TRANSCENDENTAL CANTOR SETS AND TRANSCENDENTAL CANTOR FUNCTIONS

Abstract

In this article, we consider the self-similar generalized Cantor set     ,,, n n
l C i C i i 1 , and we
establish the existence of probability true measure

such that
    
,..,
1
j
j s
E E
s
   
 
  1
0 1
generated by   n Ci
. The Holder order  of the set   n Ci
is   logn s
and we establish that
  ,    ,    l l x n x s i i n
       
 
  
1 1
2
for all not finite n -adic  ,..., . n
l xC i i 1
Transcendental numbers, such as e and  are a mathematical expression of nature, we introduce the
transcendental Cantor set generated by transcendental numbers, which can be defined by
 
, ,...,
lim n
k k
k
k
C  C C


 
0 1 , where the sequence   k C
is non-increasing and corresponds with the
transcendental number

, for such a set, we consider an analog of the Cantor function.