Quantifier soup

Anyone who has tried to teach Calculus I students the \varepsilon-\delta definition of the limit knows that students have a difficult time working out definitions with multiple quantifiers. In fact it is something that we must come back to again and again with our mathematics majors.

While doing a little research this week I came across the definition of something called the specification property for a discrete dynamical system—it was introduced in the early 1970’s by Rufus Bowen. The logical complexity of the definition made me laugh out loud the first time I read it. I had to reread it about 5 times to figure out what it was saying.

The dynamical system f:X\to X is said to satisfy the specification property if for any \varepsilon>0 there exists an integer M>0 such that for any k\ge 2, any k points x_1,\ldots,x_k\in X, any integers a_1\le b_1\le a_2\le b_2\le\cdots\le a_k\le b_k with a_i-b_{i-1}\le M for 2\le i\le k and for any integer p with p\ge M+b_k-a_1, there exists a point x\in X with f^p(x)=x such that d(f^n(x),f^n(x_i))\le \varepsilon for a_i\le n\le b_i, 1\le i\le k.

(In case you’re wondering, roughly speaking it says that for a dynamical system with this property, given a finite number of finite orbit segments, there is a periodic orbit that follows one orbit segment very closely, then a little while later follows the next one, then a little while later follows the next one, etc.)