PROBLEM 1: Experiment with the functions 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and the doubling function

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

SOLUTION:

We will compute the orbits for each of the functions for the seed

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

PkkiRkc2ImYqNiNJInhHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwmKiQ5JCIiIyIiIiEiI0YvRiRGJEYk

print();

LUkkc2VxRyUqcHJvdGVjdGVkRzYkLS1JI0BARzYiNiRJIkZHRilJImtHRik2IyQiIiQhIiIvRiw7IiIhIiNd

print();

QyQtSSV3aXRoRzYiNiNJJnBsb3RzRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlISIi

LUkqcG9pbnRwbG90RzYiNiU3Iy1JJHNlcUclKnByb3RlY3RlZEc2JDckSSJrR0YkLS1JI0BARzYkRilJKF9zeXNsaWJHRiQ2JEkiRkdGJEYsNiMkIiIkISIiL0YsOyIiISIjXS9JKGNvbm5lY3RHRiRJJXRydWVHRikvSSZjb2xvckdGJEkkcmVkR0Yk

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

print();

LUkkc2VxRyUqcHJvdGVjdGVkRzYkLS1JI0BARzYkRiRJKF9zeXNsaWJHNiI2JEkiR0dGK0kia0dGKzYjJCIiJCEiIi9GLjsiIiEiIz8=

print();

PkkiSEc2ImYqNiNJInhHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJC1JKnBpZWNld2lzZUclKnByb3RlY3RlZEc2JjMxIiIhOSQyRjIjIiIiIiIjLCRGMkY2MzFGNEYyMkYyRjUsJkYyRjYhIiJGNUYkRiRGJA==

print();

LUkkc2VxRyUqcHJvdGVjdGVkRzYkLS1JI0BARzYkRiRJKF9zeXNsaWJHNiI2JEkiSEdGK0kia0dGKzYjIyIiIiIiKi9GLjsiIiEiI10=

print();

Note that this is a 6-cycle. However when we use decimals, then the answer is nonsense because of truncation errors.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

LUkkc2VxRyUqcHJvdGVjdGVkRzYkLS1JI0BARzYkRiRJKF9zeXNsaWJHNiI2JEkiSEdGK0kia0dGKzYjLUkiL0dGKTYjJCIjISohIiIvRi47IiIhIiNd

print();

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

LUkqcG9pbnRwbG90RzYiNiU3Iy1JJHNlcUclKnByb3RlY3RlZEc2JDckSSJrR0YkLS1JI0BARzYkRilJKF9zeXNsaWJHRiQ2JEkiSEdGJEYsNiMjIiIiIiIqL0YsOyIiISIjXS9JKGNvbm5lY3RHRiRJJXRydWVHRikvSSZjb2xvckdGJEkkcmVkR0Yk

LUkqcG9pbnRwbG90RzYiNiU3Iy1JJHNlcUclKnByb3RlY3RlZEc2JDckSSJrR0YkLS1JI0BARzYkRilJKF9zeXNsaWJHRiQ2JEkiSEdGJEYsNiMtSSIvR0YwNiMkIiMhKiEiIi9GLDsiIiEiI10vSShjb25uZWN0R0YkSSV0cnVlR0YpL0kmY29sb3JHRiRJJHJlZEdGJA==

PROBLEM 2: Find all real fixed points of the following functions

(a) 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(b) 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(c) 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(d) 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

SOLUTION:

(a) We will solve the problem analytically and graphically.

PkkiRkc2ImYqNiNJInhHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwmKiQ5JCIiIyIiIiEiI0YvRiRGJEYk

print();

LUkmc29sdmVHNiI2Iy8tSSJGR0YkNiNJInhHRiRGKg==

print();

LUklcGxvdEc2IjYmNyRJInhHRiQtSSJGR0YkNiNGJy9GJzshIiQiIiQvSSJ5R0YkOyEiJSIiJS9JJmNvbG9yR0YkNyRJJHJlZEdGJEklYmx1ZUdGJA==

PkkiRkc2ImYqNiNJInhHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwmKiQ5JCIiJCIiIkYtISIkRiRGJEYk

print();

LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy8tSSJGR0YnNiNJInhHRidGLQ==

print();

LUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYmNyRJInhHRictSSJGR0YnNiNGKi9GKjshIiQiIiQvSSJ5R0YnOyEiJSIiJS9JJmNvbG9yR0YnNyRJJHJlZEdGJ0klYmx1ZUdGJw==

PkkiRkc2ImYqNiNJInhHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCokOSQiIidGJEYkRiQ=

print();

LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy8tSSJGR0YnNiNJInhHRidGLQ==

print();

The only real fixed points are 0,1.

LUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYmNyRJInhHRictSSJGR0YnNiNGKi9GKjshIiQiIiQvSSJ5R0YnOyEiJSIiJS9JJmNvbG9yR0YnNyRJJHJlZEdGJ0klYmx1ZUdGJw==

PkkiRkc2ImYqNiNJInhHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJComOSQiIiItSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ2I0YsRi1GJEYkRiQ=

print();

QyQ+STFfRW52QWxsU29sdXRpb25zRzYiSSV0cnVlRyUqcHJvdGVjdGVkRyEiIg==

LUkmc29sdmVHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy8tSSJGR0YnNiNJInhHRidGLQ==

print();

There are inifinitely many fixed points and they are all neutral.

LUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYmNyRJInhHRictSSJGR0YnNiNGKi9GKjshIz8iIz8vSSJ5R0YnOyEjRCIjRC9JJmNvbG9yR0YnNyRJJHJlZEdGJ0klYmx1ZUdGJw==

PROBLEM 3: For each of the following seeds, discuss the behavior of the resulting orbit under the doubling function:

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

SOLUTION:

PkkiSEc2ImYqNiNJInhHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJC1JKnBpZWNld2lzZUclKnByb3RlY3RlZEc2JjMxIiIhOSQyRjIjIiIiIiIjLCRGMkY2MzFGNEYyMkYyRjUsJkYyRjYhIiJGNUYkRiRGJA==

print();

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

LUkkc2VxRyUqcHJvdGVjdGVkRzYkLS1JI0BARzYkRiRJKF9zeXNsaWJHNiI2JEkiSEdGK0kia0dGKzYjJCIiJCEiIi9GLjsiIiEiIzU=

print();

Eventually a 4-cycle.

LUkkc2VxRyUqcHJvdGVjdGVkRzYkLS1JI0BARzYkRiRJKF9zeXNsaWJHNiI2JEkiSEdGK0kia0dGKzYjJCIiKCEiIi9GLjsiIiEiIzU=

print();

Eventually a 4-cycle.

LUkkc2VxRyUqcHJvdGVjdGVkRzYkLS1JI0BARzYkRiRJKF9zeXNsaWJHNiI2JEkiSEdGK0kia0dGKzYjIyIiIiIiKS9GLjsiIiEiIzU=

print();

Eventually fixed.

LUkkc2VxRyUqcHJvdGVjdGVkRzYkLS1JI0BARzYkRiRJKF9zeXNsaWJHNiI2JEkiSEdGK0kia0dGKzYjIyIiIiIjOy9GLjsiIiEiIzU=

print();

Eventually fixed.

LUkkc2VxRyUqcHJvdGVjdGVkRzYkLS1JI0BARzYkRiRJKF9zeXNsaWJHNiI2JEkiSEdGK0kia0dGKzYjIyIiIiIiKC9GLjsiIiEiIzU=

print();

A 3-cycle.

LUkkc2VxRyUqcHJvdGVjdGVkRzYkLS1JI0BARzYkRiRJKF9zeXNsaWJHNiI2JEkiSEdGK0kia0dGKzYjIyIiIiIjOS9GLjsiIiEiIzU=

print();

Eventually a 3-cycle.

LUkkc2VxRyUqcHJvdGVjdGVkRzYkLS1JI0BARzYkRiRJKF9zeXNsaWJHNiI2JEkiSEdGK0kia0dGKzYjIyIiIiIjNi9GLjsiIiEiIzU=

print();

A 10-cycle.

LUkkc2VxRyUqcHJvdGVjdGVkRzYkLS1JI0BARzYkRiRJKF9zeXNsaWJHNiI2JEkiSEdGK0kia0dGKzYjIyIiJCIjQS9GLjsiIiEiIzo=

print();

Eventually a 10-cycle.

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn

PROBLEM 4: Give an explicit formula for 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Can you write down a formula for 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

SOLUTION: The best way of thinking of the doubling function as it is defined above is

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

So 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 and 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Finally LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYsLUkobWZlbmNlZEdGJDYkLUYjNiMtSSVtc3VwR0YkNiUtSSNtaUdGJDYlUSJIRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiMtRjQ2JVEibkYnRjdGOi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRicvRjtRJ25vcm1hbEYnLUYsNiQtRiM2Iy1GNDYlUSJ4RidGN0Y6RkUtSSNtb0dGJDYwUSI9RidGRS8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGVC8lKXN0cmV0Y2h5R0ZULyUqc3ltbWV0cmljR0ZULyUobGFyZ2VvcEdGVC8lLm1vdmFibGVsaW1pdHNHRlQvJSdhY2NlbnRHRlQvJSVmb3JtR1EmaW5maXhGJy8lJ2xzcGFjZUdRL3RoaWNrbWF0aHNwYWNlRicvJSdyc3BhY2VHRmBvLyUobWluc2l6ZUdRIjFGJy8lKG1heHNpemVHUSlpbmZpbml0eUYnLUYxNiUtSSNtbkdGJDYkUSIyRidGRUY9RkItRk82MFEifkYnRkVGUkZVRldGWUZlbkZnbkZpbi9GXG9RIUYnL0Zfb1EkMGVtRicvRmJvRmVwRmNvRmZvLUZPNjJRJG1vZEYnLyUlYm9sZEdGOS9GO1ElYm9sZEYnLyUrZm9udHdlaWdodEdGXXFGUkZVRldGWUZlbkZnbkZpbkZicC9GX29RMG1lZGl1bW1hdGhzcGFjZUYnL0Zib0ZhcUZjb0Zmb0ZfcC1GXHA2JFEjMS5GJ0ZFRl9wRl9wIn more explicit notation

Pi1JIkBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JEkiSEdGKEYqZio2I0kieEdGKEYoNiRJKW9wZXJhdG9yR0YoSSZhcnJvd0dGKEYoLUkqcGllY2V3aXNlR0YmNiozMSIiITkkMkY3IyIiIiIiJSwkRjdGOzMxRjlGNzJGNyNGOiIiIywmRjdGOyEiIkY6MzFGQEY3MkY3IyIiJEY7LCZGN0Y7ISIjRjozMUZHRjcyRjdGOiwmRjdGOyEiJEY6RihGKEYo

print();

LS1JIkBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JEkiSEdGKEYqNiNJInhHRig=

print();

Pi1JIkBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC1GJDYkSSJIR0YoRixGLGYqNiNJInhHRihGKDYkSSlvcGVyYXRvckdGKEkmYXJyb3dHRihGKC1JKnBpZWNld2lzZUdGJjYyMzEiIiE5JDJGOSMiIiIiIiksJEY5Rj0zMUY7RjkyRjkjRjwiIiUsJkY5Rj0hIiJGPDMxRkJGOTJGOSMiIiRGPSwmRjlGPSEiI0Y8MzFGSUY5MkY5I0Y8IiIjLCZGOUY9ISIkRjwzMUZQRjkyRjkjIiImRj0sJkY5Rj0hIiVGPDMxRldGOTJGOSNGSkZDLCZGOUY9ISImRjwzMUZobkY5MkY5IyIiKEY9LCZGOUY9ISInRjwzMUZeb0Y5MkY5RjwsJkY5Rj0hIihGPEYoRihGKA==

print();

LS1JIkBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2JC1GJDYkSSJIR0YoRixGLDYjSSJ4R0Yo

print();

I am sure you can write down the general formula for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEibkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLzYmUSFGJy9GM1EmZmFsc2VGJy8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJy9GNlEnbm9ybWFsRicvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnth iterate of the doubling function.

Another nice and enlightning way of thinking of the above defined doubling function is the following:

Every real number has a binary expansion. In particular, if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUlZm9ybUdRIUYnLyUnbHNwYWNlR1EkMGVtRicvJSdyc3BhY2VHRk8vJShtaW5zaXplR1EiMUYnLyUobWF4c2l6ZUdRKWluZmluaXR5Ric=is in [0, 1) then 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