TIFR 2013 problem 21 | No fixed point Homeomorphism

Try this problem from TIFR 2013 problem 21 based on no fixed homeomorphism.

Question: TIFR 2013 problem 21

True/False?

Every homeomorphism of the 2-sphere to itself has a fixed point.

Hint:

\(z= -z\) implies \(z=0\)

Discussion:

2-sphere means \( S^2=\left \{(x,y,z)\in\mathbb{R}^3 | x^2+y^2+z^2=1 \right \} \).

i.e, \( S^2=\left \{v\in\mathbb{R}^3 | ||v||=1 \right \} \).

\(||.||\) denotes the usual 2-norm (Euclidean norm).

Let us try \(f:S^2\to S^2\) defined by \(f(v)=-v\) for all \(v\in\mathbb{R}^3\).

The only vector in \(\mathbb{R}^3\) that is fixed by \(f\) is 0, which doesn't lie in \(S^2\).

We hope \(f\) turns out to be a homeomorphism.

\(||f(v)-f(w)||=||-v+w||=||v-w||\). So f is in fact Lipshitz function, so continuous.

\(f(f(v)=v\) for all \(v\in\mathbb{R}^3\). Therefore, \(f\) itself is inverse of \(f\). Which proves that \(f\) is bijective (since, inverse function exists) and homeomorphism (inverse is also continuous).

 

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TIFR 2013 problem 20 | Sequence limit~Fixed Point

Try this problem 20 from TIFR 2013 based on Sequence Limit - Fixed Point. This problem is an application of the Banach Fixed Point Theorem.

Question: TIFR 2013 problem 20

True/False?

Consider the function \(f(x)=ax+b\) with \(a,b\in\mathbb{R}\). Then the iteration \(x_{n+1}=f(x_n)\); \(n\ge0\) for a given \(x_0\) converges to \(\frac{b}{1-a}\) whenever \(0<a<1\).

Hint:

Banach Fixed Point Theorem

Discussion:

Once the existence of limit is guaranteed, we can safely calculate the limit. If limit is \(x\) then \( x=ax+b\). In other words, \(x=\frac{b}{1-a}\).

The question really is does limit exists?

\(x_{n+1}-x_{n}=f(x_{n})-f(x_{n-1}) \)

\(=ax_{n}-ax_{n-1}=a^{2}(x_{n-1}-x_{n-2})=...=a^{n}(x_{1}-x_{0})\).

For \(n<m\),

\(x_m-x_n=x_m-x_{m-1}+x_{m-1}-x_{m-2}+...-x_{n} \)

\( =(a^{m}+...a^{n})(x_1-x_0)=\frac{a^n}{1-a}(x_1-x_0) \)

And the right hand side converges to 0 as n tends to infinity (Here,we are using the fact that \(0<a<1\)) i.e, the right hand side can be made arbitrarily small using large enough n, so \(x_n\) is a Cauchy sequence. We are in \(\mathbb{R}\), so by completeness, \(x_n\) converges.

Remark:

The above discussion really didn't make use of Banach Fixed Point Theorem. But knowledge is power. And if known that :

"Let \(T:X\to X\) be a contraction mapping, X-complete metric space. Then T has precisely one fixed point \(u\in X\). Furthermore, for any \(x\in X\) the sequence \(T^k (x)\) converges and the limit is \(u\). " (Banach Fixed Point Theorem).

the answer is immediate. The above solution runs in the line of the proof of this Theorem.

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