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Recently I have much interest in algebraic topology and algebraic geometry, I am a student of field of complex dynamic systems. According to my knowledge, my friends told me that there are many applications of algebraic geometry and algebraic topology in dimension of reduction in statistics and some other fields. I want to konw whether there exists some interesting applications of algebraic geometry and albebraic topology in dynamics system. Any comments and advice will be appreciated. Thanks.

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algebraic geometry and dynamical systems brings to mind the name of Christopher Byrnes to me. –  roy smith Dec 22 '11 at 5:04

7 Answers 7

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The first application of algebraic geometry to dynamical systems that comes to my mind is the following preprint of Gromov -- very old one : ON THE ENTROPY OF HOLOMORPHIC MAPS

http://www.ihes.fr/~gromov/PDF/10[24].pdf

A recent, excellent survey of dynamics on algebraic surfaces is :

http://perso.univ-rennes1.fr/serge.cantat/Articles/dyn-aut.pdf

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Densely scattered around the boundary of the Mandelbrot set $M$ you can find a miriad of tiny "Babybrots". We know that they are quasi-conformally isomorphic copies of $M$; that is to say, they are deformed, but not to wildly. Still, it is VERY surprising that they are visually recognizable. All the gray regions below seem to be bounded by circles and cardioids:

Surprise, surprise! The largest interior component of $M$ is indeed bounded by a cardioid, and the large disk to its left is truly round. But all other components are bounded by real algebraic curves of high degree, and are no longer true cardioids and circles:

The paper A Parameterization of the Period 3 Hyperbolic Components of the Mandelbrot Set by D. Giarrusso and Y. Fisher studies these boundaries as algebraic curves, and shows how to construct explicit parametrizations in the cases of period 1 (The Cardioid $C$), period 2 (The Disk), and period 3 (the two largest "disks" above and below $C$, plus the tiny "cardioid" visible in the middle of the left antenna).

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Methods from arithmetic algebraic work have been used in the work of Christoph Deninger on dynamical systems. References can be found on the arXiv at Cornell, for example the papers "Number theory and dynamical systems on foliated spaces" (math/0204110, published in Jber. d. Dt. Math.-Verein. 103 (2001), 79-100), and his ICM contribution "Some analogies between number theory and dynamical systems" (Doc. Math. J. DMV, Extra Volume ICM I, 1998, 23-46). More recent papers concerning dynamical systems can be found on Deninger's website.

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One place where papers on applications of algebraic topology — to dynamical systems as well as to statistics, data analysis, bio-medicine, computational geometry and other areas — gets aggregated is on the webpage of the Computational Topology group at Stanford. This page has a running listing of relevant preprints and papers that may form a good starting point for you.

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Thanks for your advice, last year a professor in Chern's institue have given a report about algebraic geometry's applications in dimension reuduction's about the accuracy test of Fisher's table test, which give me deep and powerful impression. –  yaoxiao Dec 22 '11 at 1:16

Another good place to start is to track the output of Marion Mrozek and Konstantin Mischaikow, and their various co-authors. There is a whole group at the University in Krakow centered on Mrozek doing algebraic topology applications to dynamical systems.

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I wanted to mention Konstantin's group at Rutgers (previously) at Georgia Tech which does essentially computational homology theory with applications to dynamical systems but only after I mention the the God Father, the late Charles Conley former professor at the University of Wisconsin and the Ph.D. thesis advisor of Konstantin Mischaikow. His survey paper on now famous Conley's Index theory should be the first read for anybody interested in involving heavy algebraic topology in dynamical systems. –  Predrag Punosevac Dec 22 '11 at 20:48
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Also Rick Mockel (another former Ph.D. student of Charles Conley) and his students have found many interesting applications of Algebraic Topology in Celestial Mechanic. One of Conley's students (Kris MeCoord) wrote a bunch of celestial mechanics papers with Ken Mayer and my advisor Qiudong Wang at the University of Cincinnati using nothing but Algebraic Topology techniques. –  Predrag Punosevac Dec 22 '11 at 20:52

The emerging field of arithmetic dynamics has seen a lot of attention lately, and it is very intimately connected to algebraic geometry (and, to a lesser degree, algebraic topology as well). I think you would enjoy reading Silverman's ``The Arithmetic of Dynamical Systems''.

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Perhaps Algebraic Dynamics and Arithmetic Dynamics are more applications of Dynamical Systems methods in Algebraic Geometry and not vice versa? –  Mahdi Majidi-Zolbanin Dec 21 '11 at 15:30

You should look at the papers of Rafal Komendarczyk, and references therein (just look for his last name in google scholar).

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