The Topology of Chaos: libtop.txt
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1) DATA FORMAT
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a) INVARIANTS
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Example (another example with RRR information can be found at the end
of this documentation):
\begin{data}
5
0 1 1
1 1 2
5 3 4
8 4 5
8 3 5
1 2 1
1 3 2
1 4 2
1 5 2
2 3 3
2 4 4
2 5 4
3 4 8
3 5 8
4 5 10
* * *
\end{data}
# line1 : number of orbits (let us say n)
# lines 2 to n+1 : one-orbit invariants, in the form
slk tors per
where slk is the self-linking number, tors the local torsion of the
orbit, and per the period. The period is mandatory, but if you do not
know exactly slk or tors, you can put a * character. Beware that if
both slk and tors are unspecified, the program is likely to have a
hard time, because it will have to try quite a number of different
symbolic names.
IMPORTANT : ORBITS MUST BE SORTED BY PERIOD !
If self-relative rotation rates for an orbit are knwon, you can them
at the end of the line. This will help the program, e.g.
slk tors per rrr_1 rrr_2 rrr_3 rrr_p
assuming that the period is p. the rrr_i numbers are to be normalized
so that they sum up to slk (i.e. multiply real rrrs by the period).
rrr_1 is always 0. If P_1, P_2, ...,P_n are the periodic points
belonging to the orbit, rrr_i is the number of full turns the
trajectory originating from P_1 winds around the trajectory
originating from P_i before they return to their initial conditions.
example: the line for the period-4 orbit 0111 of the Smale's horsehoe
should be
5 3 4 0 2 1 2
-------
rrr part
which means that the self-linking number is 5, the torsion is 3, the
period 4, and the rrrs are 0,2/4,1/4,2/4
Note that order in which rrrs appear is important, because it is left
invariant by regular isotopy.
If one of the SRRR is uncertain, include none.
#line n+2 : to be left blank
#lines n+3 to end of file : linking number information in the form
n1 n2 lk(n1,n2)
where n1 and n2 are the number of two orbits (orbits are numbered
according to their order of appearance in the file, i.e. orbit#1
appears on the first line after the line giving the number of orbits).
If you do not know the linking number of two orbits, just do not put
the corresponding line, or put a * character in place of the linking
number.
lines may appear in any order since the first two numbers tell us
which orbits the third corresponds to.
If rrr information is known, you can add it to the end of the line.
This can help the program.
Let us assume that two orbits have periods p1 and p2. Their RRRs will
take at most gcd(p1,p2) different values. THUS put here gcd(p1,p2)
values normalized so that they sum up to the linking number (i.e.
multiply the real rrrs by the lcm of the periods).
order is not important as far as I know, and values will be internally
sorted.
example: linking information for a period-2 and a period-4
orbits.
2 7 3 1 2
------
rrr information
In this file, they appear as orbits #2 and 7, their linking number is
3, and since the gcd(2,4)=2, rrrs can take only two different values
which are 1/4 and 2/4 , since lcm(2,4)=4. As they are 8 RRRS, this
means that each value corresponds to 4 RRRs. The information on how
many RRRs correspond to one value has not to be fed to the program, as
it can get it from the periods.
Again, if one of the rrrs is not known, include none.
#last line : a line with three * characters :
* * *
indicates the end of the data
IMPORTANT: if you find that this file format is not convenient and
could be simplified, do not hesitate to send me a message, and I will
try to improve it before I release the source code.
b) TEMPLATES
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The notation used is more or less the one proposed by Mindlin et al.
Example : the Smale's horsehoe template
| 0 0 |
| 0 1 |
( 0 1 )
The | | part is the template matrix. Diagonal elements M_{ii} give the
torsions of the period-1 orbit "i", in units of half-turns. Off-diagonal
elements M_{ij} are twice the linking number lk("i","j") of the
period-1 orbits "i" and "j".
The ( ) part corresponds to the layering terms. The higher the number,
the closer to you, or the higher on the branch line, or the
frontmost....
c) SYMBOLIC NAMES
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example:
| 0 0 |
| 0 1 |
( 0 1 )
There are 1 solutions
orbit # 1 : 1
orbit # 2 : 01
orbit # 3 : 0111
orbit # 4 : 01111
orbit # 5 : 01011
The matrices indicate for which template the symbolic names have been
determined. On each line, one finds one or several names, which are
compatible with the topological invariants.
The symbols in the symbolic names refer to the template branches,
numbered with base 0 from left to right as they appear in the template
matrix.
For example, the 01111 orbit goes once through the first branch (with
torsion 0), then 4 times through the second branch (with torsion 1).
The exact meaning is the following: The program has searched for all
possible sets of symbolic names yielding invariants identical to the
experimental ones. For a given orbit, the names listed are those which
the orbit had in the various correct sets of symbolic names found.
For example
orbit # 18 : 010110111 010111011
tells us that in some sets the orbit with the same invariants as orbit
#18 had the name 010110111, in some other sets the name was 010111011
Marc Lefranc