Expansive systems - davidar/scholarpedia GitHub Wiki
A discrete invertible (the case we shall mainly refer to) expansive system is a dynamical system such that every point of the underlying space has a distinctive behaviour. A homeomorphism <math>f</math> from the compact metric space <math>M</math> onto <math>M</math> is expansive if there exists <math>\alpha >0\ ,</math> (called expansivity constant of <math>f</math>) such that if <math>x,y\in M</math> and <math>dist(f^{n}(x),f^{n}(y))\leq \alpha</math> for every <math>n\in Z</math> then, <math>x=y\ .</math> Thus, if <math>x\neq y\ ,</math> then for some <math>n,</math> <math>dist(f^{n}(x),f^{n}(y))>\alpha .</math>
Expansive systems are then wholly sensitive to initial conditions and therefore, in this sense, chaotic.
Assume the dynamics of <math>f</math> is observed with a precision that permits to distinguish points at a distance larger than <math>\alpha ,</math> meanwhile, points at a distance less than <math>\delta >0,</math> <math>\delta <<\alpha,</math> are not distinguished. Then, a <math>\delta</math>-small neighbourhood of, say, <math>x\in M</math> with infinite points, will be seen -at present- as only one point. However, for some <math>n\in Z\ ,</math> the <math>n</math>-iterate through <math>f</math> of this point, will show many of them, since points at a distance larger than <math>\alpha</math> are distinguished by the observer (see [B]).
Since <math>M</math> is compact, on account of the expansiveness of <math>f,</math> it is not difficult to show that given a <math>\delta</math> like in the preceding paragraph, <math>\delta <\alpha /2\ ,</math> there is a <math>C^{0}-</math>neighbourhood <math>N</math> of <math>f\ ,</math> such that if <math>g\in N\ ,</math> and <math>dist(g^{n}(x),g^{n}(y))\leq \alpha </math> for every <math>n\in Z,</math> then <math>dist(g^{n}(x),g^{n}(y))\leq \delta</math> for all <math>n\in Z.</math> Therefore the relation <math>\mathcal{R}_{\delta }=\{(x,y)\in M\times M:dist(g^{n}(x),g^{n}(y))\leq \delta ,\;\;n\in Z\} </math> is an equivalence relation on <math>M,\ .</math> The canonical projection <math>\pi:M\to M/\mathcal{R}_{\delta }</math> is closed and consequently <math>M/\mathcal{R}_{\delta }</math> is a Hausdorff compact topological space and therefore the <math>M/\mathcal{R}_{\delta }</math> is a compact metrizable space, and <math>g^{\ast }: M /\mathcal{R}_{\delta }\rightarrow M /\mathcal{R}_{\delta }</math> defined by <math>g^{\ast }(\pi (x))=\pi (g(x))\ ,</math> is an expansive homeomorphism of <math>M/ \mathcal{R}_{\delta }.</math>
Again, an observer that can not distinguish points at a distance less than <math>\delta </math> will see the motion as taking place in <math>M/ \mathcal{R}_{\delta }</math> (instead of <math>M</math>) under the action of <math>g^{\ast }\ .</math>
Clearly, homeomorphisms conjugate to an expansive one, are also expansive.
Consider <math>\ 2^{Z}=\left\{ \left( a_{n}\right)
- a_{n}=0\text{ or }1,n\in Z\right\}</math> and the distance:
With this metric, which induces the product toplogy, <math>2^{Z}</math> is compact. Let <math>\sigma :2^{Z}\rightarrow 2^{Z}</math> be defined by <math>\sigma (a_{n})=(b_{n}),</math> where <math>b_{n}=a_{n-1}\ .</math> <math>\sigma</math> is the usual shift homeomorphism. If <math>(a_{n}) \neq ( b_{n})</math> then, for some <math>K\in Z,a_{K}\neq b_{K}\ ,</math> and , therefore
<math>dist(\sigma ^{K}(( a_{n})) ,\sigma ^{K} ((b_{n})) )\geq 1,</math>
showing that the shift is an expansive homeomorphism.
Take a rotation of <math>S^{1}</math> by an angle <math>2\pi \alpha\ ,</math>
where <math>\alpha</math> is irrational, and replace the points of a dense orbit, say <math>\left\{ x_{n},n\in Z\right\}\ ,</math>with arcs of diameter decreasing with <math>| n|</math> in order to get a new space also homeomorphic to <math>S^{1}\ .</math> The Denjoy map, <math>f\ ,</math> may be defined by assigning to each point not
on the added arcs, the former image under the rotation, and mapping (length) linearly the arc replacing <math>x_{n}</math> onto the one replacing <math>x_{n+1},n\in Z\ .</math> It is easy to see that this map is a homeomorphism of <math>S^{1}\ ,</math> and that the set <math>D</math> of points not lying in the interior of the added arcs is compact and invariant under the Denjoy map (in fact this set is homeomorphic to the Cantor set). A non-trivial arc whose end points lie on this set contains some of the added arcs, and, consequently, some iterate of this arc will include the one replacing <math>x_{0}</math> of diameter, say <math>d\ .</math> Thus, <math>d</math> will be an expansivity constant for the restriction of <math>f</math> to <math>D\ .</math>
Let <math>f</math> be a diffeomorphism of a compact, Riemannian, smooth manifold <math>M</math> onto itself; <math>f</math> is Anosov if there exitsts <math>L>0,0<\lambda <1\ ,</math> and continuous non trivial <math>Tf</math> invariant sub-bundles <math>S,U</math> of <math>TM,\ ,</math> such that <math>S\oplus U=TM\ ,</math> <math>\left\| Tf^{n}(s)\right\| \leq L\lambda ^{n}</math> for <math>s\in S,n\geq 0\ ,</math> and <math>\| Tf^{-n}(u)\| \leq L\lambda ^{n}</math> and <math>u\in U,n\geq 0\ .</math>
If <math>A</math> is a compact <math>f</math>-invariant subset of <math>M</math> and the above decomposition holds on <math>A\ ,</math> <math>A</math> is called a hyperbolic set. The restriction <math>f|A\ ,</math> of <math>f</math> to a hyperbolic set <math>A</math> is also expansive. Anosov difeeomorphisms may also be characterized in a different way (see [L1]). Let <math>B:TM\rightarrow R</math> be a continuous quadratic form, i.e, <math>B_{x}=B|T_{x}M</math> is a quadratic form on the vector space <math>T_{x}M</math> that depends continuously on <math>x\in M\ .</math> A diffeomorphism <math>f:M\rightarrow M</math> is quasi-Anosov if there exists such a <math>B</math> with the property <math>B_{f(x)}((Tf)_{x}(v))-B_{x}(v)>0,</math> for every <math>x\in M\ ,</math> and each <math>v\in T_{x}M,\| v\| \neq 0\ .</math> A diffeomorphism <math>f</math> is Anosov if and only if it is quasi-Anosov and <math>B_{x}</math> is non-degenerate for all <math>x\in M\ .</math> There are quasi-Anosov diffeomorphisms that fail to be Anosov (see [FR], the examples in this paper have a strange attractor and a strange repeller [M] and the motion of most points evolves to the attractor and comes from the reppeller). This characterization of quasi-Anosov (Anosov) diffeomorphisms permits to conclude the existence of a <math>C^{1}</math> neighbourhood of <math>f</math> such that any finite composition of diffeomorphisms in that neighbourhhod is also quasi-Anosov (Anosov). We shall see below that Anosov and quasi-Anosov diffeomorphisms (and hyperbolic sets) are expansive.
Let <math>f</math> be a homeomorphism of an oriented compact surface <math>M</math> of genus larger than 1 onto itself. The map <math>f</math> is pseudo-Anosov if there exist two <math>f</math>-invariant, transversal foliations with singularities (see figure 1) <math>W^{S},W^{U},</math> and also two transversal measures <math>\mu _{S},\mu _{U}</math> (defined on the space of (stable, unstable) leaves of <math>W^{S},</math> respectively <math>W^{U}</math>) and <math>\lambda >1</math> such that <math>f^{\ast }(\mu _{U})=\lambda \mu _{U}</math> and <math>f^{\ast }(\mu _{S})=\lambda ^{-1}\mu _{S}\ .</math> The existence and expansivity of these homeomorphisms is proved in [T], [FLP].
Let <math>f:T^{2}\rightarrow T^{2}</math> be defined by
<math>\label{(1)} f(x,y)=(2x+y-\frac 12\pi c\sin (2\pi x),\;x+y-\frac 12\pi c\sin (2\pi x)).</math>
For <math>0\leq c<1,</math> <math>f</math> is Anosov (for <math>c=0</math> <math>f</math> is linear), but for <math>c=1\ ,</math> <math>f</math> is expansive but is neither Anosov nor quasi-Anosov since <math>Tf_{0}</math> has no non-trivial invariant subspaces.
Answer:
Theorem [U]. Let <math>M</math> be a compact metric space and <math>f:M\rightarrow M</math> be an homeomorphism such that there is <math>\alpha >0</math> with the property that for <math>x,y\in M,x\neq y,</math> <math>dist(f^{n}(x),f^{n}(y))>\alpha </math> for some <math>n\geq 0\ .</math> Then, <math>M</math> is finite.
Let <math>f:M\rightarrow M</math> be a homeomorphism; for <math>x\in M\ ,</math> the stable set of <math>x</math> is
<math>W^{S}(x)=\{ y\in M:dist(f^{n}(x),f^{n}(y))\rightarrow 0 \mbox{ if } n\rightarrow +\infty \} ,</math> and the unstable set is
<math>W^{U}(x)=\{ y\in M:dist(f^{n}(x),f^{n}(y))\rightarrow 0 \mbox{ if } n\rightarrow -\infty \} .</math>
The local stable (unstable) sets of <math>x,</math> are defined as follows: given <math>\varepsilon >0,</math> <math>W_{\varepsilon }^{S}(x)=\left\{ y\in M:dist(f^{n}(x),f^{n}(y))\leq\varepsilon ,n\geq 0\right\}</math> <math>W_{\varepsilon }^{U}(x)=\left\{ y\in M:dist(f^{n}(x),f^{n}(y))\leq \varepsilon ,n\leq 0\right\}</math>
Let now <math>f</math> be expansive. May the stable set contain a neighbourhood of <math>x</math> for every <math>\varepsilon >0\ ?</math> In other words : may <math>x</math> be Lyapunov stable in the future? The answer is yes; it is easy to find a shift invariant subset of <math>2^{Z}</math> for which <math>0</math> is Lyapunov stable in the future. Nevertheless,
Theorem [L2]. If <math>M</math> is locally connected there are no stable points (either in the future or in the past).
Corollary. If <math>M</math> is locally connected, for every <math>\varepsilon >0,</math> there is <math>r>0\ ,</math> such that for every <math>x\in M\ ,</math> <math>W_{\varepsilon }^{S}(x)</math> and <math>W_{\varepsilon }^{U}(x)</math> contain a compact connected set of diameter larger than <math>r\ .</math>
(Compare with the Denjoy map <math>f|D\ ;</math> for points not lying on the added arcs the local stable (unstable) sets are trivial.)
Application. There are no expansive homeomorphisms of <math>S^{1}.</math>
Proof: Assume by contradiction that there exist an expansive homoemorphism on <math>S^1\ .</math> Then by the previous Corollary there are non-trivial stable open sets (a connected set of <math>S^1</math> contains an open arc) and every point of it is a stable point, in contradiction with the above Theorem.
Theorem [L1]. Let <math>f</math> be a homeomophism of <math>M\ ,</math> then <math>f</math> is expansive if and only if there exist a neighbourhood <math>N</math> of the diagonal in <math>M\times M</math> and a real continuous function <math>V</math> (Lyapunov) defined on <math>N\ ,</math> vanishing on the diagonal and such that for <math>(x,y)\in N,x\neq y,</math> <math>V(f(x),f(y))-V(x,y)>0.</math>
In order to proof expansivity for Anosov and quasi-Anosov diffeomorphisms, the quadratic form <math>B</math> mentioned in the section Anosov and quasi-Anosov diffeomorphisms, can be used to construct a Lyapunov function. In fact, for <math>y</math> close to <math>x\ ,</math> the Lyapunov function is <math>V(x,y)=B_{x}(u)\ ,</math> where <math>\exp _{x}(u)=y\ .</math> The expansivity of pseudo-Anosov maps may be shown also using Lyapunov functions [L2]. For the examples in (<figref>Fig_3_pseudoorbita.png</figref>), choose
<math>V(x,y)=V((x_{1},x_{2}),(y_{1},y_{2}))=(y_{1}-y_{2})((x_{1}-y_{1})-(x_{2}-y_{2})).</math>
Classification Theorem ([Hi], [L3]). Let <math>f</math> be an expansive homeomorphism of a compact connected oriented boundaryless surface <math>M\ .</math> Then,
- <math>S^{2}</math> does not support such a homeomorphism,
- if <math>M=T^{2},</math> <math>f</math> is conjugate to an Anosov diffeomorphism
- if the genus of <math>M</math> is larger than 1, then <math>f</math> is conjugate to a pseudo-Anosov homeomorphism.
Those properties are consequences of the description of the local stable (unstable) sets of <math>f\ .</math>
Usually, the study of local stable (unstable) sets are made on the basis of strong assumptions on the dynamics of <math>Tf\ ,</math> as for Anosov diffeomorphisms, hyperbolic sets, etc. In our case, even for expansive diffeomorphims, we only have the dialogue between the topology of <math>M</math> and the dynamics of <math>f\ .</math> Nevertheless, after showing the local connectedness of the connected component containing <math>x</math> of <math>W_{\varepsilon}^{S}(x)(W_{\varepsilon }^{U}(x))</math> the following theorem is proved.
Theorem. For <math>x\in M,\; W_{\varepsilon }^{S}(x)(W_{\varepsilon }^{U}(x))</math> is the union of a finite number <math>r</math> of arcs, <math>(r\geq 2)</math> that meet only at <math>x\ .</math> Stable (unstable) sectors (the sets limited by two consecutive stable (unstable) arcs) are separated by unstable (resp. stable) arcs. If at <math>x \in M\ ,</math> <math>r\geq 3\ ,</math> <math>x</math> is called a singular point; the set of singular points is finite.
When <math>r=2\ ,</math> as it is always the case for Anosov diffeomorphisms, <math>x</math> has a neighbourhood <math>N</math> such that if <math>y</math> and <math>z</math> belong to <math>N\ ,</math> <math>W_{\varepsilon }^{S}(y)\cap W_{\varepsilon }^{U}(z)</math> is not void. This is not the case for singular points (see figure 2).
Now a very brief mention of some steps of the proof of the Classification Theorem is given. For <math>r\geq2\ ,</math> if <math>y</math> and <math>z</math> lie in a sector then <math>W_{\varepsilon }^{S}(y)</math> and <math>W_{\varepsilon }^{U}(z)</math> meet only once. The set of these intersections includes, by the Theorem of invariance of domain, a neighbourhood of <math>x</math> in the sector (local product structure). This implies that singular points can not accumulate and then, their number is finite. Let now <math>M^{\ast }</math> be the universal cover of <math>M\ .</math> It is not difficult to show that the lifting to <math>M^{\ast }</math> of a stable or an unstable set is closed and that the union of the lifting of a stable arc and an unstable one can not be homeomorphic to <math>S^{1}\ .</math>
If <math>S^{2}</math> supported an expansive homemorphism, and <math>W^{S}(x)</math> does not contain singular points, it is homeomorphic to <math>S^{1}\ ,</math> and this in turn, implies the existence of stable points; a contradiction.
That expansive homeomorphisms <math>f</math> of surfaces of genus <math>\geq 1\ ,</math> are conjugate to Anosov or to pseudo-Anosov maps follows from the following two Lemmas.
Lemma An expansive homeomorphism <math>f</math> on a surface <math>M</math> of genus <math>\geq 1</math> is isotopic to an Anosov (if <math>M=T^2</math>) or to a pseudo-Anosov map (genus <math>\geq 1</math>).
Proof. It follws from [L3] on account of Thurston's results [T].
Definition. Let <math>f,g</math> be homeomorphisms of the compact metric space <math>M\ ;</math> <math>f</math> is semi-conjugate to g if there exists <math>h:M\rightarrow M</math> continuous and surjective, such that <math>h\circ f=g\circ h\ .</math>
Lemma If the expansive homeomorphism <math>f</math> of the surface <math>M</math> is isotopic to an Anosov diffeomorphism, or to a pseudo-Anosov homeomorphism <math>g\ ,</math> then <math>f</math> is semi-conjugate to <math>g</math>
Proof. See [F], [L3].
In both cases, <math>h^*:M^*\to M^*\ ,</math> a lifting of the semi-conjugacy <math>h</math> is a proper map, and this fact is an essential tool to prove that the semi-conjugacy is, actually, a conjugacy.
Consider now expansive homeomorphisms <math>f</math> defined on compact boundaryless manifolds <math>M</math> of dimension larger than 2. In the case of surfaces, it follows from the Classification theorem that periodic points are dense on the surface, and , moreover, that on an open and dense set, <math>r=2\ .</math> Thus for points <math>x</math> in that set, <math>W_{\varepsilon }^{S}(x)</math> includes a topological 1-dimensional manifold and <math>W_{\varepsilon }^{U}(x)</math> another such manifold, topologically transversal to the first one at <math>x\ .</math> The results concerning <math>dim M\geq 3</math> assume the existence of a dense set of periodic points <math>p</math> such that <math>W_{\varepsilon }^{S}(p)</math> contains a topological manifold of dimension <math> d,\;1\leq d<\dim M\ ,</math> and <math>W_{\varepsilon }^{U}(p)</math> a manifold of complementary dimension, topologically transversal to <math>W_{\varepsilon }^{S}(p)</math> at <math>p\ .</math> Points <math>x</math> with such a behaviour of <math>W_{\varepsilon }^{S}(x)</math> and <math>W_{\varepsilon }^{U}(x)</math> are called topologically hyperbolic.(This is the case for Anosov diffeomorphisms at every <math>x\in M</math>).
Theorem([ABP], [V1], [V2]). Let <math>f</math> be an expansive homeomorphism of <math>M</math> with a dense set of topologically hyperbolic periodic points. Then there is an open and dense set with local product structure. Furthermore if <math>\dim M\geq 3,</math> and for some topologically hyperbolic periodic point <math>p\ ,</math> either <math>W_{\varepsilon }^{S}(p)</math> or <math>W_{\varepsilon }^{U}(p)</math> is one-dimensional, <math>M</math> is a torus and <math>f</math> is conjugate to a linear Anosov diffeomorphism.
Therefore, in this case, in contrast with what happens for surfaces, there are no singularities. This is, essentially, a consequence of the fact that if <math>\dim M\geq 3\ ,</math> say, <math>W_{\varepsilon }^{S}(p)</math> separates small balls centered at <math>p\ ,</math> meanwhile <math>W_{\varepsilon }^{U}(p)</math> does not. Of course, if we do not assume that one of this dimensions is one, the result is false: take the product of two pseudo-Anosov maps.
Let <math>f</math> be a homeomorphism of a compact metric space <math>M</math> onto itself.
<math>f</math> is persistent if for any <math>\varepsilon >0</math> there exists a <math>C^{0}</math>-neighbourhood <math>N</math> of <math>f</math> such that for <math>g\in N</math> and <math>x\in M\ ,</math> there exists <math>y\in M</math> with the following property. <math>dist(f^{n}(x),g^{n}(y))\leq \varepsilon ,\;\;n\in Z</math>
<math>f</math> is topologically stable if for <math>\varepsilon >0\ ,</math> there exists,<math>N\ ,</math> a <math>C^{0}-</math>neighbourhood of <math>f\ ,</math> such that any <math>g\in N</math> is semi-conjugate to <math>f</math> (see 4)) and <math>dist(x,h(x))<\varepsilon\ .</math>
A <math>\delta</math> pseudo-orbit for <math>f</math> is a sequence <math>\{ x_{n}:n\in Z\}</math> such that <math>dist(f(x_{n}),x_{n+1})<\delta\ ,</math> <math>n\in Z\ .</math> Such a pseudo-orbit is <math>\varepsilon</math> shadowed if there is <math>y\in M</math> such that <math>dist(x_{n},f^{n}(y))\leq \varepsilon ,n\in Z.</math>
Clearly b) implies a) since the semi-conjugacy <math>h</math> is surjective, but a) does not imply b). All three properties are invariant under conjugacy. Anosov diffeomorphisms satisfy b) ([W1]) and, since because of the classification theorem, every expansive homeomorphism of <math>T^{2}</math> is conjugate to an Anosov, then all expansive homeomorphisms of <math>T^{2}</math> sastisfy b). A pseudo-Anosov homeomorphism <math>f</math> satisfies a) (see [H]) but not b); because, according to [W2], for expansive systems b) is equivalent to c) and figure 3 shows an <math>f</math> pseudo-orbit shadowed by no <math>f</math>-trajectory; thus <math>f</math> does not satisfy c).
The quasi-Anosov diffeomorphisms are not even persistent. However each semi-trajectory is persistent : given <math>x\in M\ ,</math> and <math>\varepsilon >0</math> there is, <math>N_{x}\ ,</math> a <math>C^{0}</math>-neighbourhood of <math>f</math> such that for any <math>g\in N_x,</math> there is <math>y\in M\ ,</math> with the property <math>dist(f^{n}(x),g^{n}(y))\leq \varepsilon,\;\;n\geq 0.</math>
This is the <math>f</math> persistence of <math>x</math> in the future. We define similarly persistence in the past. A point <math>x</math> could be <math>f</math> persistent in the future and in the past without being persistent on both sides. This is the case of many points in a quasi-Anosov diffeomorphism. An open question is: are all the semi-trajectories of an expansive system persistent?
Let <math>M</math> be a compact boundaryless smooth manifold, and let <math>E</math> be the set of all expansive diffeomorphisms of <math>M\ .</math>
Theorem [Ma]. The <math>C^{1}</math>-interior of <math>E</math> is the set of quasi-Anosov diffeomorphisms of <math>M\ .</math>
On surfaces , quasi- Anosov diffeomorphisms are Anosov, and since in case <math>M</math> has genus larger than 1, <math>M</math> does not support Anosov diffeomorphisms, the interior mentioned in the theorem, is, in this case, void. Thus, there are expansive diffeomorphisms which are not approximated by Anosov . Consider now the case <math>M=T^{2}\ ,</math> where we do have Anosov diffeomorphisms. Since every expansive homeomorphism <math>f</math>is conjugate to a linear Anosov diffeomorphism <math>l\ ;</math> <math>f=hlh^{-1}</math> and according to [Mu] <math>h</math> may be <math>C^{0}</math>-approximated by a diffeomorphism <math>g</math> it follows easily, as <math>glg^{-1}</math> is Anosov, that <math>f</math> has arbitrarily <math>C^{0}</math>-close Anosov diffeomorphisms. However, it is not known, whether the <math>C^{1}</math>-closure of the <math>C^{1}</math>-interior of the expansive diffeomorphisms of <math>T^{2}</math> includes all the expansive diffeomorphisms of the 2-torus. In other words: Is every expansive diffeomorphism the <math>C^{1}</math>-limit of Anosov diffeomorphisms? On the other hand , according to the results in [K], it is possible to conclude that such an expansive diffeomorphism has a dense set of periodic hyperbolic points.
We consider flows with no equilibrium points. Such a flow <math>\varphi _{t}:M\rightarrow M,t\in R,</math> is expansive if there exist <math>\alpha ,\sigma >0,</math> such that if <math>x,y\in M,</math> and <math>dist(\varphi _{t}(x),\varphi _{\tau (t)}(y))\leq \alpha</math> for every <math>t\in R,</math> then <math>y=\varphi _{t_{0}}(x)</math> for some <math>t_{0},0\leq \left| t_{0}\right| \leq \sigma\ .</math> Here <math>\tau
- R\rightarrow R</math> is a re-parametrization of the flow through <math>y\ ,</math> i.e, a surjective homeomorphism with <math>\tau (0)=0\ .</math> This definition is somewhat more complicated than the one for discrete expansive systems as a consequence of the fact that we ask for geometric (instead of kinematic) separation. Important examples of expansive flows are geodesic flows on compact smooth Riemannian manifolds of negative curvature.
- R. Bowen, P. Walters. On expansive one-parameter flows. J. Diff Eq. 12(1972) 180-193
- M. Brunella. Expansive flows on Seifert manifolds and on Torus bundles.Bol. Soc. Brasil. Mat. (N.S.) 24(1993),89-104
- M. Brunella. Surfaces of section for expansive flows on three-manifolds.J.Math.Soc.Japan 47(1995), 491-501
- K. Moriyasu, K. Sakai, W. Sun. <math>C^{1}-</math>stably expansive flows. J. Differential Equations 213(2005) 352-367.
- J. Lewowicz. Lyapunov functions and Stability of Geodesic Flows. Springer Lecture Notes in Math. 1007(1981),463-480.
- M. Paternain. Expansive flows and the fundamental group. Bol.Soc.Brasil. Mat.(N.S.)24(1993), 179-199
- M. Paternain. Expansive geodesic flows on surfaces. Ergodic Theory Dynam. Systems 13(1993),153-165
- R. Ruggiero, V. Rosas. On the Pesin set of expansive geodesic flows in manifolds with no conjugate points Bol.Soc. Brasil. Mat. (N.S.)34(2003), 263-274
- R. Ruggiero. The accessibility property of expansive geodesic flows without conjugate points. Ergodic Theory Dynam. Systems 28(2008), 229-244.
This section refers to continuous maps <math>f</math> of a compact metric space <math>M</math> to itself that are not necessarily one-to-one. For those maps, a natural analogue to the notion of expansiveness is positive expansiveness.
A map <math>f</math> is positively expansive if <math>dist(f^{n}(x),f^{n}(y))\leq \alpha \; ;n\geq 0\ ,</math> implies <math>x=y\ .</math> A simple example of such a map is <math>f:S^{1}\rightarrow S^{1};\ ,</math> <math>f(z)=z^{n},</math> <math>n>1,</math> where <math>S^{1}</math> is the set of complex numbers <math>z</math> of modulus 1.
As in the preceding section we mention a short list of papers concerning, mainly, positively expansive maps.
- E.Coven and W. Reddy. Positively expansive maps on compact manifolds. Lecture notes in Math 819, Springer Verlag, 1980, 96-110
- K. Hiraide. Positively expansive open maps of Peano spaces, Topology and its Appl. 37 (1990), 213-220
- K. Hiraide. Nonexistence of positively expansive maps on compact connected manifolds with boundary, Proc. Amer. Math.Soc. 110 (1990), 565-568
- M.Nasu. Endomorphisms of Expansive systems on compact metric spaces and the pseudo-orbit tracing property. Trans. of the Am. Math Soc. 352(2000),10, 4731-4757
- W. Reddy. Expanding maps on compact metric spaces. Toplogy and its Appl. 13 (1982) 327-334
- D. Richeson and J. Wiseman. Positively expansive dynamical systems. Topology and its Appl. 154(3), (2007), 604-613
- M.Shub. Endomorphisms of compact differentiable manifolds. Amer. J. Math 91 (1969), 175-199.
[ABP] A. Artigue, J. Brum, R. Potrie. Local product structure for expansive homeomorphisms. Toplogy and its Applications, (2008) (To appear).
[B] J.L. Borges. Tigres azules. Obras Completas (3). Emece Editores (1989), 381-388
[FLP}] A. Fathi, F. Laudenbach, V. Poenaru. Travaux de Thurston sur les surfaces. Asterisque (1979)66-67
[F] J. Franks. Anosov Diffeomorphisms.Proceedings of the Symposium in pure mathematics. 14(1970), 61-94
[F,R}] J. Franks, C. Robinson. A quasi-Anosov diffeomorphism that is not Anosov. Trans. Am. Math.Soc. 283(1976), 267-278.
[H] M. Handel. Global Shadowing of pseudo-Anosov diffeomorphisms. Ergodic Theory Dynam. Systems 5(1985)373.377
[Hi] K. Hiraide. Expansive diffeomorphisms of surfaces are pseudo-Anosov. OsakaJ. Math.27(1990), 117-162.
[K] A. Katok. Lyapunov exponents, entropy and periodic orbits of diffeomorphisms. Publ. Marh. IHES 51 (1980).
[L1] J. Lewowicz. Lyapunov functions and Topological Stability. Journal of Diff. Equations. 38(1980) 192-209.
[L2] J. Lewowicz. Persistence in expansive systems. Ergodic Theory Dynam. Systems 3(1983), 567-578.
[L3] J. Lewowicz. Expansive Homeomorphisms of surfaces.Bol. Soc. Bras. Math. 20(1989), 113-133.
[Ma] R. Mañe. Expansive Diffeomorphisms. Lecture Notes in Math.468 (1975), 162-174
[Mu] J. Munkres. Obstructions to the smoothing of piece-wise differentiable homeomorphisms. Ann. of Math. 72(3)(1960), 521-554
[T] W. Thurston. On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Am. Math. Soc. 19 (1988), 417-431
[U] W. Utz. Unstable homeomorphisms. Proc. Am. Math. Soc. 1(1950), 769-774
[V1] J. Vieitez.Three Dimensional expansive homeomorphisms. Pitman Research Notes in Math.285(1993),299-323.
[V2] J. Vieitez. Expansive homeomorphisms and hyperbolic diffeomorphisms on three manifolds. Ergodic Theory Dynam. Systems 16(1996), 591-622.
[W1] P. Walters. Anosov diffeomorphisms are topologically stable.Topology 9(1970), 71-78
[W2] P. Walters. On the pseudo-orbit tracing property and its relation to stability. Lecture Notes in Math. 668 (1978), 231-244.
Internal references
- John W. Milnor (2006) Attractor. Scholarpedia, 1(11):1815.
- Yuri A. Kuznetsov (2007) Conjugate maps. Scholarpedia, 2(12):5420.
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
- Boris Hasselblatt and Yakov Pesin (2008) Hyperbolic dynamics. Scholarpedia, 3(6):2208.
- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
[M] J. Milnor. Attractor. Scholarpedia 1(11):1815 (2006),1-9.
- A. Katok, B. Hasselblatt. Introduction to the Modern theory of Dynamical Systems. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995. ISBN: 0-521-34187-6
- J. Milnor. Dynamical Lectures. (These are not finished notes)