202 research outputs found
Integers in number systems with positive and negative quadratic Pisot base
We consider numeration systems with base and , for quadratic
Pisot numbers and focus on comparing the combinatorial structure of the
sets and of numbers with integer expansion in base
, resp. . Our main result is the comparison of languages of
infinite words and coding the ordering of distances
between consecutive - and -integers. It turns out that for a
class of roots of , the languages coincide, while for other
quadratic Pisot numbers the language of can be identified only with
the language of a morphic image of . We also study the group
structure of -integers.Comment: 19 pages, 5 figure
Rational numbers with purely periodic -expansion
We study real numbers with the curious property that the
-expansion of all sufficiently small positive rational numbers is purely
periodic. It is known that such real numbers have to be Pisot numbers which are
units of the number field they generate. We complete known results due to
Akiyama to characterize algebraic numbers of degree 3 that enjoy this property.
This extends results previously obtained in the case of degree 2 by Schmidt,
Hama and Imahashi. Let denote the supremum of the real numbers
in such that all positive rational numbers less than have a
purely periodic -expansion. We prove that is irrational
for a class of cubic Pisot units that contains the smallest Pisot number
. This result is motivated by the observation of Akiyama and Scheicher
that is surprisingly close to 2/3
Combinatorial and Arithmetical Properties of Infinite Words Associated with Non-simple Quadratic Parry Numbers
We study arithmetical and combinatorial properties of -integers for
being the root of the equation . We determine with the accuracy of the maximal number of
-fractional positions, which may arise as a result of addition of two
-integers. For the infinite word coding distances between
consecutive -integers, we determine precisely also the balance. The word
is the fixed point of the morphism and . In the case the corresponding infinite word is
sturmian and therefore 1-balanced. On the simplest non-sturmian example with
, we illustrate how closely the balance and arithmetical properties of
-integers are related.Comment: 15 page
A new method for constructing Anosov Lie algebras
It is conjectured that every manifold admitting an Anosov diffeomorphism is,
up to homeomorphism, finitely covered by a nilmanifold. Motivated by this
conjecture, an important problem is to determine which nilmanifolds admit an
Anosov diffeomorphism. The main theorem of this article gives a general method
for constructing Anosov diffeomorphisms on nilmanifolds. As a consequence, we
give counterexamples to a corollary of the classification of low-dimensional
nilmanifolds with Anosov diffeomorphisms and a correction to this statement is
proven. This method also answers some open questions about the existence of
Anosov diffeomorphisms which are minimal in some sense.Comment: 18 pages, some small revisions according to referee report, to appear
in Transactions of the AM
Entropy in Dimension One
This paper completely classifies which numbers arise as the topological
entropy associated to postcritically finite self-maps of the unit interval.
Specifically, a positive real number h is the topological entropy of a
postcritically finite self-map of the unit interval if and only if exp(h) is an
algebraic integer that is at least as large as the absolute value of any of the
conjugates of exp(h); that is, if exp(h) is a weak Perron number. The
postcritically finite map may be chosen to be a polynomial all of whose
critical points are in the interval (0,1). This paper also proves that the weak
Perron numbers are precisely the numbers that arise as exp(h), where h is the
topological entropy associated to ergodic train track representatives of outer
automorphisms of a free group.Comment: 38 pages, 15 figures. This paper was completed by the author before
his death, and was uploaded by Dylan Thurston. A version including endnotes
by John Milnor will appear in the proceedings of the Banff conference on
Frontiers in Complex Dynamic
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