45,852 research outputs found
Symbolic extensions in intermediate smoothness on surfaces
We prove that maps with on a compact surface have
symbolic extensions, i.e. topological extensions which are subshifts over a
finite alphabet. More precisely we give a sharp upper bound on the so-called
symbolic extension entropy, which is the infimum of the topological entropies
of all the symbolic extensions. This answers positively a conjecture of
S.Newhouse and T.Downarowicz in dimension two and improves a previous result of
the author \cite{burinv}.Comment: 27 page
Counting points of fixed degree and bounded height
We consider the set of points in projective -space that generate an
extension of degree over given number field , and deduce an asymptotic
formula for the number of such points of absolute height at most , as
tends to infinity. We deduce a similar such formula with instead of the
absolute height, a so-called adelic-Lipschitz height
Markov extensions and lifting measures for complex polynomials
For polynomials on the complex plane with a dendrite Julia set we study
invariant probability measures, obtained from a reference measure. To do this
we follow Keller in constructing canonical Markov extensions. We discuss
``liftability'' of measures (both -invariant and non-invariant) to the
Markov extension, showing that invariant measures are liftable if and only if
they have a positive Lyapunov exponent. We also show that -conformal
measure is liftable if and only if the set of points with positive Lyapunov
exponent has positive measure.Comment: Some changes have been made, in particular to Sections 2 and 3, to
clarify the exposition. Typos have been corrected and references update
Fields with the Bogomolov property
A field is said to have the Bogomolov property relative to a height function h’ if and only if h’ is either zero or bounded from below by a positive constant for all for all elements in this field. In this thesis we study this property according to canonical heights associated to rational functions introduced by Call and Silverman in 1994. In the first part we will translate known results into the dynamical setting. Then we prove an effective version of a theorem of Baker which states that the Néron-Tate height of an elliptic curve with multiplicative reduction at a finite place v is bounded from below by a positive constant at points which are unramified over v. In the last section of this thesis we give a complete classification of rational functions f defined over the algebraic numbers such that the maximal totally real field has the Bogomolov property relative to the canonical height associated to f
Unbounded subnormal weighted shifts on directed trees
A new method of verifying the subnormality of unbounded Hilbert space
operators based on an approximation technique is proposed. Diverse sufficient
conditions for subnormality of unbounded weighted shifts on directed trees are
established. An approach to this issue via consistent systems of probability
measures is invented. The role played by determinate Stieltjes moment sequences
is elucidated. Lambert's characterization of subnormality of bounded operators
is shown to be valid for unbounded weighted shifts on directed trees that have
sufficiently many quasi-analytic vectors, which is a new phenomenon in this
area. The cases of classical weighted shifts and weighted shifts on leafless
directed trees with one branching vertex are studied.Comment: 32 pages, one figur
Extensions of Positive Definite Functions: Applications and Their Harmonic Analysis
We study two classes of extension problems, and their interconnections: (i)
Extension of positive definite (p.d.) continuous functions defined on subsets
in locally compact groups ; (ii) In case of Lie groups, representations of
the associated Lie algebras by unbounded skew-Hermitian
operators acting in a reproducing kernel Hilbert space (RKHS)
.
Why extensions? In science, experimentalists frequently gather spectral data
in cases when the observed data is limited, for example limited by the
precision of instruments; or on account of a variety of other limiting external
factors. Given this fact of life, it is both an art and a science to still
produce solid conclusions from restricted or limited data. In a general sense,
our monograph deals with the mathematics of extending some such given partial
data-sets obtained from experiments. More specifically, we are concerned with
the problems of extending available partial information, obtained, for example,
from sampling. In our case, the limited information is a restriction, and the
extension in turn is the full positive definite function (in a dual variable);
so an extension if available will be an everywhere defined generating function
for the exact probability distribution which reflects the data; if it were
fully available. Such extensions of local information (in the form of positive
definite functions) will in turn furnish us with spectral information. In this
form, the problem becomes an operator extension problem, referring to operators
in a suitable reproducing kernel Hilbert spaces (RKHS). In our presentation we
have stressed hands-on-examples. Extensions are almost never unique, and so we
deal with both the question of existence, and if there are extensions, how they
relate back to the initial completion problem.Comment: 235 pages, 42 figures, 7 tables. arXiv admin note: substantial text
overlap with arXiv:1401.478
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