C^∞ 3D flows with positive topological entropy admit only finitely many ergodic measures of maximal entropy, which are rapid mixing on a dense open set.
Zang, Measures of maximal entropy forC ∞ three-dimensional flows,Preprint
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For C^r surface diffeomorphisms with h_top(f) ≥ λ⁺(f)/r, h_top(f) equals lim (1/n) log ∫_M ||Df^n_x|| dx.
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Thermodynamics formalism for singular flows
C^∞ 3D flows with positive topological entropy admit only finitely many ergodic measures of maximal entropy, which are rapid mixing on a dense open set.
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Entropy formula for surface diffeomorphisms
For C^r surface diffeomorphisms with h_top(f) ≥ λ⁺(f)/r, h_top(f) equals lim (1/n) log ∫_M ||Df^n_x|| dx.