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Posts Tagged ‘hyperbolicity’

A major part of dynamical system theory studies the iteration of diffeomorphisms under various assumptions, geometric or analytic or otherwise. It is also interesting to study them collectively. In particular, Diff^r(M), the set of C^r diffeomorphisms of some manifold M and some number r\geq1, is a group. It is for instance of interest to know when it is simple.

Todd Fisher gave a talk in Orsay about the (non)triviality of the centralizer for many elements. Recall that the centralizer of an element g of a group G is Z(g):=\{h\in G:hg=gh\}, the set of all elements that commute with g. It is also the set of all elements h that conjugate g with itself, i.e., it can be interpreted as the set of symmetries of g.

Z(g) may be very large, e.g., equal to the whole of G if this group is Abelian. It is said to be trivial if it is reduced to \{g^n:n\in\mathbb Z\} (which is always part of it). Note that such a diffeomorphism cannot be embedded in a smooth flow.

One expects that the set \mathcal T^r(M) of C^r-diffeomorphisms of M with trivial centralizer is very big (residual or even open and dense). To show this for all compact manifolds and all 1\leq r\leq\infty) was one of Smale’s questions for the the twenty-first century.

Results are known in dimension one or C^1-smoothness under hyperbolicity assumptions:

It has turned out that the above is really only generic and fails to hold on an open and dense set according to Bonatti, Crovisier, Vago and Wilkinson. On the circle, this holds only on a set with empty interior. This last result extends to tori in dimensions up to 4 according to a work in progress of Bakker and Fisher.

Hyperbolicity also allows some results:

  • a generic C^1-axiom A diffeomorphisms with no cycles has trivial centralizer (Tugawa 1978);
  • there is an open and dense set of surface C^\infty axiom-A diffeomorphisms with no cycles having a trivial C^1-centralizer (Palis-Yoccoz 1989). This even holds for arbitrary dimensions if one assumes the existence of a sink or a source.

On surfaces, Fisher has extended the last Palis-Yoccoz result to intermediate smoothness, i.e., C^r, 2\leq r\leq \infty and axiom A with no-cycles, i.e., \Omega-stable, diffeomorphisms instead of the above ones with strong transversality, i.e., the structurally stable ones.

The proof of Palis and Yoccoz (and of its later generalizations) relies on a local theorem showing that on each basin, two commuting maps must be iterates one of the other and a global theorem that connects the basins and identify the iterates. The last is the delicate part of the proof.

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I have shown that, like diffeomorphisms, piecewise affine surface homeomorphisms are approximated in entropy by horseshoes, away from their singularities. It follows in particular that their topological entropy is lower-semicontinuous: a small perturbation cannot cause a macroscopic drop in entropy.

The continuity of the entropy for such maps had been an open problem for some time. Rigorous numerical estimates by Duncan SANDS and Yutaka ISHII seemed to suggest some discontinuous drops, but investigation at a small scale suggested these drops to be steep yet continuous variations.

Izzet B. YILDIZ has solved this question by finding for Lozi maps f_{a,b}(x,y)=(1-a|x|+by,x) on \mathbb R^2, small numbers \epsilon_1,\epsilon_2>0 such that, setting (a,b)=(1.4+\epsilon_1,0.4+\epsilon_1), for all 0<\epsilon<\epsilon_2:

  • h_{top}(f_{a,b})=0;
  • h_{top}(f_{a+\epsilon,b})>\frac14\log\frac12(\sqrt{5}+1).

The verification turns out to be quite simple (once you know where to look!). The non-wandering set of f_{a,b} is shown to be reduced to be reduced to the fixed points of its fourth iterates, yielding the zero entropy immediately. f_{a+\epsilon,b} on the other hand is shown to admit 2 disjoint closed quadrilaterals U,V such that f^4(U) hyperbolically crosses both U and V and f^4(V) hyperbolically crosses U. This means that the sides of U and V can be branded alternatively s and u with the following property. The image of a u side crosses each of U,V it meets, intersecting both their s sides and none of their u sides. This again yields the entropy estimate.

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