# Asymptotic cycles

Post-publication activity

Curator: Sol Schwartzman

Suppose $$X$$ is a compact polyhedron and $$F$$ is a uniformly continuous function from $$[0,\infty)$$ into $$X\ .$$ If for each $$t_0 ,$$ we choose a curve going from $$F(t_0)$$ to $$F(0)$$ and we tack this on to $$F$$ restricted to $$[0,t_0]$$ in an obvious way we get a closed curve which we may denote by $$C_{t_0}\ .$$ Let $$\bar{C_{t_0}}$$ be the element of $$H_{1}(X,\mathbb{R})$$ determined by this closed curve.

So long as we restrict our choice of the curves going from $$F(t_0)$$ to $$F(0)$$ to curves that are uniformly bounded in length with respect to some metric on $$X \ ,$$ if $$\lim_{T\to\infty}\frac{1}{T}\bar{C}_T$$ exists for one choice of these curves it exists and has the same value for any other choice. If this limit exists we call it the asymptotic cycle determined by $$F$$ and denote it by $$A_F\ .$$

## Asymptotic cycles and winding numbers

If we are given a flow on $$X$$ we say $$p \in X$$ is quasi-regular provided for any continuous real valued function $$f$$ on $$X \ ,$$ $\lim_{T\to\infty}\frac{1}{T}\int_{0}^{T} f(t\cdot p)dt$ exists.

If for any quasi-regular point $$p$$ we let $$F_p$$ be the function from $$[0,\infty)$$ into $$X$$ sending $$t$$ into $$t\cdot p$$ then $$F_p$$ will determine an asymptotic cycle, which we will denote by $$A_p\ .$$

This is connected with the fact that if $$\phi$$ is any continuous complex valued function of absolute value one on $$X$$ and $$\phi(t\cdot p) = \exp 2 \pi i l(t)$$ where $$l(t)$$ is a continuous real valued function, then $\lim_{T\to \infty}[l(T)-l(0)]/T$ exists when $$p$$ is quasi-regular.

Note that such a continuous $$l(t)$$ always exists and that it is unique up to an additive integer. Thus $$[l(T)-l(0)]$$ does not depend on the particular $$l$$ we chose. We call $\lim_{T\to \infty} [l(T)-l(0)]/T$ the winding number determined by $$\phi$$ and the quasi-regular point $$p\ .$$

The set of quasi-regular points has measure one with respect to any invariant probability measure $$\mu$$ and is an invariant set. Moreover there is an invariant probability measure $$\mu_p$$ associated with our quasi-regular point $$p$$ such that for any continuous $$f\ ,$$ $\lim_{T \to \infty}\frac{1}{T}\int_{0}^{T}f(t\cdot p)dt = \int_{X} f(x)d\mu_p(x).$

To get the connection of this with 1-dimensional homology we need to consider the Bruschlinski group $$B(X)\ ,$$ which is the quotient of the multiplicative group of continuous complex valued functions of absolute value one modulo the subgroup consisting of all such functions that have continuous logarithms.

For any quasi-regular point $$p$$ the winding number associated with a given $$\phi$$ depends only on the element $$\bar{\phi}$$ in $$B(X)$$ determined by $$\phi\ .$$ Thus we get a map $$W_p$$ of $$B(X)$$ into the reals sending $$\bar{\phi}$$ into its winding number. This map is a homomorphism which we will call the winding number homomorphism determined by $$p\ .$$

It is known if we assign $$B(X)$$ to each compact Hausdorff space $$X$$ we get a contra-variant functor that is naturally equivalent to the one-dimensional Ĉech cohomology functor $$\hat{H}^1(X,Z)\ .$$

For compact polyhedra $$\text{Hom}(\hat{H}^1(X,Z),\mathbb{R})$$ is naturally equivalent to $$H_1(X,\mathbb{R})\ .$$ Thus $$W_p\in \text{Hom}(B(X),\mathbb{R})$$ corresponds to an element $$A_p \in H_1(X,\mathbb{R})\ .$$ This is the asymptotic cycle associated with $$p$$ that is described in the introduction.

Since $$H_1(X,\mathbb{R})$$ is finite dimensional for our polyhedron, if $$\mu$$ is any invariant probability measure, $$\int_X A_p d\mu(p)$$ is defined. (Recall that $$A_p$$ is defined for almost all $$p\ .$$) We call this element of $$H_1(X,\mathbb{R})$$ the asymptotic cycle determined by $$\mu$$ and denote it by $$A_{\mu}\ .$$ Of course $$A_p = A_{\mu_p}$$ for any quasi-regular point $$p\ .$$

Suppose now that $$X$$ is a compact oriented differentiable $$n$$-manifold and that our flow arises from a smooth vector field $$V\ .$$ If $$w$$ is a positive $$n$$-form on $$X$$ it is known that the measure determined by $$w$$is invariant if and only if the interior product of $$w$$ with $$V$$ is a closed $$(n-1)$$ form. It was observed by Arnold if the measure determined by $$w$$ is an invariant probability measure $$\mu$$ the associated $$\mu$$-asymptotic cycle is obtained by Poincaré duality from the $$(n-1)$$ dimensional cohomology class determined by the interior product of $$w$$ with $$V\ .$$

## Rotation numbers and asymptotic cycles

Suppose $$X$$ is a connected compact metric space, $$F$$ is a continuous map of $$X$$ into itself, and that $$\nu$$ is an invariant probability measure for $$F\ .$$ Let $$B_F(X)$$ be the subgroup of the Bruschlinski group $$B(X)$$ consisting of those elements in $$B(X)$$ sent into themselves by the endomorphism of $$B(X)$$ induced by $$F\ ,$$ and let $$\phi$$ be a continuous function of absolute value one such that the element $$\bar{\phi}$$ in $$B(X)$$ to which $$\phi$$ belongs is in $$B_F(X)\ .$$

Then $$\phi(F(X))/\phi(X)$$ can be written in the form $$\exp(2\pi i \theta(X))$$ where $$\theta$$ is a continuous real valued function on $$X\ .$$ Since $$X$$ is connected $$\theta$$ is uniquely determined up to an additive constant that is an integer. It follows that $$\exp(2\pi i \int_X \theta(x)d\nu(x))$$ does not depend on the particular $$\theta$$ we have chosen. Moreover if $$\phi_1=\phi \exp(2 \pi i \alpha(x))$$ where $$\alpha$$ is a continuous real valued function, then $\phi_1(F(x))/\phi_1(x)=[\phi(F(x))/\phi(x)]\exp(2 \pi i [\alpha(F(x))-\alpha(x)])$ Therefore $\int_X \frac{1}{2\pi i} \ln (\phi_1(F(x))/\phi_1(x))d\nu = \int_X \frac{1}{2 \pi i} \ln (\phi(F(x))/\phi(x)) d\nu,$ so we actually get a map $$R_{\nu}$$ of $$B_F(X)$$ into the multiplicative group of complex numbers of absolute value one. The map sending $$\lambda \in B_F(X)$$ into $$R_{\nu}(\lambda)$$ is a homomorphism. We call $$R_{\nu}(\lambda)$$ the $$(\nu)$$-rotation number of $$\lambda$$ for the map $$F\ .$$ If $$\bar{\phi}$$ is the element of $$B_F(X)$$ determined by $$\phi$$ we call $$R_{\nu}(\bar{\phi})$$ the $$(\nu)$$-rotation number of $$\phi\ .$$

We will say that $$p\in X$$ is quasi-regular provided for every continuous real valued function $$f$$ on $$X\ ,$$ $\lim_{N\to \infty}\frac{1}{N}\sum_{n=0}^{N-1}f\left(F^{(n)}(p)\right)$ exists.

If $$p$$ is quasi-regular there is a probability measure $$\nu_p$$ invariant under $$F$$ such that this limit equals $$\int_X f(x)d\nu_p(x)\ .$$ The set of quasi-regular points is invariant under $$\phi$$ and has measure one with respect to any invariant probability measure.

If $$p$$ is quasi-regular and $$\phi\left(F(X)\right)/\phi(X)=\exp(2 \pi i \theta(X))$$ where $$\theta(x)$$ is a continuous real valued function , then we let $$\delta_{\phi}(x)=\theta(F(x))-\theta(x)$$ for any $$x\ .$$ $$\delta_{\phi}(x)$$ does not depend on our choice of $$\theta\ ,$$ since $$X$$ is connected. If $$x$$ is quasi-regular and $$\phi \in B_F(X)\ ,$$ $R_{\nu}(\phi)=\exp 2 \pi i \left( \lim_{N\to\infty}\frac{1}{N}[\theta(F^{(n-1)}(x))-\theta(x)]\right)$ where $$F^{(n)}(x)$$ is the $$n$$th iterate of $$F$$ applied to $$x\ .$$

Let us now assume that $$B_F(X)=B(X)\ .$$ For any invariant probability measure $$\nu\ ,$$ the map $$R_{\nu}$$ sends $$B(X)$$ into $$T^1\ ,$$ the multiplicative group of complex numbers of absolute value one. This map is a homomorphism and if we make the additional assumption that $$X$$ is a polyhedron $$R_{\nu}\in \text{Hom}(B(X),T^1)$$ may be identified with an element of $$H_1(X,T^1)\ .$$ We call this element the $$(\nu)$$-asymptotic cycle for $$F$$ and denote it by $$a_{\nu}\ .$$

The coefficient homomorphisms $$0\to Z \to \mathbb{R} \to T^1 \to 0$$ induce maps $$H_1(X,Z) \to H_1(X,\mathbb{R}) \to H_1(X,T^1)\ .$$ However $$H_1(X,\mathbb{R})$$ can be identified with $$\text{Hom}(H^1(X,Z),\mathbb{R})\ .$$ Since $$H^1(X,Z)$$ is a finitely generated free abelian group, $$\text{Hom}(H^1(X,Z),\mathbb{R})$$ gets sent onto $$\text{Hom}(H^1(X,Z),T^1)$$ via the natural map. Thus $$H_1(X,T^1)$$ can be identified with $$H_1(X,\mathbb{R})/H_1(X,Z)\ .$$

Notice that if $$F$$ is homotopic to the identity, $$B_F(X)=B(X)\ .$$

## Global cross sections and asymptotic cycles

Let $$X$$ be a compact connected differentiable manifold and let $$V$$ be a smooth vector field on $$X$$ defining a flow. The flow will be said to possess a global cross-section provided there exists an equivariant homeomorphism of $$X$$ with a flow that differs from the suspension of some homeomorphism only in that the time it takes to return to the space on which this homeomorphism takes place, instead of equaling one can be any positive continuous function. An equivariant homeomorphism of $$X$$ with such a flow determines in an obvious way a global cross section $$K\subseteq X\ ,$$ a time of return $$\rho$$ as a continuous function on $$K\ ,$$ and a homeomorphism $$F$$ of $$K$$ onto itself sending $$p\in K$$ into $$(\rho(p))\cdot p\ .$$

We can also associate with $$K$$ a continuous complex valued function $$\phi$$ of absolute value one. This function is to send the orbital segment of $$X$$ from any point $$p \in K$$ to $$(\rho(p))\cdot p$$ onto the unit circle in the complex plane so that both $$p$$ and $$(\rho(p))\cdot p$$ get sent into 1 and so that the orbital segment wraps in the obvious way around the unit circle.

Since $$B(X)$$ is naturally isomorphic to $$H^1(X,Z)$$ this enables us to associate with $$K$$ an element of $$H^1(X,Z)$$ which we can call the cohomology class for $$K\ .$$

If $$(K_1,F_1)$$ is a global cross section with return map $$F_1$$ and $$(K_2,F_2)$$ is another such pair, then if $$K_1$$ and $$K_2$$ have the same cohomology class associated with them there must exist an equivariant homeomorphism of $$(K_1,F_1)$$ with $$(K_2,F_2)\ .$$ Moreover if $$\lambda \in H^1(X,Z)$$ has a cross section associated with it, it must have a cross section associated with it that is a smooth sub-manifold of $$X$$ for which the return time is a smooth function on this sub-manifold.

We will call an element $$\lambda$$ of $$H^1(X,Z)$$ indivisible provided there does not exist an integer $$n>1$$ and an element $$\lambda_1\in H^1(X,Z)$$ such that $$\lambda=n\cdot\lambda_1\ .$$ Because $$H^1(X,Z)$$ is a finitely generated free abelian group any non-zero element of $$H^1(X,Z)$$ is uniquely representable as a positive integer times an indivisible element.

If there exists a global cross section corresponding to $$\lambda \in H^1(X,Z)$$ and $$\lambda=n\lambda_1$$ where $$\lambda_1$$ is indivisible then there exists a cross section corresponding to $$\lambda_1\ .$$ The cross section corresponding to $$\lambda$$ is connected if and only if $$\lambda$$ is indivisible.

Suppose that $$K$$ is a connected global cross section of the flow on $$X$$ with return map $$F\ .$$ Given an invariant probability measure $$\nu$$ on $$K$$ there is an obvious way to associate with it an invariant probability measure $$\mu$$ on $$X\ .$$ In this way we get a 1-1 correspondence between invariant probability measures on $$X$$ and $$K\ .$$

If $$\phi$$ is a continuous complex valued function of absolute value one on $$K$$ a necessary and sufficient condition for $$\phi(F(x))/\phi(X)$$ to possess a continuous logarithm on $$K$$ is that there exist an extension of $$\phi$$ to a continuous complex valued function $$\psi$$ of absolute value one on $$X\ .$$

Suppose now that $$K$$ is a connected global cross section for the flow on $$X\ .$$ Let $$\nu$$ be an invariant probability measure for the return map on $$K\ ,$$ and let $$\mu$$ be the corresponding invariant probability measure on $$X\ .$$ Further let $$\phi$$ and $$\psi$$ be continuous complex valued functions of absolute value one on $$K$$ and $$X$$ respectively and suppose that $$\psi$$ is an extension of $$\phi\ .$$ Let $$A_{\mu}$$ be the $$\mu$$ asymptotic cycle for the flow on $$X\ .$$

If we identify $$A_{\mu}$$ with the corresponding element of $$\text{Hom}(B(X),\mathbb{R})\ ,$$ then $R_{\nu}(\bar{\phi})=\exp(2 \pi i A_{\mu}(\bar{\psi})).$

Next let $$C$$ be the collection of elements $$\lambda$$ in $$H_1(X,\mathbb{R})$$ such that there exists an invariant probability measure $$\mu$$ with $$\lambda=A_{\mu}\ .$$ Then $$C$$ is a compact convex subset of $$H_1(X,\mathbb{R})\ .$$

If $$\alpha \in H^1(X,Z)$$ then $$\alpha$$ determines an element of $$H_1(X,\mathbb{R})$$ and therefore an element of $$\text{Hom}(H_1(X,\mathbb{R}),\mathbb{R})\ .$$ Then a necessary and sufficient condition for there to exist a connected cross section corresponding to $$\alpha$$ is that $$\alpha$$ be indivisible and that the corresponding element of $$\text{Hom}(H_1(X,\mathbb{R}),\mathbb{R})$$ be positive on $$C\ .$$ Consequently a necessary and sufficient condition that there exist a global cross section for $$X$$ is that 0 does not belong to $$C\ .$$

Gelfand and Shapiro give an example of a smooth flow on a compact manifold such that a $$\mu$$-asymptotic cycle equals zero. Since the flow they consider makes $$X$$ a minimal set, this gives a counter example to the conjecture that every compact minimal set possesses a global cross section.