# Rotation theory

Post-publication activity

Curator: Michał Misiurewicz

Rotation Theory is a part of the Dynamical Systems Theory. It deals with ergodic averages and their limits, not only for almost all points, like in Ergodic Theory, but for all points. It grew from the theory of rotation numbers for circle homeomorphisms, developed by Poincaré. It has applications to many classes of dynamical systems, for instance to continuous circle maps homotopic to the identity, annulus and torus homeomorphisms isotopic to the identity, subshifts of finite type, and continuous interval maps. When it deals with periodic orbits, it is a powerful tool in Combinatorial Dynamics. Another part of it creates links between Dynamical Systems and Algebraic Topology.

More details on Rotation Theory can be found in Alsedà, Llibre and Misiurewicz  or Misiurewicz  and in Athanassopoulos .

## Classical case

Let $$\mathbb{T}=\mathbb{R}/\mathbb{Z}$$ be the circle and denote by $$\pi$$ the natural projection $$\mathbb{R}\to\mathbb{T}$$. Then a continuous map $$f\colon \mathbb{T}\to\mathbb{T}$$ has a lifting $$F\colon \mathbb{R}\to\mathbb{R}$$, that is, a continuous map such that $$\pi\circ F=f\circ\pi$$ (there are countably many of them, differing by an integer, so we fix one). There exists an integer $$d$$ such that $$F(x+1)=F(x)+d$$ for all $$x\in\mathbb{R}$$; it is called the degree of $$f\ .$$

By a theorem of Poincaré, if $$f\colon\mathbb{T}\to\mathbb{T}$$ is an orientation preserving homeomorphism (so its degree is 1) then the limit of $$(F^n(x)-x)/n$$ as $$n\to\infty$$ exists for every $$x\in\mathbb{R}$$ and does not depend on $$x$$. It is called the rotation number of $$f$$ (more precisely, of $$F$$).

## Circle Maps of Degree One

For general continuous circle maps of degree 1 rotation number $$\rho(F,x)$$ can depend on the point $$x$$ and for many points it may not exist. Therefore one can speak of the pointwise rotation set $$\rho_p(F)$$ consisting of all rotation numbers $$\rho(F,x)$$ that do exist. This set is a closed interval, perhaps degenerated to a point.

For every number $$a$$ from the rotation interval there exists an invariant set $$M_a\subset \mathbb{T}$$ such that $$f$$ restricted to $$M_a$$ is minimal and preserves order. Moreover, the rotation number of every $$x\in M_a$$ is $$a$$. In particular, if $$a=p/q$$ with $$p,q$$ coprime, then $$M_a$$ is a cycle of $$f$$ of period $$q$$. If we drop the assumption that $$p,q$$ are coprime, still a cycle of period $$q$$ and rotation number $$p/q$$ exists, except perhaps when $$a$$ is the endpoint of the rotation interval.

If the rotation interval of $$f$$ is nondegenerate then the topological entropy of $$f$$ is positive. The best estimates of the entropy given the rotation interval are known.

Another class of maps for which similar results hold is the class of discontinuous circle maps that have liftings satisfying $$F(x+1)=F(x)+1$$ and discontinuities only with left-hand-side limit larger than the right-hand-side limit. The best known examples of such maps are Lorenz-like maps (see Alsedà, Llibre, Misiurewicz and Tresser ).

## Annulus

The same construction can be made in the case of the annulus $$\mathbb{A}=\mathbb{T}\times [0,1]$$ instead of the circle. Assume that $$f\colon \mathbb{A}\to \mathbb{A}$$ is a homeomorphism isotopic to the identity. Then the lifting $$F$$ maps $$\mathbb{R}\times [0,1]$$ to itself and to define the rotation number instead of $$(F^n(x)-x)/n$$ one takes its first coordinate. This time the pointwise rotation set is not necessarily an interval, but nevertheless it is compact. If $$p/q\in\rho_p(F)$$ with $$p,q$$ coprime, then $$f$$ has a cycle of period $$q$$ and rotation number $$p/q\ .$$ Moreover, for all except (perhaps) finitely many $$a\in\rho_p(F)$$ there exists a compact invariant set $$X_a$$ such that $$\rho(F,x)=a$$ for all $$x\in X_a$$.

The reason why the results are weaker than for circle maps is that although considering homeomorphisms instead of continuous maps offsets the increase of dimension by 1, one measures displacement only in one direction, leaving the space for maneuver in the other dimension. To prevent it, often stronger assumptions are made on $$f\ .$$ An annulus homeomorphism $$f$$ isotopic to the identity is called twist if for each $$x\in \mathbb{R}$$ the first coordinate of $$F(x,y)$$ is a strictly monotone function of $$y$$ (in other words, vertical segments are mapped to curves slanted in one direction). It has graph intersection property if the image under $$f$$ of the graph $$\gamma$$ of every continuous function from $$\mathbb{T}$$ to $$[0,1]$$ intersects $$\gamma\ .$$ Note that if $$f$$ preserves the Lebesgue measure then it has graph intersection property. The Aubry-Mather Theorem (Aubry and Le Daeron , Mather ) states that if $$f$$ is twist and has graph intersection property, then for every number $$a$$ between the rotation numbers of $$f$$ on the components $$\mathbb{T}\times \{0\}$$ and $$\mathbb{T}\times \{1\}$$ of the boundary there exists an invariant set $$M_a\subset \mathbb{A}$$ such that its projection on the first coordinate is one-to-one, and $$f$$ restricted to $$M_a$$ is minimal and preserves order on the first coordinate. Moreover, the rotation number of every $$x\in M_a$$ is $$a$$. In particular, if $$a=p/q$$ with $$p,q$$ coprime, then $$M_a$$ is a cycle of $$f$$ of period $$q$$. Such a cycle is called a Birkhoff periodic orbit.

## General Theory

Let $$X$$ be a compact metric space, $$f:X\to X$$ a continuous map and $$\varphi:X\to\mathbb{R}^d$$ a Borel bounded (usually continuous) function (an observable). If for $$x\in X$$ the limit

$\rho(f,\varphi, x)=\lim_{n\to\infty}\frac1n\sum_{j=0}^{n-1}\varphi(f^j(x))$

exists, we call it the rotation vector of $$x$$. The set $$\rho_p(f,\varphi)$$ of all rotation vectors of points of $$X$$ is the pointwise rotation set of $$f$$ for the observable $$\varphi$$. The (general) rotation set $$\rho(f,\varphi)$$ of $$f$$ for the observable $$\varphi$$ is the set of all limits of the sequences of the form

$\frac1{n_i}\sum_{j=0}^{n_i-1}\varphi(f^j(x_i)),$

where $$x_i\in X$$ and $$n_i\to\infty\ .$$ For an ergodic invariant probability measure $$\mu$$, its rotation vector $$\rho(f,\varphi,\mu)$$ is the integral $$\int\varphi\,d\mu\ .$$ The set $$\rho_m(f,\varphi)$$ of all rotation vectors of ergodic invariant probability measures is the measure rotation set of $$f$$ for the observable $$\varphi\ .$$

To apply this formalism to circle maps, one takes as $$\varphi$$ the displacement function $$\varphi(x)=F(y)-y\ ,$$ where $$y$$ is a lifting of $$x$$. For annulus homeomorphisms, $$F$$ is replaced by its first component.

Clearly, $$\rho_p(f,\varphi)\subset\rho(f,\varphi)\ .$$ For an ergodic measure $$\mu$$, by the Birkhoff Ergodic Theorem, $$\rho(f,\varphi,x)=\rho(f,\varphi,\mu)$$ for $$\mu$$-almost every $$x$$, and thus $$\rho_m(f,\varphi)\subset \rho_p(f,\varphi)\ .$$

If $$P$$ is a cycle of $$f$$ of period $$n$$ then for $$x\in P$$

$\rho(f,\varphi,x)=\frac1n\sum_{j=0}^{n-1}\varphi(f^j(x))=\int\varphi\,d\mu_P$

for the probability measure $$\mu_P$$ equidistributed on $$P$$.

## Flows

Let $$X$$ be a compact metric space and let $$\Phi$$ be a continuous flow on $$X$$. That is, $$\Phi:\mathbb{R}\times X\to X$$ is a continuous map such that $$\Phi(0,x)=x$$ and $$\Phi(s+t,x)=\Phi(t,\Phi(s,x))$$ for every $$x\in X\ ,$$ $$s,t\in\mathbb{R}\ .$$ We will often write $$\Phi^t(x)$$ instead of $$\Phi(t,x)\ .$$ Let $$\xi$$ be a time-Lipschitz continuous observable cocycle for $$(X,\Phi)$$ with values in $$\mathbb{R}^m$$, that is, a continuous function $$\xi:\mathbb{R}\times X\to\mathbb{R}^m$$ such that $$\xi(s+t,x)=\xi(s,\Phi^t(x))+\xi(t,x)$$ and $$\|\xi(t,x)\|\le Lt$$ for some constant $$L$$ independent of $$t$$ and $$x$$.

The (general) rotation set $$R$$ of $$(X,\Phi,\xi)$$ is the set of all limits $$\lim_{n\to\infty}\frac{\xi(t_n,x_n)}{t_n}\ ,$$ where $$\lim_{n\to\infty}t_n=\infty.$$ It is compact and if $$X$$ is connected, so it $$R$$. The set $$R$$ is equal to the rotation set of the time-one-map $$\Phi^1$$ and the observable $$\varphi$$ defined by $$\varphi(x)=\xi(1,x)\ .$$

One can define rotation vectors of points and ergodic measures, the pointwise rotation set and the measure rotation set for flows in a similar way. They all will be equal to the analogous objects for $$\Phi^1$$ and $$\varphi$$. Moreover, instead of a flow, one can consider a semiflow.

## Asymptotic Cycles

Let $$X$$ be a compact metric space and let $$\Phi$$ be a continuous flow on $$X$$. For a continuous function $$\psi:X\to\mathbb{T}$$ one can define a continuous cocycle $$\tilde\psi:\mathbb{R}\times X\to\mathbb{T}$$ with values in $$\mathbb{T}$$, by $$\tilde\psi(t,x)=f(\Phi^t(x))-f(x)$$ (mod 1). This cocycle can be lifted to a continuous cocycle $$\xi$$ with values in $$\mathbb{R}$$ (that is, we have $$\xi=\tilde\psi$$ (mod 1)). Cocycle $$\xi$$ is time-Lipschitz continuous. Set $$\varphi_f(x)=\xi(1,x)$$ and for any invariant probability measure $$\mu$$ on $$X$$ consider $$\int_X\varphi_f\;d\mu\ .$$ In particular, if $$\mu$$ is ergodic, this is the rotation number of $$\mu$$.

Now, instead of fixing $$f$$ and varying $$\mu$$, one fixes $$\mu$$ and varies $$f$$. If $$f_1$$ and $$f_2$$ are homotopic, then the corresponding integrals $$\int_X\varphi_{f_i}\;d\mu$$ are equal. The group of homotopy classes of continuous functions from $$X$$ to $$\mathbb{T}$$ is isomorphic to the first Čech cohomology group with integer coefficients $$\check H^1(X;\mathbb{Z})\ ,$$ so one gets a group homomorphism $$A_\mu:\check H^1(X;\mathbb{Z})\to\mathbb{R}$$ defined by $$A_\mu[f]=\int_X\varphi_f\;d\mu\ .$$ This homomorphism is called the $$\mu$$-asymptotic cycle of the flow. The image $$A_\mu(\check H^1(X;\mathbb{Z}))\subset\mathbb{R}$$ is called the $$\mu$$-winding numbers group of the flow.

## Torus Homeomorphisms

The formalism described earlier can be applied in a natural way to the maps of tori in any dimension with the displacement function (in the lifting) as the observable. However, except for the circle maps, the only strong results can be obtained for homeomorphisms of a 2-dimensional torus $$\mathbb{T}^2$$ isotopic to the identity. Then the general rotation set is convex and is equal to the convex hull of the pointwise and measure rotation sets. Results similar to those for circle maps and to the Aubry-Mather Theorem hold, with the exception that one cannot guarantee the existence of points with rotation vectors from the boundary of the rotation set, except the extreme vectors. Thus, if the rotation set is a polygon, then all its interior points and vertices are "good," but the points on the sides which are not vertices can be "bad."

It is easy to construct homeomorphisms of $$\mathbb{T}^2$$ isotopic to the identity with the rotation set being a convex polygon with rational vertices. There must be many other rotation sets, since $$\rho(F)$$ depends continuously on $$F$$ (in the Hausdorff topology), as long as $$\rho(F)$$ has nonempty interior. However, while there is an explicit example of a rotation set which is a "polygon" with infinitely many vertices (Kwapisz ), it is even not known whether the rotation set can be a strictly convex figure other than a singleton.

## Other observables

The observable which is used in the definition of the rotation set does not have to be a displacement function. For instance, for a continuous interval map $$f$$ one can take the observable $$\psi(x)$$ equal to $$1/2$$ if $$(f^2(x)-f(x))(f(x)-x)\le 0$$ and 0 otherwise. The rotation numbers obtained in such a way are called over-rotation numbers and are used in Combinatorial Dynamics.

By taking as the observable the identity, one gets as the rotation vectors centers of mass for various invariant measures. An interesting example is an interval map $$f_a(x)=axe^{-x}$$ for $$a>e$$ and $$x\in [a^2e^{-1-a/e},a/e]$$. Then all centers of mass are at $$\ln a$$.

## Symbolic Dynamics

If for a map $$f\colon X\to X$$ with an observable $$\varphi\colon X\to\mathbb{R}^d$$ one can find a Markov partition for $$f\ ,$$ then quite often the methods of Symbolic Dynamics work. The basic idea is to replace the system by a subshift of finite type $$\sigma:\Sigma\to\Sigma$$ which is nearly conjugate to the original map. Then if the observable $$\varphi$$ is "natural" enough, it can be replaced by an observable $$\psi:\Sigma\to\mathbb{R}^d$$ which is constant on cylinders of length 2 (depends only on the zeroth and first coordinates), takes values from $$\mathbb{Z}^d$$ and $$(\sigma,\psi)$$ gives (almost) the same rotation numbers as $$(f,\varphi)\ .$$

Dealing with a subshift of finite type with the observable as above is usually much simpler than with the original system. Let $$\tau_1,\dots,\tau_s$$ be all the elementary (not passing more than once through any vertex) loops in the transition graph of $$\sigma$$ and let $$\rho_1,\ldots,\rho_s$$ be their rotation vectors. Then the rotation set $$\rho(\sigma,\psi)$$ is equal to the convex hull of $$\rho_1,\ldots,\rho_s$$, and again the results similar to the ones listed before hold. However, while one obtains cycles with desired rotation vectors, the periods of those cycles can be larger than expected.

## Other Rotation Objects

The theory of rotation vectors for torus homeomorphisms homotopic to the identity can be generalized to the case when the torus is replaced by a compact surface of genus zero (Franks ) or of negative Euler characteristic (Franks ). Then the homological rotation vectors live in the first homology group with real coefficients.

The theory of asymptotic cycles can be used to define $$\nu$$-rotation number maps and $$\nu$$-rotation number groups for a homeomorphism $$h$$ of a compact metric space and an invariant probability measure $$\nu$$ (Athanassopoulos ). The flow for which the asymptotic cycles are considered is the suspension of $$h$$.

Bernadette and Mitchell  generalized the theory of asymptotic cycles to the non-commutative setting (homotopy rather than homology).

Rotation sets of periodic orbits of maps $$z\mapsto z^d$$ of the unit circle have been defined by Goldberg  and used by Goldberg and Milnor  in holomorphic dynamics.

## History and Applications

Rotation numbers for circle homeomorphisms have been defined by Poincaré. They are useful in the number of applications, most notably in the KAM Theory and Complex Dynamics. Rotation numbers were defined for annulus homeomorphisms by Birkhoff and they are important for instance in the theory of billiards. The next generalization was by Newhouse, Palis and Takens  for circle maps of degree 1 in connection with the saddle-node bifurcation of diffeomorphisms. It found important applications in Combinatorial Dynamics. Asymptotic cycles were introduced by Schwartzman . They play essential role in Asymptotic Homology theory, that provides a bridge between Dynamical Systems (mainly its part dealing with flows) and Algebraic Topology. Rotation vectors for torus homeomorphisms has been studied for the first time by Kim, MacKay and Guckenheimer , Llibre and McKay  and Herman ; the definition of the general rotation set was introduced by Misiurewicz and Ziemian . Rotation sets for subshift of finite type have been introduced by Ziemian  (some elements of this approach can be found in Fried ) and they are essential if symbolic dynamics is to be used for studying rotation sets for other systems. Over-rotation numbers were defined by Blokh and Misiurewicz  and are used in Combinatorial Dynamics.