tessel@tum.bot
Jul1-04, 04:47 PM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>On Sun, 27 Jun 2004, Sanjiv Ramachandran wrote:\n\n> the author has criticized the dynamical systems approach to understand\n> turbulence.\n\n[snip]\n\n> 3. This question might sound stupid. If people agree that some phenomena\n> is described by differential equations and that dynamical systems is the\n> study of differential equations, where is the controversy ?? Unless of\n> course, we doubt the validity of the differential equation itself\n> (Navier-Stokes, in this case).\n\nI was not aware of any controversy here (probably because I know very\nlittle about turbulence!), but you should be aware that any claim that\n"dynamical systems is the study of differential equations" is completely\nincorrect. If one -had- to give a one-sentence characterization, it would\nprobably be better to say that "most notions of a dynamical system involve\nsomewhere the notion of an appropriate action by some group or semigroup\nor groupoid". Some of the most important can indeed be related to\ndifferential equations; e.g. a system of autonomous ODEs may arise from\nthe phase space analysis of a system of reaction-diffusion equations (a\nsystem of PDEs), or, as in your case, to the Navier-Stokes equations.\n\nHere is some very sketchy background information, some advice, and a short\nlist of suggested references for further background reading, which I hope\nyou will find valuable or at least intriguing:\n\nFirst, a bit of background: there is no universally agreed upon definition\nof "dynamical system", although various formal definitions have been\noffered by Steve Smale and many subsequent authors. There are however\nseveral broad classes of definitions:\n\n(a) There are various types of "discrete" dynamical system in terms of a\ntransformation on some space, with certain properties, which is iterated:\n\n(i) in ergodic theory and symbolic dynamics one encounters\n"measure-theoretic dynamical system" (X,M,mu,T), where mu is a T-invariant\nBorel probability measure (if you know what that is),\n\n(ii) a "topological dynamical system" is often defined as something like a\ncompact Hausdorff space X with a homeomorphism T:X--> X (or even just a\ncontinuous map) which is iterated.\n\n(b) "Discrete" dynamical systems can each be considered as an object in an\nappropriate "category"; one can set this up in terms of categories induced\nby assuming that T: X--> X is measure-preserving or continuous or smooth\n(or several of these, usually with compatibility of structure assumptions)\non Z-spaces or N-spaces (Z the additive group of integers, N the additive\nsemigroup of natural numbers); these can be readily generalized to\nZ^d-spaces. These are often called "Z-shifts" or "Z^d-shifts"\n(topological dynamical systems) or "Markov chains" (the analogous\nmeasure-theoretic dynamical systems). If you have heard of "cellular\nautomatons", btw, these are usually just Z^d-shifts with one extra level\nof structure.\n\n(c) One can generalize to actions by topological groups or even Lie\ngroups---most often abelian groups, in fact, almost always just R or R^d\nas an additive group--- in particular, re my recent post on certain\n"tiling dynamical systems", these are often defined as something like\nR^d-spaces with a continuous action on some tiling space which has been\nmade into a compact Hausdorff space (there are several equivalent and\nnatural ways of doing this, just as for discrete group actions); now going\nin the opposite direction (toward increasingly special classes of\ndynamical systems), we have:\n\n(i) "smooth dynamical systems" often involve something like an\n"equicontinuous action by a family of diffeomorphisms"; in particular,\nmany authors consider something like an autonomous system of ODEs (c.f.\n"phase space methods") to comprise a "dynamical system",\n\n(ii) "complex analytic dynamical systems", which usually involve iterating\na holomorphic or meromorphic map on the Riemann sphere or something like\nthat (c.f. "Julia set", "Mandlebrot set", if you\'ve heard of those).\n\nHmm... oh dear, I don\'t think I said that very well.\n\nLet me try again to briefly summarize, in slightly different words, some\nof the most imporant "rough classifications" of the immense variety of\ndynamical system one could study:\n\nThe definition of a given class of dynamical systems usually involves some\nkind of "group action", although this fact may be hidden from view. The\ngroup G which acts can be a discrete group, or a topological group, or\neven a Lie group. In the case of topological or Lie groups, G can have\nvarious dimensions. It can be abelian or nonabelian. The action effected\nby G can be measure-preserving, or continuous, or smooth, or analytic.\nThe space X on which G acts can be a measure space, or a topological\nspace, or a smooth or even analytic manifold (the Riemann sphere is a\nfamiliar and very important example of an analytic manifold). One cannot\nchoose these attributes with complete independence, of course: it only\nmakes sense to say that T is a continuous map if X is a topological space,\netc.\n\nAs an important special case, one can study the group G = <T> generated by\niterating a single invertible map T on some "space" X, where X can be a\nmeasure space, or a topological space (often a compact Hausdorff space, or\neven a compact metric space), or even a topological or smooth manifold.\nThe map T can be measure-preserving, or continuous, or smooth, or even\nanalytic (c.f. "Julia sets" and the "Mandlebrot set" and all that). In\ndefining this type of dynamical system, many authors ignore the presence\nof a group action by Z, the integers. Actually, you are just as likely to\nencounter -noninvertible- maps T in this context; iterating T then defines\na semigroup <T>, i.e. an "action" by N, the natural numbers. But at first\nit is often best to stick to groups and group actions.\n\nA basic idea here is that measure spaces are more general than topological\nspaces, which are more general than topological manifolds, which are more\ngeneral than smooth manifolds, which are more general than analytic\nmanifolds. Similarly, measureable maps--- are in a sense--- more general\nthan continuous maps, which are more general than smooth maps, which are\nmore general than analytic maps. Going the other way, invertible maps are\nless general than possibly non-invertible maps, and groups are less\ngeneral than semigroups.\n\nAnother basic idea here is that when some pair (X,T) possesses more than\none level of structure--- in particular, both topological and\nmeasure-theoretic structure--- one usually demands that these structures\nbe in some sense "compatible" with one another. In particular, there will\nbe many invariant measures mu associated with a given continuous map T on\na compact Hausdorff space X, and thus many measure-theoretic dynamical\nsystems (X,T,mu) sharing the same underlying topological dynamical system\n(X,T). In this situation, one can define the topological entropy h(X,T)\nof the common underlying topological dynamical system (X,T), and also\nmeasure-theoretic entropies H(X,T,mu) of the various measure-theoretic\ndynamical systems. Then, an important theorem says that maximizing\nH(X,T,mu) over the various mu always gives h(X,T)! (This is not at all\nobvious from the definitions.)\n\n(The measures mu above are assumed to be Borel measures, which amounts to\nsaying we only consider measures which are compatible with the topological\nstructure, in an appropriate sense. Also, I am ignoring here some\ntechnical issues about sigma-algebras here which can obscure the point I\nam making, and some additional assumptions about X which are far less\nrestrictive than they might at first appear.)\n\nThis is all rather informal, but hopefully gives the idea, even if you\nhave never seen some of the technical terms I mentioned. BTW, if you\nlearn even a tiny bit about "category theory", it will become -much-\neasier to understand how all these concepts are organized.\n\nNext, some advice: I see you are at PSU. Well, if you don\'t often visit\nthe math department, you should, because you are fortunate enough to share\na campus with perhaps the foremost group of dynamical systems people in\nthe U.S., or even the world! In particular, I believe that A. Katok is\nstill at PSU; he is hugely knowledgeable, and a leading figure in the\nmathworld side of dynamical systems. And another fairly accessible math\ndepartment (U Maryland at College Park) boasts another leading research\ngroup working on dynamical systems (or -two- groups, depending upon how\nyou count).\n\nPerhaps I should warn you that many "rigorists" are wont to complain about\nthe activity of a large school of people writing papers on "applied\nchaos", who, the rigorists feel, ignore enormous progress made by\nmathematicians toward correcting very serious errors in older ideas on\nthings like various kinds of "fractal dimensions" (often related to\nvarious "entropies"), especially in the context of simulations of\ncomplicated systems in applied physics, etc. OTH, the "rough-and-ready"\ncrowd may complain that the rigorists underestimate the difficulty\nnonmathematicians experience in learning how to even -state- the latest\ntheorems, much less apply them. Representatives of both camps can be\nfound at UM, and probably at PSU, so you will probably encounter further\ndifferences of approach to the highly complicated and extremely broad\ntopic of "chaos" etc. in "dynamical systems".\n\nBut don\'t let the sometimes heated debate between these two camps deter\nyou from trying to take the best of what each has to offer: keep an open\nmind and I claim you will find that both "camps" make some valid points;\ntheir lamentable failure to talk to each other as much as they could and\nshould, is IMHO mostly based on mutual misunderstanding, misappreciation,\nand underestimation.\n\nAs (I guess) an engineering/applied physics student at PSU, you have a\nheaven-sent opportunity to try to become an "ambassador" between the\n"ruffians" and the "rigorists". (You might not have known it, but you\'re\nprobably a ruffian by institutional affiliation and perhaps by\nbackground.)\n\nThis would not be easy--- more because of the difficulty of learning\nenough to appreciate so many and higly varied points of view, than because\nof conflicting ideologies/personalities--- but if you succeed in learning\nenough to explain in useful terms to the ruffians what the rigorists have\naccomplished, or if you can help the rigorists cook up\ndefinitions/theories more useful to the ruffians, this could eventually\nmake you a person Highly Valued by both camps, so looking into the stuff I\nam talking about could be a smart (and highly lucrative) career move--- at\nleast, if you are very capable and very ambitious.\n\nBut more immediately, there are very real scientific dividends which\nshould -quickly- accrue from learning more math. One reason for this is\nthat, as a rule, "ruffians" who trying to simulate the behavior of some\ndynamical system--- perhaps not as end in itself, but only as part of\ndoing something else, like "simulated annealing"--- are very likely\nperforming their simulations in fundamentally incorrect ways, and thus\nobtaining wildly misleading results. This makes rigorists froth at the\nmouth. Often, they say, if only the ruffians knew a few simple facts,\nthey would be in a much better position to appreciate the fact that\n-correct- simulation methods are\n\n(a) readily available,\n\n(b) easy to learn,\n\n(c) easy to use in practice.\n\nA specific example might help here:\n\nMany engineers, computer scientists, or other applied scientists often\nneed to simulate a "Markov chain". Here, a fundamental fact, as even your\naverage ruffian may appreciate :-/ is that many such chains are "ergodic".\nThere are some nice criteria for knowing when a Markov chain is ergodic.\nIf it is, one can apply the immensely powerful ergodic theorems. But\nthere is a fundamental problem here (probably known only to rigorists):\nwhile you know the convergence occurs, it might occur very slowly. (I am\noversimplifying, but in some very common situations, including simulated\nannealing, this -is- a serious concern).\n\nFortunately, a very important recent advance by the rigorists is the\ndiscovery of an amazing idea which goes by the name "sampling from the\npast". This amounts to an easily implemented algorithm which completely\ncircumvents the problem, allowing one to simulate the unknown invariant\nmeasure rather than the process of convergence to that measure.\n\nIn short, if you didn\'t use sampling from the past (or one of the even\nmore recent and even better "perfect sampling" algorithms) for your Markov\nchain simulation (this could happen only if you\'ve never heard of these\nalgorithms, since the simplest ones are no harder to implement than a\nnaive simulation), you would actually not be modeling what you think you\nare at all--- and in general you -cannot- tell just by looking at the\noutput how accurate your simulation is! But if you use "sampling from the\npast" you never need to worry about just what your simulation is actually\nsimulating.\n\nTo admit the obvious, to make a point I am here somewhat overstating\nthings. Every advance in understanding raises new questions which lead to\nthe recognition of new shortcomings. Hence the more recent "perfect\nsampling" algorithms. But this doesn\'t invalidate my point: the\nrigorists have here moved far beyond most ruffians, yet this great leap\nforward is by no means inaccessible to nonmathematicians, and a real\ndialog between ruffians and rigorists here could have a highly beneficial\nimpact on the further development of this field. I can think of analogous\nsituations where rigorists have gone off in directions which are of little\ninterest to ruffians, but only because they didn\'t know enough about\nrelevant applications to see the superior potential of another theoretical\napproach. BTW, in my experience, rigorists are usually -very- interested\nin possible applications of their theories, and they are very capable of\nbeing influenced to change the direction of their theoretical\nexplorations, should an applied scientist demonstrate serious interest in\ntheir work. So, here lies a significant opportunity for both theoretical\nand applied mathscis, hopefully not to be missed.\n\nFinally, here are some references for further reading:\n\nTo get a quick impression of the huge scope of modern definitions of\n"dynamical system", see\n\nauthor = {Anatole Katok and Boris Hasselblatt},\ntitle = {Introduction to the Modern Theory of Dynamical Systems},\npublisher = {Cambridge University Press},\nyear = 1995}\n\n(Last time I checked, Katok was in the math department at PSU, so you\nmight want to find out if he is still there.)\n\nA much easier introduction (aimed at engineers and scientists) is:\n\nauthor = {Robert C. Hilborn},\ntitle = {Chaos and Nonlinear Dynamics: An Introduction for Scientists\nand Engineers},\npublisher = {Oxford University Press},\nyear = 1994}\n\nEven easier (but much sketchier and far less comprehensive):\n\ntitle = {Coping with Chaos: Analysis of Chaotic Data and Exploitation\nof Chaotic Systems},\neditor = {Edward Ott and Tim Sauer and James A. Yorke},\nseries = {Nonlinear Science},\npublisher = {Wiley},\nyear = 1994}\n\nIf these books look too scary, try the following "picture book" (again\nwritten for engineers and scientists), which offers a faily broad overview\nwhile passing over many of the more abstract/general concepts:\n\nauthor = {E. Atlee Jackson},\ntitle = {Perspectives of Nonlinear Dynamics},\nnote = {Two Volumes},\npublisher = {Cambridge University Press},\nyear = 1991}\n\nA very nice, recent, and easy book on Markov chains is\n\nauthor = {Olle Haggstrom},\ntitle = {Finite {M}arkov chains and algorithmic applications},\nseries = {London Mathematical Society student texts},\nvolume = 52,\npublisher = {Cambridge University Press},\nyear = 2002}\n\nThe highlight of this short book is an introduction to sampling from the\npast--- see above for why every scientist should know about this! An\neasier book which will provide neccessary background is\n\nauthor = {John G. Kemeny and J. Laurie Snell},\ntitle = {Finite Markov Chains},\npublisher = {Van Nostrand},\nyear = 1960}\n\n(there are many alternatives if you can\'t locate a copy of this book).\n\nMarkov chains, as I mentioned above, are a special type of\nmeasure-theoretic dynamical system. Underlying very Markov chain is a\ntopological dynamical system called a "shift space" (or better, a\nZ-shift). These are the bread-and-butter of (one-dimensional) "symbolic\ndynamics", which is the essential background to information theory\n(wherein the fundamental ergodic theorem was stated by Shannon in his\nfoundational 1948 paper) and many other important parts of applied\nmath/engineering.\n\nAs you may already know, such shifts often turn out to be hiding inside\n(and in some sense controlling the behaviour of) more complicated\ndynamical systems, including ones defined in terms of some system of\ndifferential equations. (Indeed, this was one the fundamental points\naddressed by Smale in the very first definition of a "dynamical system"---\nSmale noticed that shifts and systems of differential equations, which\ninitially may seem like very different things, are each instances of a\nmore abstract notion which he called a dynamical system.)\n\nFor an introduction to shift spaces, here is a textbook which is aimed in\npart at engineers and other non-mathematicians:\n\nauthor = {Douglas Lind and Brian Marcus},\ntitle = {Introduction to Symbolic Dynamics and Coding},\npublisher = {Cambridge University Press},\nyear = 1995}\n\n(If you don\'t like this book, I can give alternative references.)\n\nIf possible, it is important to understand the relationship between\nmeasure-theoretic dynamical systems and their underlying topological\ndynamical systems, as very sketchily described above. If you know a bit\nabout measure-theory, the following graduate textbook is particularly good\nfor learning about this:\n\nauthor = {Peter Walters},\ntitle = {Introduction to Ergodic Theory},\npublisher = {Springer},\nyear = 1981}\n\nAn inexpensive short text on ergodic theory and these two fundamental\nbroad classes of dynamical systems is:\n\nauthor = {M. Pollicott and M. Yuri},\ntitle = {Ergodic Theory and Dynamical Systems},\npublisher = {London Mathematical Society},\nseries = {Student Texts},\nnumber = 40,\nyear = 1998}\n\nI alluded above to the Birkhoff (pointwise), von Neumann (mean), and\nShannon-MacMillan-Breiman ergodic theorems; such ergodic theorems are\nlikely to apply to essentially any dynamical system you are ever likely to\nmeet in theory or applications, although justifying this claim in\nparticular instances can be quite hard. For these ergodic theorems, see\nany of the above and also, for an important discussion of the possibly\nvery slow ergodic convergence to an unknown invariant measure, see:\n\nauthor = {Karl Petersen},\ntitle = {Ergodic Theory},\npublisher = {University of Cambridge Press},\nseries = {Cambridge Series in Advanced Mathematics},\nvolume = 2,\nyear = 1983}\n\nAnother very important phenomenon every scientist should be aware of is\nthe fact that -number theory- is often hiding even in smooth dynamical\nsystems like the quasi-Hamiltonian dynamical systems which often arise in\nphysics/astronomy! This will probably seem quite surprising, but it is\nnot terribly hard to begin to see why this claim is much more plausible\nthan it might sound at first. For an easy introduction to some of the\nbasic ideas, try this article:\n\nauthor = {J. C. Lagarias},\ntitle = {Number Theory and Dynamical Systems},\nbooktitle = {The Unreasonable Effectiveness of Number Theory},\neditor = {Burr, Stefan A.},\nseries = {Proceedings of Symposia in Applied Mathematics},\nvolume = 46,\npublisher = {American Mathematical Society},\naddress = {Providence, Rhode Island},\nyear = 1991}\n\n(Several other articles in this book would also be good reading for an\nengineer, e.g. the article by McIlroy.)\n\nAnother possibly surprising connection: a good deal of mathematical\nlogic--- even, in some abstract sense, all mathematical discussion\nwhatever--- can be encompassed within the study of the more abstract types\nof dynamical systems such as tiling dynamical systems! (I can give some\nrelevant references if desired, but this is probably not of immediate\ninterest to engineers, even though it is hiding not very far from the\nsurface in topics like communication theory.)\n\nLast but not least, for a first look at complex dynamical systems, try the\nAMS short course:\n\neditor = {Brodil Branner and Robert L. Devaney},\ntitle = {Complex dynamical systems : the mathematics\nbehind the Mandelbrot and Julia sets},\naddress = {Providence, R.I.},\npublisher = {American Mathematical Society},\nyear = 1994}\n\n(I can give more references if this grabs anyone.) Again, AFAIK this is\nnot yet of direct interest in engineering or applied physics, although I\'d\nwelcome correction if anyone knows of "dirty applications" of Julia sets\nand all that! But it is a very beautiful topic nonetheless.\n\nOK, that was a bit disorganized but I hope it will help. Wrt your other\nquestions--- there are so many "dimensions" which are often discussed in\ndynamical system theory, even in the very special context you appear to be\nworking in, that I think you need to clarify which you mean.\n\n"T. Essel" (hiding somewhere in cyberspace)\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>On Sun, 27 Jun 2004, Sanjiv Ramachandran wrote:
> the author has criticized the dynamical systems approach to understand
> turbulence.
[snip]
> 3. This question might sound stupid. If people agree that some phenomena
> is described by differential equations and that dynamical systems is the
> study of differential equations, where is the controversy ?? Unless of
> course, we doubt the validity of the differential equation itself
> (Navier-Stokes, in this case).
I was not aware of any controversy here (probably because I know very
little about turbulence!), but you should be aware that any claim that
"dynamical systems is the study of differential equations" is completely
incorrect. If one -had- to give a one-sentence characterization, it would
probably be better to say that "most notions of a dynamical system involve
somewhere the notion of an appropriate action by some group or semigroup
or groupoid". Some of the most important can indeed be related to
differential equations; e.g. a system of autonomous ODEs may arise from
the phase space analysis of a system of reaction-diffusion equations (a
system of PDEs), or, as in your case, to the Navier-Stokes equations.
Here is some very sketchy background information, some advice, and a short
list of suggested references for further background reading, which I hope
you will find valuable or at least intriguing:
First, a bit of background: there is no universally agreed upon definition
of "dynamical system", although various formal definitions have been
offered by Steve Smale and many subsequent authors. There are however
several broad classes of definitions:
(a) There are various types of "discrete" dynamical system in terms of a
transformation on some space, with certain properties, which is iterated:
(i) in ergodic theory and symbolic dynamics one encounters
"measure-theoretic dynamical system" (X,M,\mu,T), where \mu is a T-invariant
Borel probability measure (if you know what that is),
(ii) a "topological dynamical system" is often defined as something like a
compact Hausdorff space X with a homeomorphism T:X--> X (or even just a
continuous map) which is iterated.
(b) "Discrete" dynamical systems can each be considered as an object in an
appropriate "category"; one can set this up in terms of categories induced
by assuming that T: X--> X is measure-preserving or continuous or smooth
(or several of these, usually with compatibility of structure assumptions)
on Z-spaces or N-spaces (Z the additive group of integers, N the additive
semigroup of natural numbers); these can be readily generalized to
Z^d-spaces. These are often called "Z-shifts" or "Z^d-shifts"
(topological dynamical systems) or "Markov chains" (the analogous
measure-theoretic dynamical systems). If you have heard of "cellular
automatons", btw, these are usually just Z^d-shifts with one extra level
of structure.
(c) One can generalize to actions by topological groups or even Lie
groups---most often abelian groups, in fact, almost always just R or R^d
as an additive group--- in particular, re my recent post on certain
"tiling dynamical systems", these are often defined as something like
R^d-spaces with a continuous action on some tiling space which has been
made into a compact Hausdorff space (there are several equivalent and
natural ways of doing this, just as for discrete group actions); now going
in the opposite direction (toward increasingly special classes of
dynamical systems), we have:
(i) "smooth dynamical systems" often involve something like an
"equicontinuous action by a family of diffeomorphisms"; in particular,
many authors consider something like an autonomous system of ODEs (c.f.
"phase space methods") to comprise a "dynamical system",
(ii) "complex analytic dynamical systems", which usually involve iterating
a holomorphic or meromorphic map on the Riemann sphere or something like
that (c.f. "Julia set", "Mandlebrot set", if you've heard of those).
Hmm... oh dear, I don't think I said that very well.
Let me try again to briefly summarize, in slightly different words, some
of the most imporant "rough classifications" of the immense variety of
dynamical system one could study:
The definition of a given class of dynamical systems usually involves some
kind of "group action", although this fact may be hidden from view. The
group G which acts can be a discrete group, or a topological group, or
even a Lie group. In the case of topological or Lie groups, G can have
various dimensions. It can be abelian or nonabelian. The action effected
by G can be measure-preserving, or continuous, or smooth, or analytic.
The space X on which G acts can be a measure space, or a topological
space, or a smooth or even analytic manifold (the Riemann sphere is a
familiar and very important example of an analytic manifold). One cannot
choose these attributes with complete independence, of course: it only
makes sense to say that T is a continuous map if X is a topological space,
etc.
As an important special case, one can study the group G = <T> generated by
iterating a single invertible map T on some "space" X, where X can be a
measure space, or a topological space (often a compact Hausdorff space, or
even a compact metric space), or even a topological or smooth manifold.
The map T can be measure-preserving, or continuous, or smooth, or even
analytic (c.f. "Julia sets" and the "Mandlebrot set" and all that). In
defining this type of dynamical system, many authors ignore the presence
of a group action by Z, the integers. Actually, you are just as likely to
encounter -noninvertible- maps T in this context; iterating T then defines
a semigroup <T>, i.e. an "action" by N, the natural numbers. But at first
it is often best to stick to groups and group actions.
A basic idea here is that measure spaces are more general than topological
spaces, which are more general than topological manifolds, which are more
general than smooth manifolds, which are more general than analytic
manifolds. Similarly, measureable maps--- are in a sense--- more general
than continuous maps, which are more general than smooth maps, which are
more general than analytic maps. Going the other way, invertible maps are
less general than possibly non-invertible maps, and groups are less
general than semigroups.
Another basic idea here is that when some pair (X,T) possesses more than
one level of structure--- in particular, both topological and
measure-theoretic structure--- one usually demands that these structures
be in some sense "compatible" with one another. In particular, there will
be many invariant measures \mu associated with a given continuous map T on
a compact Hausdorff space X, and thus many measure-theoretic dynamical
systems (X,T,\mu) sharing the same underlying topological dynamical system
(X,T). In this situation, one can define the topological entropy h(X,T)
of the common underlying topological dynamical system (X,T), and also
measure-theoretic entropies H(X,T,\mu) of the various measure-theoretic
dynamical systems. Then, an important theorem says that maximizing
H(X,T,\mu) over the various \mu always gives h(X,T)! (This is not at all
obvious from the definitions.)
(The measures \mu above are assumed to be Borel measures, which amounts to
saying we only consider measures which are compatible with the topological
structure, in an appropriate sense. Also, I am ignoring here some
technical issues about \sigma-algebras here which can obscure the point I
am making, and some additional assumptions about X which are far less
restrictive than they might at first appear.)
This is all rather informal, but hopefully gives the idea, even if you
have never seen some of the technical terms I mentioned. BTW, if you
learn even a tiny bit about "category theory", it will become -much-
easier to understand how all these concepts are organized.
Next, some advice: I see you are at PSU. Well, if you don't often visit
the math department, you should, because you are fortunate enough to share
a campus with perhaps the foremost group of dynamical systems people in
the U.S., or even the world! In particular, I believe that A. Katok is
still at PSU; he is hugely knowledgeable, and a leading figure in the
mathworld side of dynamical systems. And another fairly accessible math
department (U Maryland at College Park) boasts another leading research
group working on dynamical systems (or -two- groups, depending upon how
you count).
Perhaps I should warn you that many "rigorists" are wont to complain about
the activity of a large school of people writing papers on "applied
chaos", who, the rigorists feel, ignore enormous progress made by
mathematicians toward correcting very serious errors in older ideas on
things like various kinds of "fractal dimensions" (often related to
various "entropies"), especially in the context of simulations of
complicated systems in applied physics, etc. OTH, the "rough-and-ready"
crowd may complain that the rigorists underestimate the difficulty
nonmathematicians experience in learning how to even -state- the latest
theorems, much less apply them. Representatives of both camps can be
found at UM, and probably at PSU, so you will probably encounter further
differences of approach to the highly complicated and extremely broad
topic of "chaos" etc. in "dynamical systems".
But don't let the sometimes heated debate between these two camps deter
you from trying to take the best of what each has to offer: keep an open
mind and I claim you will find that both "camps" make some valid points;
their lamentable failure to talk to each other as much as they could and
should, is IMHO mostly based on mutual misunderstanding, misappreciation,
and underestimation.
As (I guess) an engineering/applied physics student at PSU, you have a
heaven-sent opportunity to try to become an "ambassador" between the
"ruffians" and the "rigorists". (You might not have known it, but you're
probably a ruffian by institutional affiliation and perhaps by
background.)
This would not be easy--- more because of the difficulty of learning
enough to appreciate so many and higly varied points of view, than because
of conflicting ideologies/personalities--- but if you succeed in learning
enough to explain in useful terms to the ruffians what the rigorists have
accomplished, or if you can help the rigorists cook up
definitions/theories more useful to the ruffians, this could eventually
make you a person Highly Valued by both camps, so looking into the stuff I
am talking about could be a smart (and highly lucrative) career move--- at
least, if you are very capable and very ambitious.
But more immediately, there are very real scientific dividends which
should -quickly- accrue from learning more math. One reason for this is
that, as a rule, "ruffians" who trying to simulate the behavior of some
dynamical system--- perhaps not as end in itself, but only as part of
doing something else, like "simulated annealing"--- are very likely
performing their simulations in fundamentally incorrect ways, and thus
obtaining wildly misleading results. This makes rigorists froth at the
mouth. Often, they say, if only the ruffians knew a few simple facts,
they would be in a much better position to appreciate the fact that
-correct- simulation methods are
(a) readily available,
(b) easy to learn,
(c) easy to use in practice.
A specific example might help here:
Many engineers, computer scientists, or other applied scientists often
need to simulate a "Markov chain". Here, a fundamental fact, as even your
average ruffian may appreciate :-/ is that many such chains are "ergodic".
There are some nice criteria for knowing when a Markov chain is ergodic.
If it is, one can apply the immensely powerful ergodic theorems. But
there is a fundamental problem here (probably known only to rigorists):
while you know the convergence occurs, it might occur very slowly. (I am
oversimplifying, but in some very common situations, including simulated
annealing, this -is- a serious concern).
Fortunately, a very important recent advance by the rigorists is the
discovery of an amazing idea which goes by the name "sampling from the
past". This amounts to an easily implemented algorithm which completely
circumvents the problem, allowing one to simulate the unknown invariant
measure rather than the process of convergence to that measure.
In short, if you didn't use sampling from the past (or one of the even
more recent and even better "perfect sampling" algorithms) for your Markov
chain simulation (this could happen only if you've never heard of these
algorithms, since the simplest ones are no harder to implement than a
naive simulation), you would actually not be modeling what you think you
are at all--- and in general you -cannot- tell just by looking at the
output how accurate your simulation is! But if you use "sampling from the
past" you never need to worry about just what your simulation is actually
simulating.
To admit the obvious, to make a point I am here somewhat overstating
things. Every advance in understanding raises new questions which lead to
the recognition of new shortcomings. Hence the more recent "perfect
sampling" algorithms. But this doesn't invalidate my point: the
rigorists have here moved far beyond most ruffians, yet this great leap
forward is by no means inaccessible to nonmathematicians, and a real
dialog between ruffians and rigorists here could have a highly beneficial
impact on the further development of this field. I can think of analogous
situations where rigorists have gone off in directions which are of little
interest to ruffians, but only because they didn't know enough about
relevant applications to see the superior potential of another theoretical
approach. BTW, in my experience, rigorists are usually -very- interested
in possible applications of their theories, and they are very capable of
being influenced to change the direction of their theoretical
explorations, should an applied scientist demonstrate serious interest in
their work. So, here lies a significant opportunity for both theoretical
and applied mathscis, hopefully not to be missed.
Finally, here are some references for further reading:
To get a quick impression of the huge scope of modern definitions of
"dynamical system", see
author = {Anatole Katok and Boris Hasselblatt},
title = {Introduction to the Modern Theory of Dynamical Systems},
publisher = {Cambridge University Press},
year = 1995}
(Last time I checked, Katok was in the math department at PSU, so you
might want to find out if he is still there.)
A much easier introduction (aimed at engineers and scientists) is:
author = {Robert C. Hilborn},
title = {Chaos and Nonlinear Dynamics: An Introduction for Scientists
and Engineers},
publisher = {Oxford University Press},
year = 1994}
Even easier (but much sketchier and far less comprehensive):
title = {Coping with Chaos: Analysis of Chaotic Data and Exploitation
of Chaotic Systems},
editor = {Edward Ott and Tim Sauer and James A. Yorke},
series = {Nonlinear Science},
publisher = {Wiley},
year = 1994}
If these books look too scary, try the following "picture book" (again
written for engineers and scientists), which offers a faily broad overview
while passing over many of the more abstract/general concepts:
author = {E. Atlee Jackson},
title = {Perspectives of Nonlinear Dynamics},
note = {Two Volumes},
publisher = {Cambridge University Press},
year = 1991}
A very nice, recent, and easy book on Markov chains is
author = {Olle Haggstrom},
title = {Finite {M}arkov chains and algorithmic applications},
series = {London Mathematical Society student texts},
volume = 52,
publisher = {Cambridge University Press},
year = 2002}
The highlight of this short book is an introduction to sampling from the
past--- see above for why every scientist should know about this! An
easier book which will provide neccessary background is
author = {John G. Kemeny and J. Laurie Snell},
title = {Finite Markov Chains},
publisher = {Van Nostrand},
year = 1960}
(there are many alternatives if you can't locate a copy of this book).
Markov chains, as I mentioned above, are a special type of
measure-theoretic dynamical system. Underlying very Markov chain is a
topological dynamical system called a "shift space" (or better, a
Z-shift). These are the bread-and-butter of (one-dimensional) "symbolic
dynamics", which is the essential background to information theory
(wherein the fundamental ergodic theorem was stated by Shannon in his
foundational 1948 paper) and many other important parts of applied
math/engineering.
As you may already know, such shifts often turn out to be hiding inside
(and in some sense controlling the behaviour of) more complicated
dynamical systems, including ones defined in terms of some system of
differential equations. (Indeed, this was one the fundamental points
addressed by Smale in the very first definition of a "dynamical system"---
Smale noticed that shifts and systems of differential equations, which
initially may seem like very different things, are each instances of a
more abstract notion which he called a dynamical system.)
For an introduction to shift spaces, here is a textbook which is aimed in
part at engineers and other non-mathematicians:
author = {Douglas Lind and Brian Marcus},
title = {Introduction to Symbolic Dynamics and Coding},
publisher = {Cambridge University Press},
year = 1995}
(If you don't like this book, I can give alternative references.)
If possible, it is important to understand the relationship between
measure-theoretic dynamical systems and their underlying topological
dynamical systems, as very sketchily described above. If you know a bit
about measure-theory, the following graduate textbook is particularly good
for learning about this:
author = {Peter Walters},
title = {Introduction to Ergodic Theory},
publisher = {Springer},
year = 1981}
An inexpensive short text on ergodic theory and these two fundamental
broad classes of dynamical systems is:
author = {M. Pollicott and M. Yuri},
title = {Ergodic Theory and Dynamical Systems},
publisher = {London Mathematical Society},
series = {Student Texts},
number = 40,
year = 1998}
I alluded above to the Birkhoff (pointwise), von Neumann (mean), and
Shannon-MacMillan-Breiman ergodic theorems; such ergodic theorems are
likely to apply to essentially any dynamical system you are ever likely to
meet in theory or applications, although justifying this claim in
particular instances can be quite hard. For these ergodic theorems, see
any of the above and also, for an important discussion of the possibly
very slow ergodic convergence to an unknown invariant measure, see:
author = {Karl Petersen},
title = {Ergodic Theory},
publisher = {University of Cambridge Press},
series = {Cambridge Series in Advanced Mathematics},
volume = 2,
year = 1983}
Another very important phenomenon every scientist should be aware of is
the fact that -number theory- is often hiding even in smooth dynamical
systems like the quasi-Hamiltonian dynamical systems which often arise in
physics/astronomy! This will probably seem quite surprising, but it is
not terribly hard to begin to see why this claim is much more plausible
than it might sound at first. For an easy introduction to some of the
basic ideas, try this article:
author = {J. C. Lagarias},
title = {Number Theory and Dynamical Systems},
booktitle = {The Unreasonable Effectiveness of Number Theory},
editor = {Burr, Stefan A.},
series = {Proceedings of Symposia in Applied Mathematics},
volume = 46,
publisher = {American Mathematical Society},
address = {Providence, Rhode Island},
year = 1991}
(Several other articles in this book would also be good reading for an
engineer, e.g. the article by McIlroy.)
Another possibly surprising connection: a good deal of mathematical
logic--- even, in some abstract sense, all mathematical discussion
whatever--- can be encompassed within the study of the more abstract types
of dynamical systems such as tiling dynamical systems! (I can give some
relevant references if desired, but this is probably not of immediate
interest to engineers, even though it is hiding not very far from the
surface in topics like communication theory.)
Last but not least, for a first look at complex dynamical systems, try the
AMS short course:
editor = {Brodil Branner and Robert L. Devaney},
title = {Complex dynamical systems : the mathematics
behind the Mandelbrot and Julia sets},
address = {Providence, R.I.},
publisher = {American Mathematical Society},
year = 1994}
(I can give more references if this grabs anyone.) Again, AFAIK this is
not yet of direct interest in engineering or applied physics, although I'd
welcome correction if anyone knows of "dirty applications" of Julia sets
and all that! But it is a very beautiful topic nonetheless.
OK, that was a bit disorganized but I hope it will help. Wrt your other
questions--- there are so many "dimensions" which are often discussed in
dynamical system theory, even in the very special context you appear to be
working in, that I think you need to clarify which you mean.
"T. Essel" (hiding somewhere in cyberspace)
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