2015 CMS Summer Meeting
University of Prince Edward Island, June 5 - 8, 2015
Let $f:X\to X$ be a map. We want to consider a process, which is not a map, and represents situation when $f$ on each step uses not only current information but also some information from the past. We define for current state $x_n$ and $0<\alpha<1$: $$x_{n+1}=f(\alpha x_n+(1-\alpha)x_{n-1})\, .$$ We are interested in something we could call an "invariant measure" of the process. We consider ergodic averages $$A_f(x_0,x_{-1})=\lim_{n\to\infty}\frac 1n\sum _{i=0}^{n-1}f(x_i).$$ They are related to ergodic averages of the map $G:X\times X\to X\times X$ defined by $$G(x,y)=(y,f(\alpha y+(1-\alpha)x)\, .$$ We considered the example where $f:[0,1]\to[0,1]$ is the tent map. Computer experiments suggest that $G$ behaves in very different manners depending on $\alpha$. We conjecture:
For $0<\alpha<1/2$ map $G$ preserves absolutely continuous invariant measure.
For $\alpha=1/2$ every point of upper half of the square ($y+x\ge 1$) has period 3 (except the fixed point $(2/3,2/3)$). Every other point (except $(0,0)$) eventually enters the upper triangle.
For $1/2<\alpha<3/4$ point $2/3,2/3$ is a global attractor for map $G$.
For $\alpha=3/4$ every point of the interval $x+y=4/3$ has period 2 (except the fixed point $(2/3,2/3)$). Every other point (except $(0,0)$) is attracted to this interval.
For $3/4<\alpha<1$ map $G$ preserves an SRB measure which is not absolutely continuous (supported on an uncountable union of straight intervals).
We also consider an application to one dimensional random maps
with neutral fixed points.
In this talk we give a brief background on general results in the area of convergence theorems for random iteration of functions, with particular attention to the case of families of random contractions. After this we concentrate on some results on a simple model of time inhomogeneous random iteration. As is the case for Markov chains, allowing the dynamics to vary with time presents new complications.
Collaborators, S. Das, Y. Saiki, E. Sander