proposition 32.76 The exponents sum to the mean divergence

Rests on

• depends_on definition 32.73 Lyapunov spectrum ¶
  • depends_on equation 32.45 eq:chaos-variational ¶
• depends_on theorem 32.6 Evolution of phase volume ¶
  • depends_on definition 32.3 Dynamical system, phase space, flow ¶
    • depends_on definition 13.82 Vector field ¶
      • depends_on definition 13.81 Vector on a manifold ¶
        • depends_on definition 13.71 Differentiable curve ¶
        • depends_on definition 13.45 Differentiable map on a topological space ¶
        • depends_on definition 13.48 Differentiable manifold ¶
    • depends_on definition 9.1 Ordinary differential equation ¶
  • depends_on theorem 7.133 Gauss ¶
    • depends_on definition 7.127 Simple regions ¶
      • depends_on definition 7.98 Functions of class $C^{1}$ ¶
        • depends_on definition 7.20 Continuity at a point ¶
        • depends_on definition 7.97 Partial derivative; gradient ¶
      • depends_on definition 6.9 Compact set ¶
        • depends_on definition 6.5 Open cover ¶
    • depends_on remark 7.128 What the derivations below take as given ¶
      • depends_on definition 7.125 Multiple integral ¶
        • depends_on axiom 7.1 Completeness of $\R$ ¶
        • depends_on definition 7.39 Darboux sums and the definite integral ¶
      • depends_on theorem 7.40 Continuous functions are integrable ¶
        • depends_on definition 7.39 Darboux sums and the definite integral ¶ ↺
        • depends_on theorem 7.24 Extreme value theorem ¶
        • depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
        • proves proof ch:05-real-analysis@proof-23 ¶
      • depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶ ↺
    • depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
      • depends_on definition 7.41 Antiderivative ¶
        • depends_on corollary 7.36 cor:ana-mvt-consequences ¶
        • depends_on definition 7.26 Derivative of a function at a point ¶
      • depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
        • depends_on definition 7.26 Derivative of a function at a point ¶ ↺
        • depends_on theorem 7.40 Continuous functions are integrable ¶ ↺
        • depends_on theorem 7.24 Extreme value theorem ¶ ↺
        • proves proof ch:05-real-analysis@proof-24 ¶
      • proves proof ch:05-real-analysis@proof-25 ¶
    • proves proof ch:05-real-analysis@proof-79 ¶
  • proves proof ch:15-nonlinear-dynamics-chaos@proof-3 ¶
• proves proof ch:15-nonlinear-dynamics-chaos@proof-26 ¶

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