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## Main Question or Discussion Point

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

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

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