Recently I have much interest in algebraic topology and algebraic geometry. I am a student of the field of complex dynamical systems. According to my knowledge, my friends told me that there are many applications of algebraic geometry and algebraic topology in reduction of dimension in statistics and some other fields. I want to know whether there exist some interesting applications of algebraic geometry and algebraic topology in dynamical systems. Any comments and advice will be appreciated. Thanks.
8 Answers
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
https://www.ihes.fr/~gromov/wpcontent/uploads/2018/08/1024.pdf
A recent, excellent survey of dynamics on algebraic surfaces is :
http://perso.univrennes1.fr/serge.cantat/Articles/dynaut.pdf
Densely scattered around the boundary of the Mandelbrot set $M$ you can find a miriad of tiny "Babybrots". We know that they are quasiconformally 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).
One place where papers on applications of algebraic topology — to dynamical systems as well as to statistics, data analysis, biomedicine, 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.

1$\begingroup$ 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. $\endgroup$– yaoxiaoCommented Dec 22, 2011 at 1:16

1$\begingroup$ The link is now redirected to appliedtopology.org with preprints listed at appliedtopology.org/preprints. $\endgroup$– J WCommented Aug 22, 2020 at 13:18
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''.

1$\begingroup$ Perhaps Algebraic Dynamics and Arithmetic Dynamics are more applications of Dynamical Systems methods in Algebraic Geometry and not vice versa? $\endgroup$ Commented Dec 21, 2011 at 15:30
Another good place to start is to track the output of Marion Mrozek and Konstantin Mischaikow, and their various coauthors. There is a whole group at the University in Krakow centered on Mrozek doing algebraic topology applications to dynamical systems.

3$\begingroup$ 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. $\endgroup$ Commented Dec 22, 2011 at 20:48

1$\begingroup$ 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. $\endgroup$ Commented Dec 22, 2011 at 20:52

1$\begingroup$ I think there is a typo in one of the names: it should be Marian instead of Marion. $\endgroup$– user39719Commented Apr 28, 2015 at 20:41

1$\begingroup$ See also the 2004 book, Computational Homology by Kaczynski, Mischaikow and Mrozek. $\endgroup$– J WCommented Aug 22, 2020 at 13:12
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), 79100), and his ICM contribution "Some analogies between number theory and dynamical systems" (Doc. Math. J. DMV, Extra Volume ICM I, 1998, 2346). More recent papers concerning dynamical systems can be found on Deninger's website.
I think this expository article: Algebraic Dynamics, Canonical Heights and Arakelov Geometry of Xinyi Yuan is useful.

3$\begingroup$ While this link may answer the question, it is better to include the essential parts of the answer here and provide the link for reference. Linkonly answers can become invalid if the linked page changes. $\endgroup$ Commented Jul 11, 2015 at 22:18

$\begingroup$ @Alex Degtyarev :Thanks for your guidance. $\endgroup$– user68208Commented Jul 11, 2015 at 22:42
You should look at the papers of Rafal Komendarczyk, and references therein (just look for his last name in google scholar).

$\begingroup$ In case it saves someone else a bit of effort, here's a link to this person's google scholar: scholar.google.com/… With dozens of papers, more context in this answer would have been nice, but I think the site norms were different back in 2011. $\endgroup$ Commented Mar 10 at 12:42