Lagrange and Markov spectra for typical smooth systems

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Abstract We prove that among the set of pairs ( $C^2$ -diffeomorphism, $C^1$ -potential), there exists a $C^1$ -open and dense subset such that either the Lagrange spectrum is finite and the dynamics is a Morse–Smale diffeomorphism or the Lagrange spectrum has positive Hausdorff dimension and the system has positive topological entropy. We also prove that such dichotomy does not hold for typical systems when replacing the Lagrange by the Markov spectrum.