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| | '''Suspension''' is a construction passing from a map to a flow. Namely, let <math>X</math> be a metric space, <math>f:X\to X</math> be a continuous map and <math>r:X\to\mathbb{R}^+</math> be a function (roof function or ceiling function) bounded away from 0. Consider the quotient space |
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| | :<math>X_r=\{(x,t):0\le t\le r(x),x\in X\}/(x,r(x))\sim(fx,0). </math> |
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| | The suspension of <math>(X,f)</math> with roof function <math>r</math> is the semiflow<ref>M. Brin and G. Stuck, ''Introduction to Dynamical Systems'', Cambridge University Press, 2002.</ref> <math>f_t:X_r\to X_r</math> induced by the time-translation <math>T_t:X\times\mathbb{R}\to X\times\mathbb{R}, (x,s)\mapsto (x,s+t)</math>. |
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| | If <math>r(x)\equiv 1</math>, then the quotient space is also called the [[mapping torus]] of <math>(X,f)</math>. |
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| | ==References== |
| | <!--- See http://en.wikipedia.org/wiki/Wikipedia:Footnotes on how to create references using <ref></ref> tags which will then appear here automatically --> |
| | {{Reflist}} |
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| | <!--- Categories ---> |
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| | [[Category:Dynamical systems]] |
Revision as of 16:57, 14 December 2013
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Suspension is a construction passing from a map to a flow. Namely, let be a metric space, be a continuous map and be a function (roof function or ceiling function) bounded away from 0. Consider the quotient space
The suspension of with roof function is the semiflow[1] induced by the time-translation .
If , then the quotient space is also called the mapping torus of .
References
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- ↑ M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, 2002.