Markov partition
E911355
A Markov partition is a special decomposition of a dynamical system’s phase space into regions whose transitions are governed by a Markov process, enabling symbolic coding and analysis of the system’s behavior.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
concept in dynamical systems
ⓘ
mathematical concept ⓘ |
| appliesTo |
continuous-time flows via Poincaré sections
ⓘ
discrete-time dynamical systems ⓘ |
| characterizedBy |
Markov property for transitions between partition elements
ⓘ
boundaries of measure zero in many constructions ⓘ finite collection of sets covering almost all of phase space ⓘ rectangles with product structure in hyperbolic systems ⓘ |
| enables |
analysis via Markov chains
ⓘ
coding of trajectories by sequences of symbols ⓘ computation of topological entropy ⓘ construction of invariant measures ⓘ construction of subshifts of finite type ⓘ symbolic dynamics ⓘ thermodynamic formalism ⓘ |
| field |
dynamical systems
ⓘ
ergodic theory ⓘ symbolic dynamics ⓘ |
| hasOutcome | symbolic model that is a shift of finite type ⓘ |
| hasProperty |
can be refined to obtain more detailed codings
ⓘ
can yield conjugacy for certain systems ⓘ gives semi-conjugacy to a symbolic shift ⓘ often not unique ⓘ |
| hasPurpose |
to decompose phase space into regions with Markov transition properties
ⓘ
to enable symbolic coding of orbits ⓘ to facilitate computation of dynamical invariants ⓘ to reduce continuous dynamics to a shift on symbols ⓘ |
| hasRepresentation | adjacency matrix describing allowed transitions ⓘ |
| introducedInContextOf | hyperbolic dynamics of Smale ⓘ |
| relatedTo |
Markov chain
NERFINISHED
ⓘ
Markov process NERFINISHED ⓘ Perron–Frobenius operator NERFINISHED ⓘ Smale’s spectral decomposition NERFINISHED ⓘ hyperbolic set ⓘ stable and unstable manifolds ⓘ subshift of finite type ⓘ topological Markov chain NERFINISHED ⓘ transition matrix ⓘ |
| usedFor |
computing zeta functions of dynamical systems
ⓘ
proving mixing properties ⓘ studying chaotic behavior ⓘ studying invariant sets ⓘ studying periodic orbits ⓘ |
| usedIn |
Anosov diffeomorphisms
NERFINISHED
ⓘ
Axiom A systems ⓘ Smale horseshoe NERFINISHED ⓘ geodesic flows on negatively curved manifolds ⓘ hyperbolic dynamical systems ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.