Random interval diffeomorphisms

Figure 1.  The first frame depicts the graphs of $g_1,g_2$, the diffeomorphisms on $\mathbb{I}$ that are conjugate to the maps $y \mapsto y \pm 1$ that generate the symmetric random walk. The second frame shows a time series of the iterated function system generated by $g_1,g_2$, both picked with probability $1/2$.

Figure 3.  The first frame shows a numerically computed histogram for a time series of the iterated function systems generated by the same diffeomorphisms $f_1^{-1}$ and $f_2^{-1}$ used in Figure 2. The second frame indicates asymptotic convergence of orbits within fibers: it depicts time series for three different initial conditions in $\mathbb{I}$ with the same $\omega$.

Figure 2.  With $r = 1/2$, the diffeomorphisms $f_1 (x) = x - r x (1-x)$ and $f_2 (x) = x + r x (1-x)$ (picked with probabilities $1/2$) give negative Lyapunov exponents at the end points $0,1$. Depicted, in the first frame, are the graphs of the inverse diffeomorphisms $f_1^{-1} (x) = \frac{1 - r - \sqrt{(1-r)^2 +4 r x}}{-2r}$ and $f_2^{-1} (x) = \frac{1 + r - \sqrt{(1+r)^2 -4 r x}}{2r}$. The inverse maps give positive Lyapunov exponents at the end points. The second frame shows a time series for the iterated function system generated by $f_1^{-1}$ and $f_2^{-1}$.

Figure 4.  The first frame depicts the graphs of $x \mapsto f_i(x) = g_i(x) ( 1 - p(x))$, $i=1,2$, with $g_i (x)$ as in (3), (4) and $p(x) = \frac{3}{10} x (1-x)$. The corresponding step skew product system has a zero Lyapunov exponent along $\Sigma_2^+ \times \{0\}$ and a positive Lyapunov exponent along $\Sigma_2^+ \times \{1\}$. The second frame shows a time series for the iterated function system generated by these diffeomorphisms.

Figure 5.  A sequence of stopping times is defined to label subsequent iterates where $y_n$ leaves $(-\infty,K]$ or $(K,\infty)$.