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Tony and I worked over the summer with Zoe Cooperband, Elisabeth Esquivel, Elise Mihanovich, Jordan Rowley, Blaine Quackenbush, and Matt Rowley.

A Lorenz map is a piecewise monotone function of the unit interval to itself, so $T:[0,1] \to [0,1]$. Choose a starting point $x_0$ in the interval, and then define a sequence by repeatedly applying the function:
\[x_n = T(x_{n-1}), \qquad n=1,2,3,\dots\]
So $x_1 = T(x_0)$ and $x_2 = T(x_1) = T(T(x_0))$, etc. This produces a discrete-time dynamical system ($n$ represents time). We studied a family of Lorenz maps $T_p$ which are defined by two monotone increasing functions $f_0$ and $f_1$ via 
\[T_p(x) = \begin{cases} f _0(x),  & x < p \\ f _1(x), &  x > p \end{cases}\] 
(see the figure). We studied the topological entropy of these dynamical systems, which is a measure of complexity of the mapping, and managed to prove that entropy changes continuously as the point of discontinuity is varied.

Students presented their results at the JMM Poster Session in January. These results were written up and submitted to a scientific journal for publication. A preprint is available here: https://arxiv.org/abs/1803.04511.

Thu, 29 Mar 2018 17:00 GMT