30 research outputs found
Multidimensional continued fractions and a Minkowski function
The Minkowski Question Mark function can be characterized as the unique
homeomorphism of the real unit interval that conjugates the Farey map with the
tent map. We construct an n-dimensional analogue of the Minkowski function as
the only homeomorphism of an n-simplex that conjugates the piecewise-fractional
map associated to the Monkemeyer continued fraction algorithm with an
appropriate tent map.Comment: 17 pages, 3 figures. Revised version according to the referee's
suggestions. Proof of Lemma 2.3 more detailed, other minor modifications. To
appear in Monatshefte fur Mathemati
Denominator-preserving maps
Let F be a continuous injective map from an open subset of R^n to R^n. Assume
that, for infinitely many k>1, F induces a bijection between the rational
points of denominator k in the domain and those in the image (the denominator
of (a_1/b_1,...,a_n/b_n) being the l.c.m. of b_1,...,b_n). Then F preserves the
Lebesgue measure.Comment: 11 pages, 1 figur
Generic substitutions
Up to equivalence, a substitution in propositional logic is an endomorphism
of its free algebra. On the dual space, this results in a continuous function,
and whenever the space carries a natural measure one may ask about the
stochastic properties of the action. In classical logic there is a strong
dichotomy: while over finitely many propositional variables everything is
trivial, the study of the continuous transformations of the Cantor space is the
subject of an extensive literature, and is far from being a completed task. In
many-valued logic this dichotomy disappears: already in the finite-variable
case many interesting phenomena occur, and the present paper aims at displaying
some of these.Comment: 22 pages, 2 figures. Revised version according to the referee's
suggestions. To appear in the J. of Symbolic Logi
Measures induced by units
The half-open real unit interval (0,1] is closed under the ordinary
multiplication and its residuum. The corresponding infinite-valued
propositional logic has as its equivalent algebraic semantics the equational
class of cancellative hoops. Fixing a strong unit in a cancellative hoop
-equivalently, in the enveloping lattice-ordered abelian group- amounts to
fixing a gauge scale for falsity. In this paper we show that any strong unit in
a finitely presented cancellative hoop H induces naturally (i.e., in a
representation-independent way) an automorphism-invariant positive normalized
linear functional on H. Since H is representable as a uniformly dense set of
continuous functions on its maximal spectrum, such functionals -in this context
usually called states- amount to automorphism-invariant finite Borel measures
on the spectrum. Different choices for the unit may be algebraically unrelated
(e.g., they may lie in different orbits under the automorphism group of H), but
our second main result shows that the corresponding measures are always
absolutely continuous w.r.t. each other, and provides an explicit expression
for the reciprocal density.Comment: 24 pages, 1 figure. Revised version according to the referee's
suggestions. Examples added, proof of Lemma 2.6 simplified, Section 7
expanded. To appear in the Journal of Symbolic Logi
Slow continued fractions, transducers, and the Serret theorem
A basic result in the elementary theory of continued fractions says that two real numbers share the same tail in their continued fraction expansions iff they belong to the same orbit under the projective action of PGL(2,Z). This result was first formulated in Serret's Cours d'alg`ebre sup'erieure, so we'll refer to it as to the Serret theorem.
Notwithstanding the abundance of continued fraction algorithms in the literature, a uniform treatment of the Serret result seems missing. In this paper we show that there are finitely many possibilities for the subgroups Sigma of PGL(2,Z) generated by the branches of the Gauss maps in a large family of algorithms, and that each Sigma-equivalence class of reals is partitioned in finitely many tail-equivalence classes, whose number we bound. Our approach is through the finite-state transducers that relate Gauss maps to each other. They constitute opfibrations of the Schreier graphs of the groups, and their synchronizability ---which may or may not hold--- assures the a.e. validity of the Serret theorem
Decreasing height along continued fractions
The fact that the euclidean algorithm eventually terminates is pervasive in
mathematics. In the language of continued fractions, it can be stated by saying
that the orbits of rational points under the Gauss map x-->1/x eventually
reach zero. Analogues of this fact for Gauss maps defined over quadratic number
fields have relevance in the theory of flows on translation surfaces, and have
been established via powerful machinery, ultimately relying on the Veech
dichotomy. In this paper, for each commensurability class of noncocompact
triangle groups of quadratic invariant trace field, we construct a Gauss map
whose defining matrices generate a group in the class; we then provide a direct
and self-contained proof of termination. As a byproduct, we provide a new proof
of the fact that noncocompact triangle groups of quadratic invariant trace
field have the projective line over that field as the set of cross-ratios of
cusps.
Our proof is based on an analysis of the action of nonnegative matrices with
quadratic integer entries on the Weil height of points. As a consequence of the
analysis, we show that long symbolic sequences in the alphabet of our maps can
be effectively split into blocks of predetermined shape having the property
that the height of points which obey the sequence and belong to the base field
decreases strictly at each block end. Since the height cannot decrease
infinitely, the termination property follows
Purely periodic continued fractions and graph-directed iterated function systems
We describe Gauss-type maps as geometric realizations of certain codes in the
(multi)monoid of nonnegative matrices in the extended modular group. Each such
code, together with an appropriate choice of unimodular intervals in P^1(R),
determines a dual pair of graph-directed iterated function systems, whose
attractors contain intervals and constitute the domains of a dual pair of
Gauss-type maps. Our framework covers many continued fraction algorithms (such
as Farey fractions, Ceiling, Even and Odd, Nearest Integer, ...) and provides
explicit dual algorithms and characterizations of those quadratic irrationals
having a purely periodic expansion.Comment: Theorem 2.8 slightly improved and other minor modifications. 26
pages, 9 figure
Billiards on pythagorean triples and their Minkowski functions
It has long been known that the set of primitive pythagorean triples can be enumerated by descending certain ternary trees. We unify these treatments by considering hyperbolic billiard tables in the Poincare disk model. Our tables have m>=3 ideal vertices, and are subject to the restriction that reflections in the table walls are induced by matrices in the triangle group PSU^pm_1,1Z[i]. The resulting billiard map ilde B acts on the de Sitter space x_1^2+x_2^2-x_3^2=1, and has a natural factor B on the unit circle, the pythagorean triples appearing as the B-preimages of fixed points. We compute the invariant densities of these maps, and prove the Lagrange and Galois theorems: a complex number of unit modulus has a preperiodic (purely periodic) B-orbit precisely when it is quadratic (and isolated from its conjugate by a billiard wall) over Q(i). Each B as above is a (m-1)-to-1 orientation-reversing covering map of the circle, a property shared by the group character T(z)=z^-(m-1). We prove that there exists a homeomorphism Phi, unique up to postcomposition with elements in a dihedral group, that conjugates B with T; in particular Phi ---whose prototype is the classical Minkowski function--- establishes a bijection between the set of points of degree <=2 over Q(i) and the torsion subgroup of the circle. We provide an explicit formula for Phi, and prove that Phi is singular and Holder continuous with exponent equal to log(m-1) divided by the maximal periodic mean free path in the associated billiard table