Constructions in elliptic dynamics

By Bassam Fayad, A. Katok. Ergodic Theory and Dynamical Systems, 2004

141 citations recorded by the source corpus.

We present an overview and some new applications of the approximation by conjugation method introduced by Anosov and Katok more than 30 years ago (Trans. Moscow Math. Soc. 23 (1970), 1–35). Michel Herman made important contributions to the development and applications of this method beginning from the construction of minimal and uniquely ergodic diffeomorphisms jointly with Fathi (Asterisque 49 (1977), 37–59) and continuing with exotic invariant sets of rational maps of the Riemann sphere (J. London Math. Soc. (2) 34 (1986), 375–384) and the construction of invariant tori with non-standard and unexpected behavior in the context of KAM theory (Pitman Research Notes Mathematical Series 243 (1992); Proc. Int. Congr. Mathematicians (Berlin, 1998) Vol. 11, 797–808). Recently the method has been experiencing a revival. Some of the new results presented in the paper illustrate variety of uses for tools available for a long time, others exploit new methods, in particular the possibility of mixing in the context of Liouvillean dynamics discovered by the first author (Ergod. Th. & Dynam. Sys. 22 (2002) 437–468; Proc. Amer. Math. Soc. 130 (2002), 103–109).

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1c43d96388e82cbcbb64817e13baad0736daceaf

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. J. Thouvenot
2. Basak Z. Gurel
3. V. Ginzburg
4. A. Katok
5. M. Saprykina
6. Bassam Fayad
7. Cristóbal Rivas
8. Andrés Navas

In Mathematical Physics

1. J. Thouvenot
2. V. Ginzburg
3. A. Katok
4. Basak Z. Gurel
5. M. Saprykina
6. Bassam Fayad
7. Cristóbal Rivas
8. Andrés Navas

In Statistical and Nonlinear Physics

1. Basak Z. Gurel
2. J. Thouvenot
3. A. Katok
4. M. Saprykina
5. Cristóbal Rivas
6. Bassam Fayad
7. V. Ginzburg
8. Andrés Navas

Citations in the corpus

• Chapter 11 Spectral properties and combinatorial constructions in ergodic theory (2006, methodology): “49, [47,78].”
• Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary (2005, result): “We quote the following proposition of [4] that is similar to corresponding statements in [1,11].”
• The Conley Conjecture and Beyond (2014, background): “Furthermore, in the setting of the theorem, there are also strong growth results; see Franks and Handel (2003), Le Calvez (2006) and Kerman (2012) for this and other refinements of Theorem 6.”
• Rigidity for $$C^1$$C1 actions on the interval arising from hyperbolicity I: solvable groups (2013, background): “The reader is referred to [15,16] for a general panorama on this, yet the construction we give below is self-contained.”
• CONSTRUCTION OF CURIOUS MINIMAL UNIQUELY ERGODIC HOMEOMORPHISMS ON MANIFOLDS: THE DENJOY-REES TECHNIQUE (2006, unknown): “See [12] for a survey.”