926 research outputs found
Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces
Transfer operators such as the Perron--Frobenius or Koopman operator play an
important role in the global analysis of complex dynamical systems. The
eigenfunctions of these operators can be used to detect metastable sets, to
project the dynamics onto the dominant slow processes, or to separate
superimposed signals. We extend transfer operator theory to reproducing kernel
Hilbert spaces and show that these operators are related to Hilbert space
representations of conditional distributions, known as conditional mean
embeddings in the machine learning community. Moreover, numerical methods to
compute empirical estimates of these embeddings are akin to data-driven methods
for the approximation of transfer operators such as extended dynamic mode
decomposition and its variants. One main benefit of the presented kernel-based
approaches is that these methods can be applied to any domain where a
similarity measure given by a kernel is available. We illustrate the results
with the aid of guiding examples and highlight potential applications in
molecular dynamics as well as video and text data analysis
The Invariant Measures of some Infinite Interval Exchange Maps
We classify the locally finite ergodic invariant measures of certain infinite
interval exchange transformations (IETs). These transformations naturally arise
from return maps of the straight-line flow on certain translation surfaces, and
the study of the invariant measures for these IETs is equivalent to the study
of invariant measures for the straight-line flow in some direction on these
translation surfaces. For the surfaces and directions for which our methods
apply, we can characterize the locally finite ergodic invariant measures of the
straight-line flow in a set of directions of Hausdorff dimension larger than
1/2. We promote this characterization to a classification in some cases. For
instance, when the surfaces admit a cocompact action by a nilpotent group, we
prove each ergodic invariant measure for the straight-line flow is a Maharam
measure, and we describe precisely which Maharam measures arise. When the
surfaces under consideration are finite area, the straight-line flows in the
directions we understand are uniquely ergodic. Our methods apply to translation
surfaces admitting multi-twists in a pair of cylinder decompositions in
non-parallel directions.Comment: 107 pages, 11 figures. Minor improvement
Thurston's sphere packings on 3-dimensional manifolds, I
Thurston's sphere packing on a 3-dimensional manifold is a generalization of
Thusrton's circle packing on a surface, the rigidity of which has been open for
many years. In this paper, we prove that Thurston's Euclidean sphere packing is
locally determined by combinatorial scalar curvature up to scaling, which
generalizes Cooper-Rivin-Glickenstein's local rigidity for tangential sphere
packing on 3-dimensional manifolds. We also prove the infinitesimal rigidity
that Thurston's Euclidean sphere packing can not be deformed (except by
scaling) while keeping the combinatorial Ricci curvature fixed.Comment: Arguments are simplife
- …