5,623 research outputs found
Tischler graphs of critically fixed rational maps and their applications
A rational map on the Riemann
sphere is called critically fixed if each critical point
of is fixed under . In this article we study properties of a
combinatorial invariant, called Tischler graph, associated with such a map.
More precisely, we show that the Tischler graph of a critically fixed rational
map is always connected, establishing a conjecture made by Kevin Pilgrim. We
also discuss the relevance of this result for classical open problems in
holomorphic dynamics, such as combinatorial classification problem and global
curve attractor problem
A Combinatorial classification of postcritically fixed Newton maps
We give a combinatorial classification for the class of postcritically fixed
Newton maps of polynomials as dynamical systems. This lays the foundation for
classification results of more general classes of Newton maps.
A fundamental ingredient is the proof that for every Newton map
(postcritically finite or not) every connected component of the basin of an
attracting fixed point can be connected to through a finite chain of
such components.Comment: 37 pages, 5 figures, published in Ergodic Theory and Dynamical
Systems (2018). This is the final author file before publication. Text
overlap with earlier arxiv file observed by arxiv system relates to an
earlier version that was erroneously uploaded separately. arXiv admin note:
text overlap with arXiv:math/070117
A Multiscale Pyramid Transform for Graph Signals
Multiscale transforms designed to process analog and discrete-time signals
and images cannot be directly applied to analyze high-dimensional data residing
on the vertices of a weighted graph, as they do not capture the intrinsic
geometric structure of the underlying graph data domain. In this paper, we
adapt the Laplacian pyramid transform for signals on Euclidean domains so that
it can be used to analyze high-dimensional data residing on the vertices of a
weighted graph. Our approach is to study existing methods and develop new
methods for the four fundamental operations of graph downsampling, graph
reduction, and filtering and interpolation of signals on graphs. Equipped with
appropriate notions of these operations, we leverage the basic multiscale
constructs and intuitions from classical signal processing to generate a
transform that yields both a multiresolution of graphs and an associated
multiresolution of a graph signal on the underlying sequence of graphs.Comment: 16 pages, 13 figure
- …