Arnold cat map
E1046813
Anosov diffeomorphism
area-preserving map
chaotic map
discrete-time dynamical system
dynamical system
ergodic transformation
hyperbolic dynamical system
measure-preserving transformation
mixing transformation
toral automorphism
The Arnold cat map is a famous example of a chaotic, area-preserving transformation on the torus that illustrates how simple deterministic rules can produce complex, seemingly random behavior.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Arnold cat map canonical | 1 |
| Arnold’s cat map | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T13561558 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Arnold cat map Context triple: [Vladimir Arnold, knownFor, Arnold cat map]
-
A.
Smale horseshoe
The Smale horseshoe is a fundamental example in dynamical systems theory that illustrates chaotic behavior through a specific stretching-and-folding map of a square into a horseshoe-shaped region.
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B.
Poincaré map
The Poincaré map is a mathematical tool in dynamical systems theory that reduces continuous-time dynamics to a discrete map by tracking intersections of trajectories with a lower-dimensional surface.
-
C.
Lorenz attractor
The Lorenz attractor is a famous chaotic set arising from a simplified model of atmospheric convection, known for its butterfly-shaped trajectory and role as an early example of deterministic chaos in dynamical systems.
-
D.
Lyapunov fractal
The Lyapunov fractal is a complex, self-similar pattern arising from iterating logistic maps with periodically varying parameters, used to visualize stability and chaos in dynamical systems.
-
E.
Denjoy
Denjoy is a French surname most notably associated with mathematician Arnaud Denjoy, known for his contributions to real analysis and the theory of integration.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Arnold cat map Target entity description: The Arnold cat map is a famous example of a chaotic, area-preserving transformation on the torus that illustrates how simple deterministic rules can produce complex, seemingly random behavior.
-
A.
Smale horseshoe
The Smale horseshoe is a fundamental example in dynamical systems theory that illustrates chaotic behavior through a specific stretching-and-folding map of a square into a horseshoe-shaped region.
-
B.
Poincaré map
The Poincaré map is a mathematical tool in dynamical systems theory that reduces continuous-time dynamics to a discrete map by tracking intersections of trajectories with a lower-dimensional surface.
-
C.
Lorenz attractor
The Lorenz attractor is a famous chaotic set arising from a simplified model of atmospheric convection, known for its butterfly-shaped trajectory and role as an early example of deterministic chaos in dynamical systems.
-
D.
Lyapunov fractal
The Lyapunov fractal is a complex, self-similar pattern arising from iterating logistic maps with periodically varying parameters, used to visualize stability and chaos in dynamical systems.
-
E.
Denjoy
Denjoy is a French surname most notably associated with mathematician Arnaud Denjoy, known for his contributions to real analysis and the theory of integration.
- F. None of above. chosen
Statements (56)
| Predicate | Object |
|---|---|
| instanceOf |
Anosov diffeomorphism
ⓘ
area-preserving map ⓘ chaotic map ⓘ discrete-time dynamical system ⓘ dynamical system ⓘ ergodic transformation ⓘ hyperbolic dynamical system ⓘ measure-preserving transformation ⓘ mixing transformation ⓘ toral automorphism ⓘ |
| actsOn | unit square with opposite sides identified ⓘ |
| definedOn |
2-dimensional torus
ⓘ
T^2 ⓘ |
| hasCharacteristicPolynomial | λ^2 - 3λ + 1 ⓘ |
| hasDeterminant | 1 ⓘ |
| hasDomain | [0,1) × [0,1) with opposite sides identified ⓘ |
| hasEigenvalues |
(3+√5)/2
ⓘ
(3-√5)/2 ⓘ |
| hasMatrixRepresentation | [[2,1],[1,1]] modulo 1 ⓘ |
| hasProperty |
area-preserving
ⓘ
chaotic ⓘ deterministic ⓘ ergodic with respect to Lebesgue measure ⓘ exhibits exponential divergence of nearby trajectories ⓘ has dense orbits ⓘ has periodic orbits of arbitrarily large period ⓘ has positive Lyapunov exponents ⓘ has sensitive dependence on initial conditions ⓘ invertible ⓘ is a Bernoulli system (measure-theoretically conjugate to a Bernoulli shift) ⓘ is a prototype of deterministic chaos ⓘ is structurally stable ⓘ linear on the torus ⓘ preserves Lebesgue measure ⓘ topologically mixing ⓘ uniformly hyperbolic ⓘ volume-preserving on the torus ⓘ |
| hasRange | [0,1) × [0,1) with opposite sides identified ⓘ |
| hasTrace | 3 ⓘ |
| hasTypicalVisualization | iterative scrambling and reappearance of a cat image ⓘ |
| illustrates |
how simple deterministic rules can produce complex behavior
ⓘ
mixing of phase space ⓘ |
| isAlsoKnownAs |
Arnold’s cat map
ⓘ
linked to:
Arnold cat map
cat map ⓘ |
| isAreaPreservingBecause | its defining matrix has determinant 1 ⓘ |
| isDefinedByFormula |
(x',y') = (2x + y, x + y) mod 1
ⓘ
(x',y') = A(x,y) mod 1 with A = [[2,1],[1,1]] ⓘ |
| named mathematical object in dynamical systems theory},{ | Vladimir Arnold ⓘ |
| relatedTo |
Anosov systems
ⓘ
Lyapunov exponents ⓘ ergodic theory ⓘ mixing in Hamiltonian systems ⓘ symbolic dynamics ⓘ |
| usedFor |
pedagogical examples in dynamical systems courses
ⓘ
studying properties of hyperbolic toral automorphisms ⓘ visual demonstrations of chaos ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Arnold cat map Description of subject: The Arnold cat map is a famous example of a chaotic, area-preserving transformation on the torus that illustrates how simple deterministic rules can produce complex, seemingly random behavior.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Arnold cat map