GEOMETRY OF CHAOS
"Statistical Physics Out of Equilibrium",
Inst. Henri Poincaré, Paris,
10 lectures October 22-26 2007 - Lecture notes
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A brief history of motion in time
An overview of the course. Skim quickly through -
parts will make sense only in the rear view mirror, do not worry
that you do not understand them yet.
Brief history of chaos
Classical mechanics has not stood still since Newton. The formalism that
we use today was developed by Euler and Lagrange. By the end of the 1800's
the three problems that would lead to the notion of chaotic dynamics were
already known: the three-body problem, the ergodic hypothesis, and nonlinear
A recapitulation of basic notions of dynamics. The
reader familiar with the dynamics on the level of an introductory
graduate nonlinear dynamics course can safely skip this material,
chapter 14: Transporting densities. J
Discrete time dynamics arises by considering sections of a
continuous flow. There are also many settings in which dynamics is
discrete, and naturally
described by repeated applications of a map.
Review of basic concepts of local dynamics: local linear
stability for flows and maps.
[skip section 4.4.1]
Cycle stabilities are invariant
Cycle eigenvalues are flow invariants.
[skip sections 5.3, 5.4 and 5.5]
Cycling for (discrete) fundamentalists
Partitioning state space
Qualitative dynamics of simple stretching and mixing flows; Smale
horseshoes and symbolic dynamics. The topological dynamics is incoded
by means of transition matrices/Markov graphs.
[skip sections 10.1.1 and 10.5]
12. Fixed points: where are they?
13. Chaos for accountants
Topological zeta function;
determinants of graphs; fear
not infinite partitions
You learn here how to count and describe itineraries.
While computing the topological
entropy from transition matrices/Markov graphs, we encounter our first
[skip sections 13.4.1 and 13.5]
Learning how to measure
A first attempt to move the
whole phase space around - natural measure and fancy operators.
[skip section 14.5]
Not your average theory
On the necessity of studying
the averages of observables in chaotic dynamics. Formulas
for averages are cast in a multiplicative form that
motivates the introduction of evolution operators.
[skip sections 15.3 and 15.4]
All that was left was a trace
If there is one idea that one should learn about chaotic dynamics, it happens
in this chapter: the (global) spectrum of the evolution is dual to the
(local) spectrum of periodic orbits. The duality is made precise by
means of trace formulas.
[skip sections 14.2.1, 14.3 and 14.4]
We derive the spectral determinants, dynamical zeta functions.
[skip sections 15.2, 15.3.1, 15.3.2, 15.5 and 15.6]
Recycle: it's the law
Spectral eigenvalues and dynamical
averages are computed by expanding spectral determinants into
cycle expansions, expansions ordered by the topological lengths
of periodic orbits.
[skip sections 18.1.3, 18.1.4, 18.2.2, 18.2.4, 18.4 and 18.5]
Cycling for (smooth) fundamentalists
Get used to it
In the preceding chapters we have moved at rather brisk pace and derived
a gaggle of formulas. Here we slow down in order to develop some fingertip
feeling for the objects derived so far. Just to make sure that the key
message - the ``trace formulas'' and their ilk - have sunk in, we rederive
them in a rather different, more intuitive way, and extol their virtues.
This part is bedtime reading. A few special determinants are worked out
[skip sections 20.1.1, 20.1.2, 20.3 and 20.4]
svn: $Author: predrag $ - $Date: 2007-10-19 07:29:33 -0400 (Fri, 19 Oct 2007) $