68,587 research outputs found
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
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
Graphs that do not contain a cycle with a node that has at least two neighbors on it
We recall several known results about minimally 2-connected graphs, and show
that they all follow from a decomposition theorem. Starting from an analogy
with critically 2-connected graphs, we give structural characterizations of the
classes of graphs that do not contain as a subgraph and as an induced subgraph,
a cycle with a node that has at least two neighbors on the cycle. From these
characterizations we get polynomial time recognition algorithms for these
classes, as well as polynomial time algorithms for vertex-coloring and
edge-coloring
Finitely ramified iterated extensions
Let K be a number field, t a parameter, F=K(t) and f in K[x] a polynomial of
degree d. The polynomial P_n(x,t)= f^n(x) - t in F[x] where f^n is the n-fold
iterate of f, is absolutely irreducible over F; we compute a recursion for its
discriminant. Let L=L(f) be the field obtained by adjoining to F all roots, in
a fixed algebraic closure, of P_n for all n; its Galois group Gal(L/F) is the
iterated monodromy group of f. The iterated extension L/F is finitely ramified
if and only if f is post-critically finite (pcf). We show that, moreover, for
pcf polynomials f, every specialization of L/F at t=t_0 in K is finitely
ramified over K, pointing to the possibility of studying Galois groups with
restricted ramification via tree representations associated to iterated
monodromy groups of pcf polynomials. We discuss the wildness of ramification in
some of these representations, describe prime decomposition in terms of certain
finite graphs, and also give some examples of monogene number fields.Comment: 19 page
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
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