Constructions in elliptic dynamics

Abstract

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).

Non-standard smooth realizations of Liouville rotations

“…We quote the following proposition of [2] that is similar to other statements in [1,6] Proposition 1. [2, proposition 5.2] Let M be an d-dimensional compact connected C ∞ manifold with an effective circle action σ preserving a smooth volume µ.…”

“…The diffeomorphism T n is given by (2) T n := H n S αn H −1 n where α n ∈ Q and H n ∈ Diff ∞ (M, λ). We choose a sequence α n := p ′ n /q ′ n such that |α n − α| → 0 monotonically.…”

The Conley Conjecture and Beyond

“…In dimension two, there are also such examples with interesting dynamics. Namely, there exist area preserving diffeomorphisms of S 2 with exactly three ergodic measures: two fixed points and the area form; see Anosov and Katok (1970) and, e.g., Fayad and Katok (2004). These are the so-called pseudo-rotations.…”

Spectral Properties and Combinatorial Constructions in Ergodic Theory

“…Simple rigid spectrum, whether atomic, mixed or continuous, is the second type (after countable Lebesgue spectrum) ubiquitous in ergodic theory and other branches of dynamics. These spectral properties are associated with the elliptic behavior [8,Section 7] in its two manifestations, Diophantine and Liouvillean [47]. Simplicity of the spectrum relies on criteria like Theorem 1.21, rigidity on Proposition 2.10.…”