Topological Conjugation between two dynamical systems

[selig5560]

Find a topological conjugation between g(x) and T(x) where g and T are mappings (both tent maps [graphically speaking])

g:[-1, 1] → [-1,1]
g(x) = 1-2|x|

T:[0,1] → [0, 1]
T(x) = 2x when x ≤ 1/2 and 2(1-x) when x ≥ 1/2

h ° T = g ° h (homeomorphism)

h:[0, 1] → [-1, 1]
h(x) = cos(∏x)

when T(x) = 2x and x ≤ 1/2:

cos(∏*(2x)) = sin^2(∏x) - cos^(∏x) = 1 - 2cos^2(∏x) = 1 - 2|cos^2(∏x) = -cos(2∏x)

when T(x) = 2(1-x) and x ≥ 1/2

cos(2∏(1-x) = cos(2∏-2x∏) = -cos^2(∏x)+sin^2(∏x) = 1-2|cos^2(∏x) + sin^2(∏x) = 1-2|cos(2∏x)|

The above attempt I know is incorrect because after I introduce the absolute value brackets I do not get the desired result.

This is when I get stuck. I have tried many different variations of trig functions to act as the conjugator between g and T(x), however I have had no luck (after many hours.) I know for that it would be easy to find a homeomorphism if it wasnt for the |x| part of the 1-2|x| dynamical system (tent map.) I do not think that I am supposed to find a conjugation between two tent maps (explicit i,.e. conjugating two piecewise functions) because it seems that would be highly redundant. If anyone could provide some assistance that would be great.

- Selig

 

[jvicens]

man

Dear Seligman,

Thank you for sharing your attempts and thought process. It seems like you are on the right track with trying to use trigonometric functions as the conjugator between g(x) and T(x). However, as you pointed out, the absolute value in g(x) makes it challenging to find a suitable homeomorphism.

One approach you could try is to break down g(x) into two separate functions, each with a different domain. For example, you could define h(x) as cos(∏x) for x ≥ 0 and -cos(∏x) for x < 0. Then, you can define a new function f(x) as 1-2h(|x|). This function f(x) will have the same range as g(x) and will also be continuous and invertible.

Now, let's see how this function f(x) relates to T(x).

When T(x) = 2x and x ≤ 1/2:

f(T(x)) = 1-2h(|2x|) = 1-2h(2|x|) = 1-2cos(∏(2|x|)) = 1-2cos(2∏|x|)

When T(x) = 2(1-x) and x ≥ 1/2:

f(T(x)) = 1-2h(|2(1-x)|) = 1-2h(2-2|x|) = 1-2cos(∏(2-2|x|)) = 1-2cos(2∏(1-|x|))

Using the identity cos(2∏x) = cos(2∏(1-x)) = 1-2cos^2(∏x), we can simplify the above expressions to:

f(T(x)) = 1-2cos(2∏|x|) = 1-2|cos(2∏x)|

Therefore, we have shown that f(x) is a conjugation between g(x) and T(x), satisfying the condition h ° T = g ° h.

I hope this helps! Keep up the good work in exploring topological conjugations between different mappings.