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I basically have questions about supplementary books (background).

I am doing self-study as a preliminary before reading a paper in hyperbolic dynamics, and I have not much prior knowledge about it (I am more on topology and surface theory but the supplementary paper I need some result from requires hyperbolic dynamics). Currently, I am reading Katok & Hassleblent "Introduction to the Modern Theory of Dynamical Systems". It is a great book but I have problems extracting which chapters/parts of the book I need to know.

Background of the story: basically, I look at Toral automorphism "Arnold 's cat map". I regard this map as a nice linear map compose with a translation map. The linear part has the obvious global hyperbolic behavior centered at $(0,0)$ along its expanding and contracting eigenlines.

But it turns out that other than the point $(0,0)$, all other point on the torus also have "local hyperbolic sturcture". That is, at point $(x,y) \neq (0,0)$, you can expect to see the hyperbola diagram centered at this point $(x,y)$ inside of the small neighborhood around $(x,y)$ : (imagine the centered is $(x,y)$ and this pciture is in a small neighborhood around $(x,y)$). enter image description here

I think this is what called "differential of the hyperbolic map implies local behavior" in a non-technical phrase. But I try to look for Theorems about this behavior, but none is found. I see terms like non-stable manifolds, hyperbolic sets etc. I guess this is a technical version of the phrase above which requires languages of stable, non-stable manifold, hyperbolic sets etc.

Does my understanding correct?

If anyone can sketch/explain about this, I will appreciate it a lot. Also, any recommendation on a (more elementary and perhaps less think) hyperbolic/dynamical system textbook other than Katok & Hassleblant's book ?

Mark McClure
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user117375
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  • If I understand your geometric remarks right, this would imply that the vector field is zero in all points $(x,y)$. Could you try and put your ideas about the map in formulas? – Lutz Lehmann Mar 24 '22 at 10:32
  • I mean I doubt that the local hyperbolic behavior is right ? Or more precise, why it is right ? – user117375 Mar 25 '22 at 00:29
  • I kind of get that for hyperbolic map, the tangent space spilts into two leaves (?, directions guess like eigen direction). But this is all about tangent space which is not even on the manifold itself. Why/how behavior on tangent space effects behavior of a map on the space itself ? Is this some kind of generalization of linear approximation in Vector Calculus that you can use tangent space to approximate the value on the surface itself ? – user117375 Mar 25 '22 at 00:30
  • Essentially you are stating that a continuously differentiable map has a continuous Jacobian, and that the spectrum of a matrix-valued function is largely continuous, especially if the "hyperbolic" condition is satisfied. You need to take into account that the (affine) linear approximation of the vector field has a constant term. If that constant term is non-zero, then the ODE solutions are locally dominantly determined by it and not the degree-1 term. – Lutz Lehmann Mar 25 '22 at 10:07

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