Home > BV functions, Real Analysis > the Cantor function and some of its properties

the Cantor function and some of its properties

Let’s start by definining the Cantor set. Define {C_0=[0,1]} and {C_{n+1} = C_n/3 \cup (2/3+C_n/3)}. At each step we delete the middle third of all the intervals of {C_n} to obtain {C_{n+1}}. Note that we obviously have {C_{n+1} \subset C_{n}} (an easy inductive argument) and {|C_n|=(2/3)^n}. The sets {C_n} are compact and descending, therefore we can define {C=\bigcap_{n=0}^\infty C_n} which is a compact subset of {[0,1]} with zero measure and it is called the Cantor set.

Since at each step we remove a middle third of all the intervals in {C_n}, one way to look at the Cantor set is to look at the ternary representation of the points in it. In the first step, we remove all the elements of {[0,1]} which have {1} on their first position in the ternary representation. In the second step we remove those (remaining) which have {1} on the second position, and so on. In the end we are left only with elements of {[0,1]} which have only digits {0,2} in their ternary representation. Using this representation we can construct a bijection between {C} and {[0,1]} which maps

\displaystyle x=\sum_{n=1}^\infty \frac{a_n}{3^n} \mapsto \sum_{n=1}^\infty \frac{b_n}{2^n}

where {b_n=0} if {a_n=0} and {b_n=1} if {a_n=2}. This proves that the Cantor set is uncountable.

We can construct the Cantor function {g:[0,1] \rightarrow [0,1]} in the following way. Denote {R_n} the set {C_n\setminus C_{n+1}} (i.e. the set removed in step {n}). On {R_1} we let {g(x)=1/2}. On {R_2} we have two intervals: on the left one we let {g(x)=1/4} and on the right one we let {g(x)=3/4}. We continue like this iteratively, at each step choosing {g} constant on each of the intervals which construct {R_n} such that the constant on an interval is the mean of the values of neighboring interval values.

A precise formula can be given:

\displaystyle g(x) =\frac{1}{2^{N_x}}+\frac{1}{2}\sum_{n=1}^{N_x-1}\frac{a_{nx}}{2^n}

where {N_x} is the first index in the representation {x=\sum_{n=1}^\infty a_{nx}/3^n} for which {a_{nx}=1} or {\infty} if no such index exists.

the graph of the Cantor function

The Cantor function {g(x)} is monotone by construction (it is also called the devil’s staircase). Since {g(x)} is constant on each interval in each {R_n} and {\bigcup R_n =[0,1]} almost everywhere we conclude that {g'(x)} exists almost everywhere in {[0,1]} and {g'(x)=0}.

We can deduce from the formula given for {g} that the Cantor function is continuous. Indeed, if {x \rightarrow y} then {\min\{n: a_{nx}\neq a_{ny}\} \rightarrow \infty} so the difference between {g(x)} and {g(y)} will be of the order {1/2^n} with {n \rightarrow \infty}.

Since {g} is monotone, it has bounded variation, and here we can see a pathological example which illustrates the structure of a function with bounded variation. A function with bounded variation in one dimension is a function {f} defined on an interval {[a,b]} such that the quantity {\sup \sum_{i=0}^{N-1} |f(x_{i+1})-f(x_i)| <\infty} for all partitions {x_0<...<x_N} of {[a,b]} ({N} is not fixed).

An equivalent definition is

\displaystyle f \in BV(\Omega) \Leftrightarrow \sup \{ \int_\Omega f\ \text{div}\varphi: \varphi \in C^\infty(\Omega), \|\varphi\|_\infty\leq 1\}<\infty

and in this case there exists a Radon measure {\mu=Du} such that

\displaystyle \int_\Omega f\ \text{div}\varphi = -\int_\Omega \varphi d\mu,\ \forall \varphi \in C^\infty(\Omega)

The structure theorem for functions of bounded variation says that if {f} has bounded variation then {Df} can be splitted into three parts:

  • {D^a f << \mathcal{L}_n}; {Df=D^a f+D^s f} (Radon Nikodym with respect to the Lebesgue measure)
  • {D^j f=D^s f \llcorner S(u)} where {S(f)} is the jump part of {f}.
  • {D^c f=Df - D^a f-D^j f}: the rest, which is called (what a coincidence) the Cantor part of {Df}.

Therefore a bounded variation function has a part which behaves as a Sobolev function (the {n} dimensional part), it has a jump part (the {n-1} dimensional part), and it may have a third part (in between dimensions {n-1} and {n}) which usually cannot be well described.

The Cantor function {g} is differentiable almpost everywhere (in {[0,1]\setminus C}) and {g'(x)=0} for every {x \in [0,1]\setminus C}. The part where {g} is differentiable represents the first component of {g} and we see that {D^a g=0}. Since {g} is continuous, we do not have a jump part. The only part of the distributional gradient which measures the vertical displacement of {g} is concentrated on the cantor set {C} (which is to be expected, since {g} is constant on every interval outside {C}) and this is the Cantor part {D^c g} of {D g}.

Usually the Cantor part is so nasty that we can say very little for these bounded variation functions. In order to obtain a space with nicer properties, the space {SBV(\Omega)} was introduce, and its definition is exactly the {BV(\Omega)} functions such that {D^c f=0}.

Another interesting property of the Cantor function is that the length of its graph is {2} although the function is increasing, {g(0)=0} and {g(1)=1}. Intuitively this seems to be false, since {g} travels mostly horizontally. First notice that {2} is an upper bound. Any curve which increases from {(0,0)} to {(1,1)} has length at most {2}. Now let’s see that the arclength is at least {2}. First look horizontally: on each {R_n} the function {g} is piecewise constant, and {R_n} cover {[0,1]} up to a set of measure zero. Therefore the horizontal arclength is {1}. Secondly, {g} must move vertically from {0} to {1} and it does this continuously, so the vertical arclength is also {1}. As a consequence, the arclength of {g} is {2}.

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