Dynamical systems and ergodic theory Smooth and hyperbolic dynamics Example

Arnold's cat map

Builds on Torus Eigenvalue Lebesgue measure

Example

Let $\mathbb{T}^2 = \R^2/\Z^2$ be the torus and let $A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ . Arnold's cat map is $f(x, y) = (2x + y, \; x + y) \bmod 1$ . It is well defined because $A$ maps $\Z^2$ to $\Z^2$ , and it is a diffeomorphism because $\det A = 1$ , so $A^{-1}$ also has integer entries.

Check that it is an Anosov diffeomorphism. The derivative of $f$ is $A$ at every point. The characteristic polynomial is $t^2 - 3t + 1$ , so the eigenvalues are $\lambda_\pm = (3 \pm \sqrt5)/2$ , with $\lambda_+ \approx 2.618$ and $\lambda_- = 1/\lambda_+ \approx 0.382$ . Let $E^u$ and $E^s$ be the eigenlines for $\lambda_+$ and $\lambda_-$ ; since $A$ is symmetric they are orthogonal. Then $\norm{A^n v} = \lambda_-^n \norm{v}$ on $E^s$ and $\norm{A^{-n} v} = \lambda_+^{-n}\norm{v}$ on $E^u$ , which is the required hyperbolicity with $C = 1$ .

Remarks

The map preserves Lebesgue measure, since $\abs{\det A} = 1$ , and it is mixing for this measure. Its periodic points are exactly the points with rational coordinates, so they are dense. The number of fixed points of $f^n$ is $\abs{\det(A^n - I)} = \lambda_+^n + \lambda_+^{-n} - 2$ ; for $n = 1$ this is $1$ , the origin. Its topological entropy is $\log \lambda_+$ .

Reading path

To understand this page from the ground up, read these 72 pages in order. The foundations come first.

Borel sigma-algebra concept
Element definition
Euclidean space object
Outer measure concept
Proposition concept
Real numbers object
Vector space definition
Set definition
Lebesgue measure object
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Linear map definition
Empty set object
Family of sets definition
Ordered pair definition
Intersection definition
Union definition
Set-builder notation concept
Existential quantifier definition
Universal quantifier definition
Endomorphism definition
Cartesian product definition
Subset definition
Union of a family of sets definition
Uniqueness quantifier definition
Eigenvalue definition
Binary relation definition
Universal set definition
Topological space definition
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Complement definition
Function definition
Basis of a topology definition
Subspace topology definition
Equivalence relation definition
Product topology definition
Preimage of a set definition
Metric space definition
Codomain definition
Function composition definition
Injective function definition
Equivalence class definition
Open ball definition
Surjective function definition
Metric topology definition
Quotient set definition
Bijective function definition
Open set definition
Quotient topology definition
Inverse function definition
Hausdorff space definition
Open cover definition
Continuous map of topological spaces definition
Closed set definition
Compact space definition
Complement of an open set is closed statement
Homeomorphism definition
Universal property of the quotient topology statement
Continuity via closed sets statement
Compact set definition
Compactness via open covers statement
Compact subsets of a Hausdorff space are closed statement
Continuous image of a compact set is compact statement
Closed subsets of compact sets are compact statement
Continuous bijection from a compact space to a Hausdorff space is a homeomorphism statement
Circle as a quotient of the real line example
Torus example
Arnold's cat map example

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Pseudo-Anosov map

Tag	`BSCZN8Y1AN`
URI	`m2://BSCZN8Y1AN`
Kind	example
Writing	W0 stub
MSC	37D20, 37A25
Instance of	Anosov diffeomorphism
Uses	Torus, Eigenvalue, Lebesgue measure
Same as	Wikidata Q2961488 Wikipedia Arnold's cat map

Arnold's cat map

Example

Remarks

Reading path

Where this leads

Further reading

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