Source Topol. Methods Nonlinear Anal. Zentralblatt MATH identifier Glasner, Eli. Classifying dynamical systems by their recurrence properties.

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Name of resource. Problem URL. Describe the connection issue. SearchWorks Catalog Stanford Libraries. Recurrence in topological dynamics : Furstenberg families and Ellis actions. Responsibility Ethan Akin. Imprint New York : Plenum Press, c Suppose that for every finite set of integers , the tuple has the vdW property. Then also has the vdW property. This will be a reprise of the proof of Proposition 1 from the previous lecture.

Given any finite number of pairs with , we see from hypothesis that there exists depending on these pairs such that. Now, for every and , consider the expression defined by. We let be arbitrary, and set. By the minimality of p, we can find such that. We thus have. If one then sets. Suppose we wish to show that the tuple has the vdW property where we use r to denote the independent variable.

Applying Lemma 2 with , we reduce to showing that the tuples have the vdW property for all finite collections of integers. But observe that all the polynomials in these tuples are linear polynomials that vanish at the origin. By the ordinary van der Waerden theorem, these tuples all have the vdW property, and so has the vdW property also. A similar argument shows that the tuple has the vdW property whenever are linear polynomials that vanish at the origin.

Applying Lemma 1, we see that obeys the vdW property when is also linear and vanishing at the origin. Now, let us consider a tuple where is also a linear polynomial that vanishes at the origin.

## Recurrence and Topology

The vdW property for this tuple follows from the previously established vdW properties by first applying Lemma 1 to reduce to the case , and then applying Lemma 2 with. Continuing in this fashion, we see that a tuple will also obey the vdW property for any linear that vanish at the origin, for any k and l. Now the vdW property for the tuple follows from the previously established cases and Lemma 2 with.

Remark 2. Exercise 3. Define the top order monomial of a non-zero polynomial with to be. Define the top order monomials of a tuple to be the set of top order monomials of the , not counting multiplicity; for instance, the top order monomials of are. Define the weight vector of a tuple relative to one of its members to be the infinite vector , where each denotes the number of monomials of degree d in the top order monomials of.

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Thus for instance, the tuple has weight vector with respect to 0, but weight vector with respect to say. Let us say that one weight vector is larger than another if there exists such that and for all. Exercise 4. Exercise 5. Let be vector-valued polynomials thus each of the d components of each of the is a polynomial which all vanish at the origin. Show that if is finitely coloured, then one of the colour classes contains a pattern of the form for some and. Exercise 6. Show that for any polynomial sequence taking values in a torus, there exists integers such that converges to.

On the other hand, show that this claim can fail with exponential sequences such as for certain values of. Thus we see that polynomials have better recurrence properties than exponentials. Thus for instance a hypergraph of order 1 is a vertex colouring, a hypergraph of order 2 is an edge-coloured complete graph, and so forth.

## Recurrence and Topology

We say that a hypergraph G is monochromatic if the edge colouring function E is constant. If W is a subset of V, we refer to the hypergraph as a subhypergraph of G. Theorem 4 Hypergraph Ramsey theorem. Then G contains arbitrarily large finite monochromatic subhypergraphs. Remark 3. There is a stronger statement known, namely that G contains an infinitely large monochromatic subhypergraph, but we will not prove this statement, known as the infinite hypergraph Ramsey theorem.

Exercise 7.

Show that Theorem 4 implies a finitary analogue: given any finite K and positive integers k, m, there exists N such that every K-coloured hyeprgraph of order k on contains a monochromatic subhypergraph on m vertices. Hint : as in Exercise 3 from the previous lecture [note — exercises have been renumbered recently], one should use a compactness and contradiction argument as in my article on hard and soft analysis.

It is not immediately obvious, but Theorem 4 is a statement about a topological dynamical system, albeit one in which the underlying group is not the integers , but rather the symmetric group , defined as the group of bijections from V to itself which are the identity outside of a finite set.

More precisely, we have. Theorem 5. Hypergraph Ramsey theorem, topological dynamics version Let V be a countably infinite set, and let W be a finite subset of V, thus is a subgroup of. Let be a -topological dynamical system, thus is compact metrisable and is an action of on X via homeomorphisms. Let be an open cover of X, such that each is -invariant.

Then there exists an element of this cover such that for every finite set there exists a group element such that i. This claim should be compared with Theorem 2 of this lecture, or Theorem 1 of the previous lecture.

## Recurrence analysis of ant activity patterns

Exercise 8. Show that Theorem 4 and Theorem 5 are equivalent. Hint : At some point, you will need the use the fact that the quotient space is isomorphic to. Here, we will not work on the compactified integers , but rather on the compactified permutations. We will view here as a discrete group; one could also give this group the topology inherited from the product topology on , leading to a slightly coarser and thus less powerful compactification, though one which is still sufficient for the arguments here.

This is a semigroup with the usual multiplication law. Let us say that is minimal if is a minimal left-ideal of. One can show by repeating Exercise 10 from Lecture 3 that every left ideal contains at least one minimal element; in particular, minimal elements exist. Note that if W is a k-element subset of V, then there is an image map which maps a permutation to its inverse image of W.

We can compactify this to a map. Caution: is not the same thing as , for instance the latter is not even compact.

https://thambcyticlupur.tk Theorem 6.