Find a topological conjugation between g(x) and T(x) where g and T are mappings (both tent maps [graphically speaking])(adsbygoogle = window.adsbygoogle || []).push({});

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

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Topological Conjugation between two dynamical systems

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

Loading...

Similar Threads - Topological Conjugation between | Date |
---|---|

I Difference between Wronskian and other methods | Dec 6, 2016 |

Difference between Lyapunov and linear stability criteria | Feb 23, 2016 |

Whittaker's solution and separable variables | Jan 11, 2016 |

Harmonic Functions, conjugates and the Hilbert Transform | Apr 26, 2012 |

Stuck on a complex ODE (conjugate issues ) | Jul 27, 2009 |

**Physics Forums - The Fusion of Science and Community**