- Iterated monodromy group
In
geometric group theory anddynamical systems the iterated monodromy group of acovering map is a group describing themonodromy action of thefundamental group on all iterations of the covering. It encodes the combinatorics andsymbolic dynamics of the covering and is an example of aself-similar group .Definition
Let be a covering of a path-connected and locally path-connected topological space X by its subset , let be the
fundamental group of X and let be the monodromy action for f. Now let be the monodromy action of the iteration of f, .The Iterated monodromy group of f is the following
quotient group ::.The iterated monodromy group acts by automorphism on the rooted tree of pre
where a vertex is connected by an edge with .Examples
Let f be a complex
rational function and let be the union of forward orbits of its critical points (thepost-critical set ). If is finite (or has a finite set of accumulation points), then the iterated monodromy group of f is the iterated monodromy group of the covering , where is theRiemann sphere .Iterated monodromy groups of rational functions usually have exotic properties from the point of view of classical group theory. Most of them are infinitely presented, many have
intermediate growth .ee also
*
Growth rate (group theory)
*amenable group
*complex dynamics
*Julia set References
* Volodymyr Nekrashevych, [http://www.ams.org/bookstore-getitem/item=surv-117 "Self-Similar Groups"] , Mathematical Surveys and Monographs Vol. 117, Amer. Math. Soc., Providence, RI, 2005; ISBN 0-412-34550-1.
* Kevin M. Pilgrim, "Combinations of Complex Dynamical Systems", Springer-Verlag, Berlin, 2003; ISBN 3-540-20173-4.External links
* [http://www.arxiv.org/find/grp_physics,grp_nlin,grp_math/1/abs:+EXACT+iterated%5fmonodromy%5fgroup/0/1/0/all/0/1?skip=0&query_id=02226432472b51fb arXiv.org - Iterated Monodromy Group] - preprints about the Iterated Monodromy Group.
* [http://mad.epfl.ch/%7Elaurent/pub/rabbit/index.html Laurent Bartholdi's page] - Movies illustrating the Dehn twists about aJulia set .
* [http://mathworld.wolfram.com/MonodromyGroup.html mathworld.wolfram.com] - The Monodromy Group page.
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