Chapter 12.1, Problem 5E

Interpretation Introduction

Interpretation:

Consider the dynamics of the baker’s map in the area-preserving case: $\text{a =}\frac{\text{1}}{\text{2}}$.

1. To write down the binary representation of $\text{B(x,y)}$, given that $\left(\text{x,y}\right)\text{ = }\left({\text{.a}}_{\text{1}}{\text{a}}_{\text{2}}{\text{a}}_{\text{3}}\mathrm{....}\text{,}{\text{.b}}_{\text{1}}{\text{b}}_{\text{2}}{\text{b}}_{\text{3}}\mathrm{....}\right)$ is the binary representation of an arbitrary point in the square.

2. To show that B has a period-$\text{2}$ orbit, and to sketch its location in the unit square.

3. To show that B has countably many periodic orbits.

4. To show that B has uncountably many aperiodic orbits.

5. To find if there is any dense orbit or not in the baker’s map; if so, then write one down explicitly, and if not, explain why not.

Concept Introduction:

• The  baker’s mapis a chaotic map from the unit square into itself. It is known as after a kneading operation that bakers apply to the dough: the dough is cut in half, and the two halves are stacked on one another and compressed..

• The baker's map is considered by its action on the space of functions defined on the unit square.