# 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.)