Geometry, Topology and Dynamics Seminar
Lori Alvin
University of Denver
Unimodal Maps and Inverse Limit Spaces
Abstract: In this talk we investigate some of the continua (compact, connected
metric spaces) that occur as inverse limit spaces of unimodal bonding maps. The
inverse limit space of a single map $f:I\to I$ is
$$\lim_{\longleftarrow} {\bf f} = \{ x = (x_0, x_1, x_2, \ldots ) : x_n \in I \textrm{ and } f(x_{n+1}) = x_n \textrm{ for all } n\in \mathbb{N} \}$$
and has metric
$$d(x,y) = \sum_{i=0}^\infty \frac{|x_i-y_i|}{2^i}.$$
We begin by exploring the continua that arise as inverse limit spaces from a single
logistic map of the form $g_a(x) = ax(1-x)$, where $a\in [0,4]$. We are particularly
interested in drawing the inverse limits that arise from the family of logistic maps
seen within the classical period doubling bifurcation diagram. We then use the
period doubling bifurcation to gain an intuition for the inverse limit space of the
logistic map with parameter $a\approx 3.569945668$, also called the Feigenbaum
limit. This map, sometimes referred to as the $2^\infty$ map, is the unique
logistic map with period points of period $2^n$ for all $n\in \mathbb{N}$ and no
other periodic points; the action of $g|_{\omega(c,g)}$ is topologically conjugate
to a special type of map called an adding machine or odometer. We conclude by
discussing inverse limit spaces whose single bonding maps have embedded adding
machines.
Monday March 30, 2015 at 3:00 PM in SEO 636