1

Let $$A = \begin{bmatrix} 1 & 1 \\ 1 & 0\\ \end{bmatrix}.$$ Then the eigenvalues of $A$ are $1/2(1+\sqrt{5})$ and $1/2(1-\sqrt{5})$. The eigenvector corresponding to the unstable eigenvalue is the line $y = 1/2(-1+\sqrt 5)x$, whereas the stable eigenvalue has eigenvector $y = -1/2 (1+\sqrt 5)x$.

Let $L$ be a hyperbolic linear automorphism of the torus induced by $A$.

Could someone explain how we get the following graph? To simplify the question, let's jsut discuss how $L$ affects $R_1$. The upper left picture is a Markov partition of the torus. The dark regions on the second and third pictures represent $R_1$, $L(R_1)$ and $L^{-1}(R_1)$. enter image description here

user398843
  • 1,771

1 Answers1

1

Since the edges of your rectangles are exactly the eigendirections, $L$ only acts on the rectangle by stretching and squishing. You have one expanding eigenvalue (the one with absolute value grater than $1$) and one contracting eigenvalue (the one with absolute value less than 1 - note that it's negative so it also flips the rectangle).

So $L$ acts on $R_1$ by stretching it in the first eigendirection (as seen in the diagram) and squishing it in the other eigendirection, while also flipping it (which explains the new arrangement of the other rectangles around $R_1$).

GSofer
  • 4,313
  • The stretching and contracting explanation sounds reasonable, but I do not udnerstand why after stretching and contracting $R_1$, we get a graph that looks like that. Could you explain it? – user398843 May 14 '20 at 06:20
  • 1
    The shaded region is $R_1$. Notice how in one direction it contracted, and in the other it expanded. Same goes for the other rectangles - think about taking the pattern on the square on the LHS, and stretching it in on eigendirection while squishing in the other. Also, recall that this is a torus (and not just a square), and so you identify the edges, which means that when you stretch, you can move from one side of the square to the other (which explains why some parts of the rectangle seem to move or divide themselves). Does this help? – GSofer May 14 '20 at 07:26
  • @user398843 By the way, where is this picture from? I'm also interested in this exact system on the torus with the same Markov partition – GSofer May 14 '20 at 18:14
  • It helps. Thank you. It is from Devaney's An Introduction to Chaotic dYNAMICAL systems, 2nd edition. – user398843 May 14 '20 at 19:07