Description
Symbolic dynamics: represent dynamics by sequences. Full shift Σ = {0,1}^ℕ or Σ_n = {0,...,n-1}^ℕ; shift map σ(s₀s₁s₂...) = (s₁s₂s₃...). Subshift of finite type (SFT): forbidden blocks. Devaney's textbook uses symbolic dynamics to analyze chaos. Smale's horseshoe: hyperbolic invariant set conjugate to shift on two symbols. Connection: logistic map at μ=4 semiconjugate to shift; quadratic maps admit symbolic coding.
Source: Wikipedia; Devaney; Smale
Dependency Flowchart
graph TD
Chaos["Devaney Chaos"]
DefShift["Def: Full shift Σ\nSequences s = s₀s₁s₂..."]
DefSigma["Def: Shift map σ\nσ(s) = s₁s₂s₃..."]
DefSFT["Def: Subshift of finite type\nForbidden blocks"]
ThmShiftChaos["Thm: σ on Σ is chaotic\nTransitive, dense periodic, sensitive"]
ThmHorseshoe["Thm: Smale horseshoe\nHyperbolic set ~ shift on 2 symbols"]
ThmLogistic["Thm: Logistic μ=4\nSemiconjugate to shift"]
LemCoding["Lem: Symbolic coding\nItineraries for quadratic maps"]
Chaos --> DefShift
DefShift --> DefSigma
DefSigma --> DefSFT
DefSigma --> ThmShiftChaos
ThmHorseshoe --> ThmShiftChaos
ThmLogistic --> LemCoding
DefSigma --> ThmLogistic
classDef axiom fill:#ff6b6b,color:#fff,stroke:#c0392b
classDef definition fill:#b197fc,color:#fff,stroke:#9775fa
classDef theorem fill:#51cf66,color:#fff,stroke:#40c057
classDef lemma fill:#74c0fc,color:#fff,stroke:#4dabf7
class Chaos axiom
class DefShift,DefSigma,DefSFT definition
class ThmShiftChaos,ThmHorseshoe,ThmLogistic theorem
class LemCoding lemma
Color Scheme
Red Prerequisite
Blue Definitions
Teal Theorems
Purple Lemmas
Process Statistics
- Nodes: 8
- Edges: 9
- Axioms: 1
- Definitions: 3
- Lemmas: 1
- Theorems: 3