Tobias Jäger

Linearization of conservative toral homeomorphisms

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.

Towards a Classification for Quasiperiodically Forced Circle Homeomorphisms

Poincares classification of the dynamics of homeomorphisms of the circle is one of the earliest, but still one of the most elegant, classification results in dynamical systems. Here we generalize this to quasiperiodically forced circle homeomorphisms homotopic to the identity, which have been the subject of considerable interest in recent years. Herman already showed two decades ago that a unique rotation number exists for all orbits in the quasiperiodically forced case. However, unlike the unforced case, no a priori bounds exist for the deviations from the average rotation. This plays an important role in the attempted classification, and in fact we define a system as

Quasiperiodically forced interval maps with negative Schwarzian derivative

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On the structure of strange non-chaotic attractors in pinched skew products

-bounded if such deviations are bounded and as

The Denjoy type of argument for quasiperiodically forced circle diffeomorphisms

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The concept of bounded mean motion for toral homeomorphisms

-unbounded otherwise. For the

Rotation numbers for quasiperiodically forced circle maps-mode-locking vs. strict monotonicity

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How chaotic are strange non-chaotic attractors?

-bounded case we prove a close analogue of Poincares result: if the rotation number is rationally related to the rotation rate on the base then there exists an invariant strip (the appropriate analogue for fixed or periodic points in this context), otherwise the system is semi-conjugate to an irrational translation of the torus. In the

Denjoy constructions for fibered homeomorphisms of the torus

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Elliptic stars in a chaotic night

-unbounded case, where neither of these two alternatives can occur, we show that the dynamics are always topologically transitive.