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I presented how surface diffeomorphisms are Borel conjugate to Markov shifts up to zero entropy measures (joint work with Mike Boyle). I stressed the soft part of the result that applies to higher dimensional diffeomorphisms with hyperbolic measures but neglects the measures maximizing the entropy at their period.

The spectral decomposition recently obtained with Sylvain Crovisier and Omri Sarig  shows that the C infinity surface diffeos are completely classified by their measures maximizing the entropy at their periods.

 

(revised slides)

 

My talk presented the result with Crovisier and Sarig about the measures maximizing the entropy. I used slides (revised version) for the first part of the seminar and then explained the decay of the entropy along sequences of non-homoclinically related measures on the blackboard.

J’ai présenté en séminaire les résultats obtenus avec Mike BOYLE d’une part (conjugaison borélienne modulo les mesures d’entropie nulle avec une chaîne de Markov topologique), avec Sylvain CROVISIER et Omri SARIG (nombre fini de mesures d’entropie maximale dans le cas C infini):

  • Séminaire d’Analyse de Bordeaux, 3 mars 2016 (transparents)
  • Séminaire de Systèmes Dynamiques, 2 février 2016.

Updating SVN on Mac OS X

When I tried to update a local copy of a SVN repository I got an error message similar to “This client is too old to work with working copy…”. I went to http://www.wandisco.com/subversion/download#osx I clicked on: “Mac OSX 10.9 – Mavericks – Subversion 1.9.2 – Client Only” There I gave away my privacy and got a package. I launched it and in exchange for total control over my computer it installed svn 1.9. Pursuant to the last message displayed by the installation I opened a terminal window and created on the current (home) directory a file named “.profile” with the single line “export PATH=/opt/subversion/bin:$PATH”. I closed the terminal and opened it again and the command “svn –version” confirmed that the new version was installed…

La Direction des Services Informatiques a mis en place une alternative à Dropbox ouverte aux membres des laboratoires CNRS. Dans le principe, on ne peut qu’applaudir. Concrètement, après moins de deux jours d’utilisation, j’ai perdu la synchronisation avec mes fichiers, ce qui ne m’est jamais arrivé en cinq années avec Dropbox. Apparemment le service est planté…

One expects measure maximizing the entropy (m.m.e.) to be especially “interesting”, especially for dynamics with “some hyperbolicity”. For instance, under some (hyperbolicity) assumptions, one expects them to determine all the aperiodic invariant probability measure (see this expository paper). By a theorem of Newhouse (1987) based on Yomdin’s theory, C^\infty smoothness ensures the existence of some m.m.e.

Finite multiplicity is a harder question – often it can be solved only after a thorough understanding of the dynamics. It is a classical result for uniformly hyperbolic diffeomorphisms. I proved it in my thesis for C^\infty interval maps with nonzero entropy (finite smoothness is not enough, even though Ruelle’s inequality shows that all ergodic measures with lower bounded entropy have lower bounded Lyapunov exponents).

Here, at the School and Conference on Dynamical Systems at ICTP, I presented the following answer to a long standing question of Newhouse:

Theorem (B-Crovisier-Sarig). A C^\infty smooth diffeomorphism of a compact surface with nonzero topological entropy has finitely many ergodic measures maximizing the entropy.

You can see the slides here.

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