Warren Dicks and J. Porti,
On the Hausdorff dimension of the Gieseking fractal.
Topology Appl., 126 (2002), 169-186.


                                                        Addenda

                                                    May 1, 2005.
 

In this article, we studied the Hausdorff dimension  d  of the Gieseking fractal.  We can also say
that  d  is the Hausdorff dimension of the Gieseking-fractal arcs that tessellate the Euclidean plane
like this:





There is a computer experiment which currently suggets that  d   is about 1.29710... .
Let  S  denote the free semigroup on the two-letter alphabet {gf }.  Let  S  act on  N4  by
         g ( a, b, c, d )  =  ( a, b + d, a + 2b + c, d ),    
         f
( a, b, c, d )  =  ( a + b, c, b, a + c + d ).
We define the weight of an element  ( a, b, c, d )  of  N4  to be  b + c + d.  
For each  n  in  N,  let   bn  denote the number of words  w  in  S  such that the weight of  
w
( 0, 0, 1, 1 )  is at most  2 cosh(n).  If  n  is positive,  let  an = bn - bn-1.   Recall that  d  lies
between the liminf and the limsup of   ( log  an/an-1  |  n = 2, 3, ... ).   We do not know whether or
not this sequence converges.  
Here are the first 24 terms.

n

bn

an

log an/an-1

Programmer

Year

 

 

 

 

 

 W. Dicks

2000

11 

 0.8472978

 W. Dicks

2000

38 

27 

 1.3499267 

 W. Dicks

2000

153 

115 

 1.4490952

 W. Dicks

2000

526 

373 

 1.1766462 

 W. Dicks

2000

1946 

1420 

 1.3368337 

 W. Dicks

2000

7111 

5165 

 1.2912482

 W. Dicks

2000

25882 

18771 

 1.2904080 

 W. Dicks

2000

95156 

69274 

 1.3057565 

 W. Dicks

2000

10 

348335 

253179 

 1.2960270

 A. Dicks

2000

11 

1271898 

923563 

 1.2941422

 A. Dicks 

2000

12 

4658192 

3386294 

 1.2992523

 A. Dicks 

2000

13 

17040856 

12382664 

 1.2965613

 A. Dicks

2000

14 

62350742 

45309886 

 1.2972278

 A. Dicks 

2000

15 

228147404 

165796662 

 1.2972368

 A. Dicks 

2000

16 

834711907 

606564503 

 1.2970489

 J.- C. Artés 

2000

17 

3053925244 

2219213337 

 1.2970969

 P. Taylor 

2000

18 

11172970200 

8119044956 

 1.2970597

 P. Taylor

2000

19 

40878223775 

29705253575 

 1.2971113 

 P. Taylor 

2000

20 

149563132885 

108684909110 

 1.2971290

 P. Taylor 

2000

21 

547206912858 

397643779973 

 1.2971036

 J. Amoros 

2003

22 

2002063012565 

1454856099707 

 1.2971056

 J. Amoros 

2003

23 

7324919842341 

5322856829776 

 1.2971031

 J. Amoros 

2003

24

26799555593731

19474635751390

 1.2971027

 J. Amoros and
 J. Vindel

 2004

25

98051175064382

71251619470651

 1.2971046    

 J. Vindel

2004



By left conjugating by   h( a, b, c, d )  =  ( a, b, d, b + c + d )  we can rewrite the above procedure as
follows.

 Let  S  denote the free semigroup on the two-letter alphabet {gf }.  Let  S  act on  Z4  by
         g
( a, b, c, d )  =  ( a, b + c,  c,  a + 2b + c + d
                                =  ( a, b + c,  c,  (b + c) + d + (a + b)) ,    
         f
( a, b, c, d )  =   ( a + b, - b - c + d,  a - b + d,  a - b - c + 2d )
                                =   ( a + b, - (b + c) +  d,  a - b + d, - c +  d + (a - b+ d )).
For each  n  in  N,  let   bn  denote the number of words  w  in  S  such that the fourth coordinate of  
w
( 0, 0, 1, 2 )  is at most  2 cosh(n).  If  n  is positive,  let  an = bn - bn-1.   Then  d  lies
between the liminf and the limsup of   ( log  an/an-1  |  n = 2, 3, ... ).   We do not know whether or
not this sequence converges.  
The first 24 terms are as above.



                          
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