215 research outputs found
Dynamical properties of the Pascal adic transformation
We study the dynamics of a transformation that acts on infinite paths in the
graph associated with Pascal's triangle. For each ergodic invariant measure the
asymptotic law of the return time to cylinders is given by a step function. We
construct a representation of the system by a subshift on a two-symbol alphabet
and then prove that the complexity function of this subshift is asymptotic to a
cubic, the frequencies of occurrence of blocks behave in a regular manner, and
the subshift is topologically weak mixing
Modules with irrational slope over tubular algebras
Let be a tubular algebra and let be a positive irrational. Let
be the definable subcategory of -modules of slope . Then
the width of the lattice of pp formulas for is . It
follows that if is countable then there is a superdecomposable
pure-injective module of slope .Comment: minor corrections/improvements to argument
Quantitative sheaf theory
We introduce a notion of complexity of a complex of ell-adic sheaves on a
quasi-projective variety and prove that the six operations are "continuous", in
the sense that the complexity of the output sheaves is bounded solely in terms
of the complexity of the input sheaves. A key feature of complexity is that it
provides bounds for the sum of Betti numbers that, in many interesting cases,
can be made uniform in the characteristic of the base field. As an
illustration, we discuss a few simple applications to horizontal
equidistribution results for exponential sums over finite fields.Comment: v3, 68 pages; the key ideas of this paper are due to W. Sawin; A.
Forey, J. Fres\'an and E. Kowalski drafted the current version of the text;
revised after referee report
Generalizations of Sturmian sequences associated with -continued fraction algorithms
Given a positive integer and irrational between zero and one, an
-continued fraction expansion of is defined analogously to the classical
continued fraction expansion, but with the numerators being all equal to .
Inspired by Sturmian sequences, we introduce the -continued fraction
sequences and , which are related to the
-continued fraction expansion of . They are infinite words over a two
letter alphabet obtained as the limit of a directive sequence of certain
substitutions, hence they are -adic sequences. When , we are in the
case of the classical continued fraction algorithm, and obtain the well-known
Sturmian sequences. We show that and are
-balanced for some explicit values of and compute their factor
complexity function. We also obtain uniform word frequencies and deduce unique
ergodicity of the associated subshifts. Finally, we provide a Farey-like map
for -continued fraction expansions, which provides an additive version of
-continued fractions, for which we prove ergodicity and give the invariant
measure explicitly.Comment: 23 pages, 2 figure
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Mini-Workshop: The Pisot Conjecture - From Substitution Dynamical Systems to Rauzy Fractals and Meyer Sets
This mini-workshop brought together researchers with diverse backgrounds and a common interest in facets of the Pisot conjecture, which relates certain properties of a substitution to dynamical properties of the associated subshift
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