Problem 2 - A (terse) introduction to Lebesgue integration

Question

Prove that if \(\alpha / 2 \pi\) is rational, then every point of \(\mathbb{T}\) is periodic for \(T_{\alpha}\), i.e., for each \(x \in \mathbb{T}\) there is an \(n>0\) such that \(T_{\alpha}^{n}(x)=x\)

Step-by-Step Solution

Answer

If \(\alpha / 2 \pi\) is rational, every point \(x \in \mathbb{T}\) is periodic for \(T_\alpha\).

Step 1: Define the transformation

The transformation \(T_{\alpha}\) on the circle \(\mathbb{T}\) is defined by \(T_{\alpha}(x) = x + \alpha \mod 1\). We apply this definition to each point \(x\) on the circle.

Step 2: Express \(\alpha\) as a rational multiple of \(2\pi\)

Since \(\alpha / 2\pi\) is rational, we express it as \(\frac{m}{n}\) for some integers \(m\) and \(n\) with \(n > 0\). This implies \(\alpha = \frac{2\pi m}{n}\).

Step 3: Calculate \(T_{\alpha}^{n}(x)\)

Apply the transformation \(n\) times: \(T_{\alpha}^{n}(x) = x + n\alpha \mod 1\). Substitute \(\alpha = \frac{2\pi m}{n}\) into this equation to get \(T_{\alpha}^{n}(x) = x + n\left(\frac{2\pi m}{n}\right) \mod 1 = x + 2\pi m \mod 1\).

Step 4: Simplify the expression

Since adding \(2\pi m\) (for any integer \(m\)) is equivalent to making full revolutions around the circle, \(x + 2\pi m \equiv x \mod 1\). Therefore, \(T_{\alpha}^{n}(x) = x\), showing that after \(n\) applications of the transformation, the point returns to \(x\).

Step 5: Conclusion

Since for each \(x \in \mathbb{T}\) such an \(n\) exists, every point of \(\mathbb{T}\) is periodic for \(T_{\alpha}\). This completes the proof that if \(\alpha / 2 \pi\) is rational, \(T_{\alpha}\) leaves all points \(x\) periodic.

Key Concepts

Periodic Points

Periodic points in circle transformations refer to the points that return to their initial position after a certain number of transformations. This concept is quite fascinating and plays a pivotal role in understanding how transformations can behave over a circular space like the unit circle \( \mathbb{T} \).
A point \( x \) on the circle \( \mathbb{T} \) is said to be periodic under a transformation \( T_{\alpha} \) if there exists some positive integer \( n \) such that \( T_{\alpha}^{n}(x) = x \).

• This means after \( n \) transformations the point returns to its original position.
• In circular transformations like \( T_{\alpha} \), this concept helps illustrate the repetitive nature of angles and rotations on a circle.

Understanding periodic points is essential in studying the stability and long-term behavior of dynamic systems, especially those mapped on circular paths.

Rational Numbers

Rational numbers are the numbers that can be expressed as a quotient or fraction \( \frac{m}{n} \), where both \( m \) and \( n \) are integers, and \( n \) is not zero. They form a crucial part of the transformation discussed in this exercise where the angle \( \alpha \) is related to \( 2\pi \) through a rational number.
The relationship \( \frac{\alpha}{2\pi} = \frac{m}{n} \) tells us that \( \alpha \) is some fraction of a full circle. Rational numbers bring about cyclical behavior in the context of this problem because:

• They enable the repetition of a particular transformation.
• They provide a predictable pattern due to their nature of expressing divisions or segments of a whole.

Understanding rational numbers helps in comprehending repeating sequences and periodicity in mathematics.

Circle Transformations

Circle transformations involve applying a function to points on a circle. For a point \( x \) on the unit circle \( \mathbb{T} \), the transformation \( T_{\alpha} \) is defined as \( T_{\alpha}(x) = x + \alpha \mod 1 \). This means that we take the point \( x \), add an angle \( \alpha \), and find the result within the range \([0, 1)\) by taking the modulo.
The aim of these transformations is to explore how points move around the circle and under which conditions they cycle back to the start:

• The unit circle \( \mathbb{T} \) represents all real numbers from 0 to 1 in a loop.
• When \( \alpha / 2\pi \) is rational, the transformation results in every point being periodic.

These transformations are deeply linked with trigonometric functions and provide insights into periodic functions and repetitive phenomena.

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers wrap around upon reaching a certain value—the modulus. It's a fascinating branch of number theory, often referred to as "clock arithmetic," and plays a crucial role in the exercise's solution.
When we apply transformations on a circle, like \( T_{\alpha} \) on \( \mathbb{T} \), modular arithmetic is used to ensure that the value remains within the circle's bounds.

• For example, the expression \( T_{\alpha}(x) = x + \alpha \mod 1 \) ensures that after adding \( \alpha \) we find the remainder when divided by 1, keeping the rotation within \([0, 1)\).
• This concept helps maintain periodicity and predictability, as points will eventually repeat their positions.

Utilizing modular arithmetic in transformations demonstrates the periodic nature of rotational mappings on circles and how fundamental numbers behave cyclically.