## Institute for Mathematical Sciences

## Preprint ims90-3

** J. J. P. Veerman and F. M. Tangerman**
* On Aubry Mather Sets*

Abstract: Let f be a two dimensional area preserving twist map. For each irrational rotation number in a certain (non trivial) interval, there is an f-invariant
minimal set which preserves order with respect to that rotation number. For large
nonlinearity these sets are, typically, Cantor sets and they are referred to as Aubry
Mather sets.
We prove that under csome assumptions these sets are ordered vertically according
to ascending rotation number ("Monotonicity"). Furthermore, if f statisfies certain
conditions, the right hand points of the gaps in an irrational Cantor set lie on a
single orbit ("Single Gap") and diffusion through these Aubry Mather sets can be
understood as a limit of resonance overlaps (Convergence of Turnstiles). These
conditions essentially establish the existence of a hyperbolic structure and limit the
number of homoclinic minimizing orbits.
Some other results along similar lines are given, such as the continuity at irrational
rotation numbers of the Lyapunov exponent on Aubry Mather sets.

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