From: dgc@ccrwest.org (David Cantor)
Subject: Re: density of sequences on the torus
Date: 16 Jan 01 13:08:18 GMT
Newsgroups: sci.math.numberthy
Summary: Points on tori whose multiples are dense in the torus

I suspect that it's extremely difficult.

For n = 1, then M may be thought of as an integer (say > 1).  Your
oondition is certainly satisfied when x is a normal number to the
base M. The definition of such a number x is that M^nx be uniformly
distributed on the torus and that's stronger than you want.  But you
condition can be expressed as a condition on the base M expansion of
x. Namely, that every binite lock of digits between 0 and M-1 occurs in
the base M expansion of x.

When you get to higher dimensions then you have to worry about the PV
numbers.  Each component of M^k x satisfy a degree n linear recursion
(given by the characteristic polynomial of M).  The PV numbers are
those algebraic numbers y > 1 (in a real embedding) such that all
conjugates y' satisfy |y'| < 1. These numbers have the property that
if x is in the real embedding used in the definition then the sequence
{xy^n} has limit 0 on the torus.  This problem will arise when the
characteristic polynomial of M is the irreducible polynomial satisfied
by a PV number.

dgc

----------------------------------------------------------------------

Original message follows:

I was recently asked the following question, and I haven't been able
to come up with a satisfactory answer.  Can someone out there shed
some additional light on this?  Here is the question:

Let T = R^n/Z^n be the n-dimensional torus, and M be an n x n integer
matrix, which we'll also consider to be the same as the map from T
--> T given by multiplying by M on the left.  The question is to
characterize those x in T, so that the image of x under the semigroup
{M^k, k>= 0} is dense in T.

This seems like a big question, so any light that can be shed on it is
appreciated.
==============================================================================

From: pleasanp@manu.usp.ac.fj (Peter Pleasants)
Subject: Re: density of sequences on the torus
Date: 16 Jan 01 13:08:17 GMT
Newsgroups: sci.math.numberthy
Summary: [missing]

This is certainly an interesting question.
One context it is relevant to is that species of quasicrystals,
for example Penrose tilings, can be parametrized by points on
a high dimensional torus.  The map might then correspond to a
spatial symmetry or to the "inflation" that many quasicrystals
have.
I don't have an answer, but I've been interested in fixed points
of such maps so can offer some comments about points whose orbits
are NOT dense and a self-reference.

1D CASE  When m is not +/-1, the points x with {m^n x} dense in [0,1]
are those whose base m expansions contain all possible finite strings
of digits - a superset of the numbers that are normal in base m.
So in general one would guess that almost all x have dense orbits
(if any at all) but that there are uncountably many exceptions.
This would imply, for example, that iterated inflations of a "random"
Penrose tiling would approximate all Penrose tilings arbitrarily
closely.  (Something that follows from the "Fibonadic number"
parametrization of Penrose tilings introduced by Fred Lunnon.)

M HAS FINITE ORDER  Then no orbit is dense (corresponding to
m = +/-1 in the 1D case).

PERIODIC POINTS  These are solutions of (M^n - I)x = 0 (mod 1)
for some n.  The number of them is |prod(lambda^n - 1)| where
the product is over the eigenvalues of M.  Unless all eigenvalues
are roots of unity this tends to infinity with n, so there are
countably many periodic points.

M HAS A RATIONAL INVARIANT SUBSPACE  Then there is a finite number
of corresponding "subtori" invariant under M.  These subtori are
not dense on T, so give uncountably many points whose orbits under
M are not dense.  Additionally we will get subtori that, though not
invariant, are periodic under M (analogously to the periodic points
above).  In general there will be countably many of these.

The above 3 cases cover all possibilities: if M has no proper
invariant subspace and all its eigenvalues are roots of unity
then it has finite order.

The main focus of the reference below is the finite order case.
We classify all Penrose tilings with any spatial symmetry (necessarily
a subgroup of the decagonal group D_{10}) and do the same for the
icosahedral 3D extension of the Penrose tilings.  But we also count
tilings with small order under inflation - a map of infinite order.

A variant of the original question, perhaps more fundamental from
the point of view of ergodic theory, would be to characterize those
x whose orbits are uniformly distributed in T.

Finally, this message may appear to come from little Fiji (everybody
knows what .fj means) but that's just a trick of modern communications
technology and in reality I'm in Kiribati (1 N, 173 E), one of the
world's most extensive countries.

REFERENCE  M. Baake, J. Hermisson and P.A.B. Pleasants,
"The torus parametrization of quasiperiodic LI-classes"
J. Phys. A: Math. Gen. 30 (1997) 3029-3056.

                                        Peter Pleasants