Linearization of conservative toral homeomorphisms

Abstract

We give an equivalent condition for the existence of a semi-conjugacy to an irrational rotation for conservative homeomorphisms of the two-torus. This leads to an analogue of Poincaré's classification of circle homeomorphisms for conservative toral homeomorphisms with unique rotation vector and a certain bounded mean motion property. For minimal toral homeomorphisms, the result extends to arbitrary dimensions. Further, we provide a basic classification for the dynamics of toral homeomorphisms with all points non-wandering.

On Annular Maps of the Torus and Sublinear Diffusion

“…We now state a result from [18] concerning the construction of circloids. We say that a set U ⊂ R × T 1 is an upper generating set if U is bounded to the left and U is essential.…”

“…For example, in [18] it is given a Poincaré-like classification theorem for conservative pseudorotations, and in [23] a classification theorem is given for rational pseudorotations, that is, for the case that ρ( f ) is a single rational vector. Pseudorotations have been thoroughly studied.…”

Aperiodic invariant continua for surface homeomorphisms

“…This allows a sort of decomposition of the dynamics in terms of the aperiodic invariant continua. Similar concepts appear in the work of Jäger [8] (where the word circloid is used instead of frontier) when studying nonwandering homeomorphisms of the torus with bounded mean motion.…”

Diffeomorphisms having rotation sets with non-empty interior

“…Remark 2.3. Other notions related to essential sets for surface homeomorphisms were proposed recently by T. Jäger (called "circloids" in [5]) and by A. Koropecki (called "annular sets" in [9]). …”