\rightarrow 0\}=(x,h(x)),x\in E^+$, with $h$ of class $C^k$ (here $E^+$ is the invariant subspace associated with positive eigenvalues of $A$) And a similar statement for the negative eigenspace holds.

These two theorem look very similar to me. Am I correct to say that the two theorems say "nearly" the same thing for the case $k=0$? The Hartman Grobman theorem gives us an isomorphism which distorts $E^+$ and $E^-$ in a continuous way, hence $W^+$ and $W^-$ are embedded topological manifolds, which is the $k=0$ statement of Hadamard-Perron (However the Hadamard-Perron does not conversely give us the homemomorphism provided by the Hartman-Grobman).

In the sense of studying the description of the positive and negative invariant manifolds of the hyperbolic map $\phi$, can we thus say that the Hadamrd-Perron theorem is just an extension of the Hartman-Grobman theorem? Furthermore, is it possible to strenghten the statement of the Hartman-Grobman theorem to $C^k- smooth maps$, just like what does Hadamard-Perron theorem does?

Further insights to the link between the two theorems are appreciated too.

• How do you define a hyperbolic map of class $C^0$? – user539887 Apr 16 '18 at 21:06
• That would just be a continuous map (with continuous inverse) – Zhanfeng Lim Apr 16 '18 at 21:31
• Any homeomorphism? – user539887 Apr 16 '18 at 21:34
• yeap, continuous map with continuous inverse ^ – Zhanfeng Lim Apr 16 '18 at 21:41
• What is $E^+$? $E^-$? Linearization? – user539887 Apr 16 '18 at 21:43

Way too many questions, you really need to ask one by one (in here).

Sure, one can define continuous hyperbolic maps. The original reference is

V. Alekseev and M. Yakobson, Symbolic dynamics and hyperbolic dynamic systems, Phys. Rep. 75 (1981), 287-325

(although it can be considered that it is formally contained in former work of Bowen). But there is no linearization in general and so the definition is based on the stable and unstable sets (and product structure).

Basically the definition requires exactly what is required for the construction of a Markov partition (which explains the emphasis on the product structure).

PS: Actually Hadamard and Perron never really proved the theorem that bears their name. So, if we combine the Palis $\lambda$-lemma with the Hadamard-Perron theorem indeed one can say that the Hadamard-Perron theorem is a generalization of the Grobman-Hartman theorem (never seen this written anywhere though).

Peculiar way of yours of writing "Hartman-Grobman", both in respect to the alphabetical order and years of Grobman (1959, 1962) and Hartman (1960, 1963) papers.