© 2007 Veikko Keränen

The following 200 new abelian square-free endomorphisms on 4 letters were found on large grid computations at RAMK.
The programs were firstly written by Veikko Keränen & Kari Tuovinen in 1990 (Lisp & C), and, later, by Viet Pham Hoang &
Veikko Keränen in 2006-2007 (Mathematica & C++). Viet built the grid and organised the runs on it.

Firstly, we present an abelian square-free substitution on 4 letters with 12 words in the image of every letter.

The 12 a2f endomorphisms g making up the substitution (and the 12^4 = 20736 a2f endomorphisms)
have the uniform modulus of 109, and, in every case, the image words g(b), g(c), g(d)
are all obtained by the circular automorphism of {a,b,c,d}* starting from g(a).

The Parikh vectors for each g(a), g(b), g(c), g(d) are the rows of the following matrix:

{{21, 31, 29, 28},
  {28, 21, 31, 29},
  {29, 28, 21, 31},
  {31, 29, 28, 21}}

The 12 image words for g(a):s are as follows, where the dots are added just for readability.
Note the delicate mutations:

abcacdcbcdcadcdb.abcd.badacdadbdcdbdabdbcbabcbdcb.acd.bdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdb.abcd.badacdadbdcdbdabdbcbabcbdcb.adc.bdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdb.abcd.badacdadbdcdbdabdbcbabcbdcb.cad.bdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdb.abdc.badacdadbdcdbdabdbcbabcbdcb.acd.bdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdb.abdc.badacdadbdcdbdabdbcbabcbdcb.adc.bdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdb.abdc.badacdadbdcdbdabdbcbabcbdcb.cad.bdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdb.adbc.badacdadbdcdbdabdbcbabcbdcb.acd.bdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdb.adbc.badacdadbdcdbdabdbcbabcbdcb.adc.bdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdb.adbc.badacdadbdcdbdabdbcbabcbdcb.cad.bdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdb.dabc.badacdadbdcdbdabdbcbabcbdcb.acd.bdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdb.dabc.badacdadbdcdbdabdbcbabcbdcb.adc.bdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdb.dabc.badacdadbdcdbdabdbcbabcbdcb.cad.bdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca

The same words can be represented in a form that shows the "movement" of the letters d and c in the invariant background:

abcacdcbcdcadcdb.d.abcbadacdadbdcdbdabdbcbabcbdcb.c.adbdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdb.d.abcbadacdadbdcdbdabdbcbabcbdcba.c.dbdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdb.d.abcbadacdadbdcdbdabdbcbabcbdcbad.c.bdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdba.d.bcbadacdadbdcdbdabdbcbabcbdcb.c.adbdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdba.d.bcbadacdadbdcdbdabdbcbabcbdcba.c.dbdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdba.d.bcbadacdadbdcdbdabdbcbabcbdcbad.c.bdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdbab.d.cbadacdadbdcdbdabdbcbabcbdcb.c.adbdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdbab.d.cbadacdadbdcdbdabdbcbabcbdcba.c.dbdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdbab.d.cbadacdadbdcdbdabdbcbabcbdcbad.c.bdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdbabc.d.badacdadbdcdbdabdbcbabcbdcb.c.adbdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdbabc.d.badacdadbdcdbdabdbcbabcbdcba.c.dbdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca abcacdcbcdcadcdbabc.d.badacdadbdcdbdabdbcbabcbdcbad.c.bdcdadcdbcbabcbdcbcacdcacbadabcbdcbcadbabcbabdbcdbdadbdcbca

The number of a2f words over 4 letters counted up to length 60

More a2f endomorphisms over 4 letters:

a2fEndomorphismsOver4Letters_Lengths102-115_200inNumber_EditedForReadability.txt

a2fEndomorphismsOver4Letters_Lengths102-115_200inNumber_WithParikhVectors.txt

You may also copy the image words of the "old"   g85 (from 1990)  and  g98 (from 2002)   as text from below:

© 2003 V. Keränen

Please, copy from the white output cells below:

cyclicPermutation = {"a""b", "b""c", & ... ; g85c = StringReplace[g85b, cyclicPermutation] ; g85d = StringReplace[g85c, cyclicPermutation] ;

g85a g85b g85c g85d

abcacdcbcdcadcdbdabacabadbabcbdbcbacbcdcacbabdabacadcbcdcacdbcbacbcdcacdcbdcdadbdcbca

bcdbdadcdadbadacabcbdbcbacbcdcacdcbdcdadbdcbcabcbdbadcdadbdacdcbdcdadbdadcadabacadcdb

cdacabadabacbabdbcdcacdcbdcdadbdadcadabacadcdbcdcacbadabacabdadcadabacabadbabcbdbadac

dabdbcbabcbdcbcacdadbdadcadabacabadbabcbdbadacdadbdcbabcbdbcabadbabcbdbcbacbcdcacbabd

ga = "abcacdcbcdcadbdcbdbabcbdcacbabdbabcabdadcdadbdcbdbabdbcbacbcdbabdcdbdcacdbcbacbcdca ... ion] ; gd = StringReplace[gc, cyclicPermutation] ; {g98a = ga, g98b = gd, g98c = gc, g98d = gb} ;

g98a g98b g98c g98d

abcacdcbcdcadbdcbdbabcbdcacbabdbabcabdadcdadbdcbdbabdbcbacbcdbabdcdbdcacdbcbacbcdcacdcbdcdadbdcbca

dabdbcbabcbdcacbacadabacbdbadacadabdacdcbcdcacbacadacabadbabcadacbcacbdbcabadbabcbdbcbacbcdcacbabd

cdacabadabacbdbadbdcdadbacadcdbdcdacdbcbabcbdbadbdcdbdadcadabdcdbabdbacabdadcadabacabadbabcbdbadac

bcdbdadcdadbacadcacbcdcadbdcbcacbcdbcabadabacadcacbcacdcbdcdacbcadacadbdacdcbdcdadbdadcadabacadcdb


Created by Mathematica  (March 29, 2007)