WEB1. Hyperbolic toral automorphisms (cont.) Let A be d d integer matrix, with detA = 1. Consider Td = Rd=Zd (the d-dimensional torus). De nition 1.1. The map T A(x) = Ax (mod Zd) is called the toral automorphism asso-ciated with A. Proposition 1.2. T A is a group automorphism; in fact, every automorphism of Td has such a form. De nition 1.3.
WEB1.1.4. Hyperbolic toral automorphisms. Expanding maps are noninvertible. Now, we look at a system similar to expanding maps, except that these are invertible.2 Let A2SL nZ be a matrix with integer entries with det(A) = 1. De ne the map F A: Tn! Tnby: F A(x) = Ax(mod 1): Since the determinant of Ais 1, F A is invertible (in fact, the matrix A 1 ...
WEB2, so T and T 1 are hyperbolic toral automorphisms. By Proposition 2.2, lim N!1 \N n=0 T n(U)! = 0: The subset Uis not toral convex (since it is not simply connected), but U\T 1(U) is toral convex, and U\T 1(U) \T 2(U) is path-connected. Thus by Theorem 3.5, \N n=0 T n(U) is a polygon that is toral convex for all N >0. This puts severe ...
WEBHYPERBOLIC TORAL AUTOMORPHISMS OF T2 E. RYKKEN ABSTRACT. Using continued fractions, we give a direct and constructive proof for the fact that every matrix in GL( 2, Z) whose eigenvalues lie off the unit circle is similar over the integers to a matrix with all nonnegative or all nonpositive entries. This was first proven indirectly by R.F ...
WEBThe above map is called a linear hyperbolic toral automorphism. The same construction can be give on the n−torus Tn starting with an n×n matrix A with integer entries, determinant 1, and eigenvalues of norm different from 1. These hyperbolic toral automorphisms are, in some sense, diffeomorphic analogsoftheexpandingmapsofthecircle.
Dynamics of Automorphisms of the Torus and Other Groups
WEBJan 29, 2021 · Toral automorphisms in two dimensions offer an accessible look into the study of hyperbolic dynamics as well as highlight the interesting interplay between algebra and ergodic theory on compact groups. We write \ (\mathbb T^2\) to mean the quotient space \ (\mathbb R^2/\mathbb Z^2\) and refer to it as a 2-torus.
WEBFor the time 1 geodesic flow on the unit tangent bundle on a hyperbolic surface X, one can construct ergodic measures of the following form. One ergodic mea-sure is supported on a closed geodesic, and another ergodic measure is supported on another closed geodesic.
Smooth local rigidity for hyperbolic toral automorphisms
WEBMar 6, 2023 · The action of a matrix $A \in SL(N,\mathbb{Z})$ on $\mathbb{R}^N$ induces an automorphism of the torus $\mathbb{T}^N=\mathbb{R}^N/\mathbb{Z}^N$, which we denote by the same letter. An automorphism $A$ is called hyperbolic, or Anosov, if the matrix has no eigenvalues on the unit circle.
Chapter 5 - Hyperbolic toral automorphisms - Cambridge …
WEBChapter. Information. Dynamical Systems and Ergodic Theory , pp. 47 - 56. DOI: https://doi.org/10.1017/CBO9781139173049.007. Publisher: Cambridge University Press. Print publication year: 1998. Access options. Get access to the full version of this content by using one of the access options below.
WEBA toral automorphism Φ is called hyperbolic if none of the eigenvalues of the matrix has modulus 1, that is, |λ| = 1 for every eigenvalue λ. For definitions, see [5]. Let A: T2 → T2 be a hyperbolic toral automorphism. Let us call the eigenvalues λu and λs. They are both real and irrational and satisfy |λu| > 1 > |λs|.
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