8. Definition. Given a family of maps with corresponding invariant densities . Then is said to be acim-stable if implies .
The limits are taken with respect to properly chosen metrics on the space of maps and densities, respectively.
Functions of the bounded variation are continuous except at a most countable number of points, at which they have two one-sided limits.
7. Let be the 3D Heisenberg group, with . Let be a cocompact discrete subgroup (for example ). Then is a 3D nilmanifold. A general non-toral
three-dimensional nilmanifold is also of this form. Suppose we have a homomorphism , which is of the form , which induces a 2D-foliation, say on and on .
Theorem. Every Reebless foliation on is almost aligned with some .
Plante for , Hammerlindl and Potrie for .
Theorem. Every partially hyperbolic system on is accessible.
J. Rodriguez Hertz, F. Rodriguez Hertz and R. Ures (convervative), Hammerlindl and Potrie (general)
6. Let and be a function. Consider the solutions of the recurrence relation:
() for all .
Note that () is actually a finite sum of terms over . It is the derivative of formal series with respect to .
Example. Billiards, or generally twist maps, where and is the generating function, the solution gives the configuration of an orbit.
There are some conditions:
Periodicity. . So descends to a map on .
Monotone. for all and all , and for all .
Coercivity. is bounded from below and there exists such that as .
Under these conditions the () is called a monotone variational recurrence relation.
A sequence is said to be a global minimizer, if (understand as over all intervals) for all sequences . Clearly a global minimizer solves (). The collection of global minimizers is also closed under coordinately convergence.
For a real number , a sequence with is called an -minimizer, if it is minimizes among all ‘s with .
Ana-minimizers in general need not be solutions to ().
Given a rational , we consider the operator (shift and subtract ) and Birkhoff orbits of rotational number prime sum over the periodic ones .
Periodic Peierls barrier. Let be a real and be coprime. Then as
It is easy to see that
There exists a periodic minimizer with if and only if .
gives an invariant curve if and only if .
Then the Peierls barrier at a general frequency is defined as when the limit exists (see Mramor and Rink, arxiv:1308.3073).