remark 24.25 What the recurrence proof assumes about volume

Rests on

• depends_on theorem 24.24 Poincaré recurrence ¶
  • depends_on definition 24.20 Liouville volume ¶
    • depends_on definition 13.102 Wedge product ¶
      • depends_on definition 13.98 $k$-form ¶
        • depends_on definition 13.83 Covector ¶
        • depends_on definition 13.84 Tensor ¶
      • depends_on equation 13.239 eq:mfd-dual-basis ¶
    • depends_on definition 24.8 Symplectic manifold ¶
      • depends_on definition 13.105 Closed form ¶
        • depends_on definition 13.103 Exterior derivative ¶
        • depends_on definition 13.98 $k$-form ¶ ↺
      • depends_on definition 13.98 $k$-form ¶ ↺
      • depends_on definition 24.2 Symplectic vector space ¶
        • depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶
    • depends_on theorem 24.12 Darboux ¶
      • depends_on definition 24.8 Symplectic manifold ¶ ↺
      • depends_on theorem A.68 Darboux ¶
        • depends_on definition A.67 Symplectic manifold ¶
        • depends_on equation 5.144 eq:lin-symplectic-normal-form ¶
        • depends_on proposition 5.119 Normal form of a non-degenerate antisymmetric form ¶ ↺
        • proves proof app:A-long-proofs@proof-57 ¶
      • proves proof ch:07-symplectic-geometry@prooflink-1 ¶
  • depends_on theorem 24.21 Liouville ¶
    • depends_on definition 24.20 Liouville volume ¶ ↺
    • depends_on proposition 7.105 Clairaut–Schwarz ¶
      • depends_on definition 7.20 Continuity at a point ¶
        • depends_on definition 7.16 Limit ¶
        • depends_on equation 7.7 eq:ana-limit-left ¶
        • depends_on equation 7.5 eq:ana-limit-right ¶
      • depends_on theorem 7.35 Mean value theorem ¶
        • depends_on proposition 7.29 Linearity ¶
        • depends_on theorem 7.34 Rolle ¶
        • proves proof ch:05-real-analysis@proof-19 ¶
      • proves proof ch:05-real-analysis@proof-64 ¶
    • depends_on theorem 24.16 The Hamiltonian flow preserves the symplectic form ¶
      • depends_on definition 24.14 Hamiltonian vector field ¶
        • depends_on definition 24.8 Symplectic manifold ¶ ↺
        • depends_on equation 22.4 eq:ham-pdot ¶
        • depends_on equation 22.3 eq:ham-qdot ¶
      • depends_on definition 24.8 Symplectic manifold ¶ ↺
      • depends_on proposition 13.128 Cartan's magic formula ¶
        • depends_on definition 13.103 Exterior derivative ¶ ↺
        • depends_on equation 13.246 eq:mfd-leibniz-forms ¶
        • depends_on equation 13.245 eq:mfd-nilpotency ¶
        • depends_on proposition 13.127 Component formulas ¶
        • proves proof ch:11-manifolds-tensors-curvature@proof-29 ¶
      • proves proof ch:07-symplectic-geometry@proof-5 ¶
    • proves proof ch:07-symplectic-geometry@proof-8 ¶
  • proves proof ch:07-symplectic-geometry@proof-10 ¶

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