March 22, 2007
10a-1
More on the Schwarzian Derivative
We have seen some simple implications of the property of negative Schwarzian
derivative.
One can give a simple geometric description of the Schwarzian derivative
which, in turn, yields new information
February 14, 2005
10-1
The Schwarzian Derivative
There is a very useful quantity Sf dened for a C 3 one-dimensional map f ,
called the Schwarzian derivative of f .
Here is the denition.
Set
f (x) 3
Sf (x) =
f (x)
2
f (x)
f (x)
2
.
(1)
Here we use f (x), f
February 7, 2005
9-1
An approximation theorem for C r (N ) where N
is the circle or a compact real interval
Let f : N N be a C k self-map of N where N is a compact interval I or
the unit circle S 1 .
A point c N is called a turning point of f if there is
February 7, 2005
8-1
Structural Stability
We consider here self-maps f : N N where N is either a compact interval
[a, b] in the real line or the unit circle S 1 in the plane.
We rst consider various topologies for certain maps. Let I be an open
interval I
March 25, 2014
IPM-1
Invariant Probability Measures
1
Some Concepts in Measure Theory
Given a set X and a subset A X , we write Ac = X \ A for the complement
of A.
Let X be a set. A algebra of subsets of X is a collection B of subsets
of X such that
1. If
March 20, 2014
1
PF-1
Non-negative and Stochastic Matrices
An n n matrix A = (Aij ) is non-negative if each entry Aij is real and nonnegative. The matrix A is positive if each entry is a positive real number.
Similarly, we dene non-negative and positive v
February 7, 2012
7-1
Subshifts of nite type
We have seen that the full-shift automorphisms give simple models for the
dynamics of several maps. There are other symbolic systems which are simple
to dene and provide a much larger class of potential models f
February 7, 2012
6-1
Topological Conjugacy
For convenience, let us dene a dynamical endomorphism to be a piecewise
continuous self-map f : X X of a complete separable metric space. Sometimes we use the term endomorphism for short, although we do not wish
February 7, 2012
5a-1
Iterated Function Systems
Let (X, d) be a compact metric space, and let int(E ) denote the interior of
a subset of X .
Let f1 , f2 , f3 , . . . , fs be a nite sequence of contraction self-maps of X .
That is, each fi ls Lipschitz wit
January 20, 2005
5-1
Cantor sets and bounded orbits in certain tent
maps
For a subset E in a metric space, let E denote the boundary of E . This is
dened to by
E = Closure(E ) \ Interior(E ).
A subset E in a metric space X is called totally disconnected i
July 20, 2006
4-1
Some properties of the maps fr (x) = r x mod(1)
where r > 1 is a positive integer
As our next example, consider the map f (x) = 2 x mod(1) as a self-map of
the interval [0, 1). That is,
f (x) = fractional part of 2x
or 2x [2x]
where [y ]
January 12, 2011
3-1
The Baire Category Theorem
We wish to develop some measurement of largeness in metric spaces.
Let X be a metric space. A subset E X is dense if for every x X ,
and > 0, there is a point y E B (x). Here B (x) denotes the open ball
of r
January 17, 2012
2bs-1
Fixed Point Theorems
Let X be a metric space and let T : X X be a mapping. A xed point of
T is a point x X such that T (x) = x.
A self-map T of a metric space X is called a contraction (or contraction
map or mapping) if there is a c
January 15, 2014
2a-1
Some Concepts in Topology
For two sets A, B , we denote the product by
A B = cfw_(a, b) : a A, b B
Recall that a metric space is a pair (X, d) in which X is a non-empty set
and d : X X R is a function from X X to the set of real num
January 12, 2011
2-1
One dimensional discrete systems
An interval I in the real line (open, closed, half-open, etc.) is a subset I
such that if x, y I , x < y , and x < z < y , then z I .
Let f : I I be a continuous map. We can frequently gain insight int