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Smooth conjugacy classes of circle diffeomorphisms with irrational rotation number

100%

Bonatti C., Guelman N.

Fundamenta Mathematicae

2014

tom 227

nr 2

129-162

We prove the C¹-density of every $C^r$-conjugacy class in the closed subset of diffeomorphisms of the circle with a given irrational rotation number.

Generic diffeomorphisms on compact surfaces

81%

Abdenur F., Bonatti C., Crovisier S., Díaz L.

Fundamenta Mathematicae

2005

tom 187

nr 2

127-159

We discuss the remaining obstacles to prove Smale's conjecture about the C¹-density of hyperbolicity among surface diffeomorphisms. Using a C¹-generic approach, we classify the possible pathologies that may obstruct the C¹-density of hyperbolicity. We show that there are essentially two types of obstruction: (i) persistence of infinitely many hyperbolic homoclinic classes and (ii) existence of a single homoclinic class which robustly exhibits homoclinic tangencies. In the course of our discussion, we obtain some related results about C¹-generic properties of surface diffeomorphisms involving homoclinic classes, chain-recurrence classes, and hyperbolicity. In particular, it is shown that on a connected surface the C¹-generic diffeomorphisms whose non-wandering sets have non-empty interior are the Anosov diffeomorphisms.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Bonatti C.

1 Abdenur F.

1 Crovisier S.

1 Díaz L.

1 Guelman N.

1 2014

1 2005

Wyniki wyszukiwania

Smooth conjugacy classes of circle diffeomorphisms with irrational rotation number

Generic diffeomorphisms on compact surfaces