Quasi-periodic Tilings

Marco de Innocentis

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May 4, 1998, 3:00:00 AM5/4/98

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How have quasi-periodic tilings of the plane been studied so far? Can
anyone give me some useful references to consult?
Thanks,

Marco

Chris Hillman

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May 4, 1998, 3:00:00 AM5/4/98

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On Mon, 4 May 1998, Marco de Innocentis wrote:

> How have quasi-periodic tilings of the plane been studied so far? Can
> anyone give me some useful references to consult?

A good place to start is the recent undergraduate level book

Author: Senechal, Marjorie.
Title: Quasicrystals and geometry / Marjorie Senechal.
Pub. Info.: Cambridge ; New York : Cambridge University Press, 1995.
LC Subject: Quasicrystals.
Crystallography-Mathematical.

This book has voluminous references to the research literature and is
available in paperback :-)

An out-of-date (only up to to 1977--- there was a ten year delay in
publication) but still useful account of the very early work on Penrose
and Ammann tilings may be found in

Author: Grunbaum, Branko.
Title: Tilings and patterns / Branko Grunbaum, G.C. Shephard.
Pub. Info.: New York : W.H. Freeman, c1987.
LC Subject: Tiling-Mathematics.

Be careful, Shephard's account of the "essential holes theorem" of Conway
(who never published or, according to what he told me, even wrote down his
work on this) is in some respects misleading. Also, I believe that the
empires reported by Ammann for the Penrose tiling systematically missed
some tiles.

For a totally different take on the "language" of Sturmian shifts (one
dimensional analogs of Sturmian tilings, aka generalized Penrose tilings)
see "Markov and SFT Approximations" on my page

http://www.math.washington.edu/~hillman/postings.html

and "Empires in Sturmian Systems" on my page

http://www.math.washington.edu/~hillman/papers.html

My PhD thesis, when it is (knock wood) finished, will have lots more
information about various aspects of Sturmian tilings and the symbolic
(multidimensional) Sturm-Robinson shifts determined by them. I can send
you a postscript file of the introduction if you're interested.

Hope this helps!

Chris Hillman

dag.to...@lu.his.no

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May 5, 1998, 3:00:00 AM5/5/98

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Marco de Innocentis wrote:
>
> Can anyone give me some useful references to consult?

Try this page. It's interactive, so you can make your own Penrose
tiling.

http://www.geom.umn.edu/apps/quasitiler/start.html

Dag Torvanger

Marcin Marciniak

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May 5, 1998, 3:00:00 AM5/5/98

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Suppose that the operator Tf(x) = \int K(x,y)f(y)dy is continuous
on L^1(0,1). Is it true that

ess sup \int |K(x,y)|dx
{y \in (0,1)}

is equal ||T||?

I would appreciate any help in solving the problem or
pointing to any reference.

Barbara Wolnik

Martin Vaeth

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May 8, 1998, 3:00:00 AM5/8/98

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>Suppose that the operator Tf(x) = \int K(x,y)f(y)dy is continuous
>on L^1(0,1). Is it true that
>
> ess sup \int |K(x,y)|dx
> {y \in (0,1)}
>
>is equal ||T||?

I guess you additionally assume that K is product-measurable
(otherwise one may use the continuum hypothesis to construct
some K such that T=0 but K(.,y)=1 a.e. for fixed y, see e.g.
Rudin, "Real and complex analysis").
Under this additional assumption, the answer is positive:
Indeed, since the departure space is L^1(0,1) (and the
image space is a perfect Banach ideal space) the operator
T is order-bounded, and for the operator |T| we have
|| T || = || |T| || (see e.g. Theorem 2.2.16(a) in the book
S.S. Kutateladze (ed.) "Vector lattices and integral operators").
In particular, we may replace K by |K| without changing the norm of
the integral operator. Thus, without loss of generality, we may
assume that K is nonnegative. Now observe that
|| T || = sup \int \int K(x,y)f(y) dy dx.
|| f ||_1 \le 1
and f\ge 0
By Fubini-Tonelli, we may swap the order of integration and get
the statement, using the well-known formula
|| g ||_\infty = sup \int g(y)f(y) dy
|| f ||_1 \le 1
(for nonnegative g the supremum needs to be taken only over
nonnegative f, of course).