Loxodromic Aliasing

When describing the "appearance" of a sequence of states it's 
sometimes necessary to account for the phenomenon of aliasing.
Of course, if the "position" of a system within the state space
changes continuously, the path (though not the rate) of the 
system can be directly deduced from the locus of points it has 
occupied (up to direction) assuming the system never exactly 
backtracks.  However, if a system changes it's position within
the state space in discrete jumps, we can't unambiguously
determine the path of the system merely from the locus of
points it has occupied, because we don't know the sequence
in which those points were occupied.

An interesting example of this arises when we examine the apparent
positions of distant stars to an observer who is subjected to a
sequence of discrete Lorentz transformations.  As Penrose pointed
out, the angles of light rays through a given point in spacetime
have a natural correspondence with the points of the Riemann 
sphere, and if we associate each of those points via stereographic 
projection with a point in the extended complex plane, the effect 
of any given (proper) Lorentz transformation on the ray angles 
in spacetime corresponds precisely to the effect on the complex 
plane of a certain linear fractional transformation, i.e., 
w -> (aw+b)/(cw+d) where a,b,c,d are complex numbers, normalized 
so that ad-bc=1.  Conversely, every LFT corresponds to a proper
Lorentz transformation.

Now, suppose our observer is subjected the the Lorentz transformation
corresponding to to the LFT

                  (1+qi)w + (2+qi)
           w ->  -----------------
                 (-1+qi)w + (2+qi)

where q equals, say, 1/100.  We can normalize the coefficients if
we like, to make ad-bc=1.  The squared trace is obviously complex,
so this is a loxodromic LFT.  If we ask the observer to mark on a
globe the apparent position of a particular star after each 
iteration of this function, what would we expect to see when 
he shows us the globe with all the markings?

Naturally since the transformation is loxodromic, but just barely,
we expect that if our observer tracked a star that began near the 
repelling fixed point he would have a single sequence of marks 
spiraling outward from that point and then ultimately spiraling 
inwards on the attracting fixed point.  This is, in fact, what
would appear to our eyes, but whether we would SEE it that way
(at first) is questionable.  Here's a picture of the markings
we would find on the globe:

    

At first glance this looks quite different from what we expected.
Instead of a single spiraling path from one fixed point to the
other, we find 13 distinct curvy paths!  What gives?  Well, if
we ask our observer why he tracked 13 stars instead of just one
as we had agreed (a patriotic gesture?), he will explain that
he tracked only one star, and it did indeed spiral loxodromically
(and monotonically) from one fixed point to the other.  The
explanation for the apparent 13 curvy paths is simple aliasing,
i.e., we are incorrectly inferring the path between the points,
and in this case we are being strongly encouraged to do so by
the arrangement of the points, each of which is in much closer
spatial proximity to the point 13 steps ahead in the sequence 
than to either of its immediate neighbors in the sequence.

For another illustration, here are three "marked globes" for
the transformation

                  (1+qi)w + (4+qi)
           w ->  -----------------
                 (-1+qi)w + (4+qi)

with q=0.02, 0.01, and 0.005, respectively.



Again we see (or SEEM to see) 13 curvy paths, and they actually
approach being contiguous paths as q becomes smaller.

One question that might occur to us is "why 13"?  It's worth
noting that the sequence of loxodromic points _almost_ gives
just 7 curvy paths, because the cycle of pseudo-paths around 
one loop of the spiral is

   1   8   2   9   3   10   4   11   5   12   6   13   7

This pattern, and in general the tendancy for aliasing at various 
frequencies, is determined by how close the LFT is to being periodic
with various periods.  It's easy to see that the LFT (aw+b)/cw+d)
is cyclical with fundamental period m if and only if the squared 
trace equals  4cos(k pi/m)^2  for some integer k coprime to m.
Since the squared trace also uniquely determines conjugacy, it 
follows that the number of distinct cyclical LFTs (up to conjugacy)
with period m is phi(m) (where phi is Euler's totient function).

Anyway, we know the LFT (w+1)/(-w+1) is cyclical with period 4,
so we would expect that if we make this loxodromic by inserting
small imaginaly components qi to each coefficient, we should see
4 distinct apparent "paths" connecting the two fixed points,
which is indeed the case.

For those who like to speculate about the possibility of some 
kind of discretization of spacetime that nevertheless preserves 
Lorentz invariance, it might be interesting to consider whether 
the there is a special physical significance for the "cyclical" 
discrete Lorentz transformations.  Also, I think the general 
issue of aliasing is quite interesting.  Of course, aliasing has
well-known practical implications for things like signal processing 
and controls theory, but it's also interesting to consider how,
in physics, _identity_ is established between discrete appearances 
of ostensibly the same object, and how we agree upon the "correct"
sequence of events on the basis of necessarily incomplete 
information about our surroundings.

By the way, it seems that linear fractional transformations occur
just about everywhere.  For one example, see the note entitled
Compressor Stalls and Mobius Transformations, which describes 
how stall cells migrate around the face of an axial compressor in 
a pattern corresponding to the action of a Mobius transformation 
on the complex plane.  Also, for more on cyclical LFTs and the 
general closed-form expression for the nth composition of an 
arbitrary LFT, see

Linear Fractional (Mobius) Transformations

By the way, here are two images showing the effect of repeated applications
of the Lorentz transformation corresponding to the parabolic LFT
w -> w + 0.01.  The figure on the left shows the paths of points that
begin on a semi-circle of radius 0.3 on the emitting side of the fixed
point.  The figure on the left shows the paths of points that begin on
a semi-cirlce of radius 0.1, to show more detail of the short range
"flux".

   

It may be worth mentioning that this progressive transformation of the
celestial sphere can always be decomposed into a pure boost and a pure
rotation.  In other words, a second observer could travel along with
the first, always exactly matching his velocity, but orienting his field
of view always in the direction of the current boost relative to the
initial frame, and this second observer will always see a simple
contraction of the stars in front of him, i.e., an anti-podal repulsion
from the point directly behind and attraction toward the point directly
ahead.  In a physical sense this axis is probably the most natural and
significant in this context, since it is the "eigenvector" of the
celestial transformation.

In contrast, our "parabolic" observer is seeing all these cute circular
orbits, but he's really just generating them himself by varying his
orientation in a particular way that has little or no physical
significance, being distinguised only by the fact that it allows his
frame to be related to the original frame by a Lorentz transformation
that corresponds to a parabolic LFT of the complex plane.

I think the two most physically significant orientations in this context
are (1) the eigenvector of the boost, and (2) the inertial orientation
of a gyroscope as it is carried along by the observer.  Another possibly
significant orientation might be based on the acceleration experienced
by the observer.  The "parabolic" orientation  is certainly neither (1)
nor (2), because it rotates through 180 degrees relative to the fixed
stars and is not in general parallel to the boost.  Also, although I
haven't checked, I doubt that the parabolic orientation is related in
any natural way to the acceleration experienced by the observer. Thus,
it is evidently just a mathematical artifact, albeit a cute one.

Here's a program for DOS or Windows PCs to illustrate the effects of
compositions of Lorentz/Mobius transformations on the angles of
incidence of light rays at a point in spacetime:

STARMOB.EXE


This is a self-extracting .zip file, containing the executable program
and two star-maps, one of which is just a uniform distribution of points
on the sphere, and the other is the actual 300 most visible stars (down
to magnitude 3.5).