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.
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
A recent, excellent survey of dynamics on algebraic surfaces is :
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''.
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.
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.
You should look at the papers of Rafal Komendarczyk, and references therein (just look for his last name in google scholar).
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.
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).
I think this expository article: Algebraic Dynamics, Canonical Heights and Arakelov Geometry of Xinyi Yuan is useful.