Intrinsic ergodicity of smooth interval maps
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We generalize the technique of Markov Extension, introduced by F. Hofbauer  for piecewise monotonic maps, to arbitrary smooth interval maps. We also use A. M. Blokh’s  Spectral Decomposition, and a strengthened version of Y. Yomdin’s  and S. E. Newhouse’s  results on differentiable mappings and local entropy.
In this way, we reduce the study ofC r interval maps to the consideration of a finite number of irreducible topological Markov chains, after discarding a small entropy set. For example, we show thatC ∞ maps have the same properties, with respect to intrinsic ergodicity, as have piecewise monotonic maps.
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- J. Buzzi,Intrinsic ergodicity of affine maps on [0, 1]d, Monatshefte für Matematik (to appear).Google Scholar
- M. Gromov,Entropy, homology and semi-algebraic geometry, Séminaire Bourbaki663, 1985–1986.Google Scholar
- B. M. Gurevič,Topological entropy of enumerable Markov chains, Soviet Mathematics Doklady10 (1969), 911–915.Google Scholar
- B. M. Gurevič,Shift entropy and Markov measures in the path space of a denumerable graph, Soviet Mathematics Doklady11 (1970), 744–747.Google Scholar
- S. E. Newhouse,On some results of F. Hofbauer on maps of the interval, inDynamical Systems and Related Topics (K. Shiraiwa, ed.), Proc. Nagoya 1990, World Scientific, Singapore, 1991, pp. 407–422.Google Scholar
- K. Petersen,Ergodic Theory, Cambridge University Press, 1983.Google Scholar
- I. A. Salama,Topological entropy and classification of countable chains, Ph.D. thesis, University of North Carolina, Chapel Hill, 1984.Google Scholar