Any Continuous Map is Homotopic to a Constant Map

Continuous Map

One defines a continuous map from A to Dp by describing each point in A in terms of the amount of time its trajectory spends in the neighborhood of the pth Morse set and its value under the abstract Lyapunov function of Theorem 2.11.

From: Handbook of Dynamical Systems , 2002

REVIEW OF FUNDAMENTAL NOTIONS OF ANALYSIS

YVONNE CHOQUET-BRUHAT , CÉCILE DEWITT-MORETTE , in Analysis, Manifolds and Physics, 2000

Two continuous maps f and f ₁, between topological spaces X and X′ are homotopic if they can be deformed into each other, that is, more precisely, if there is a continuous map F: X × I → X′, I = [0, 1], such that F(., 0) = f, F(., 1) = f ₁ (cf. the case X = [0,1] p. 20, homotopic paths). It is straightforward to check that this relation is reflexive, symmetric and transitive, and hence is an equivalence relation, which we denote f ⋍ f ₁.

homotopic

1)

Show that if X is homeomorphic to X ₁ and X′ to X ^′ ₁, then there is a bijective correspondence between the homotopy classes of maps X → X′ and X ₁ → X′ ₁.

2)

Let φ be an homeomorphism of X homotopic to the identity mapping. Show that the mappings f: X → X′ and f о φ are in the same homotopy class.

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The Stable Homotopy Theory of Finite Complexes

Douglas C. Ravenel , in Handbook of Algebraic Topology, 1995

2.1 Homotopy

A basic problem in homotopy theory is to classify continuous maps up to homotopy. Two continuous maps from a topological space X to Y are homotopic if one can be continuously deformed into the other. A more precise definition is the following.

DEFINITION 2.1.1

Two continuous maps f ₀ and f ₁ from X to Y are homotopic if there is a continuous map (called a homotopy)

$X\times \left[0,1\right]\stackrel{h}{\to }Y$

such that for t = 0 or 1, the restriction of h to X × {t} is f_t . If f ₁ is a constant map, i.e. one that sends all of X to a single point in Y, then we say that f ₀ is null homotopic and that h is a null homotopy. A map which is not homotopic to a constant map is essential. The set of homotopy classes of maps from X to Y is denoted by [X, Y].

For technical reasons it is often convenient to consider maps which send a specified point x ₀ ∈ X (called the base point) to a given point y ₀ ∈ Y, and to require that homotopies between such maps send all of {x ₀} × [0, 1] to y ₀. Such maps and homotopies are said to be base point preserving. The set of equivalence classes of such maps (under base point preserving homotopies) is denoted by [(X, x ₀), (Y, y ₀)].

Under mild hypotheses (needed to exclude pathological cases), if X and Y are both path-connected and Y is simply connected, the sets [X, Y] and [(X, x ₀), (Y, y ₀)] are naturally isomorphic.

In many cases, e.g., when X and Y are compact manifolds or algebraic varieties over the real or complex numbers, this set is countable. In certain cases, such as when Y is a topological group, it has a natural group structure. This is also the case when X is a suspension (2.3.1 and 3.1.2).

In topology two spaces are considered identical if there is a homeomorphism (a continuous map which is one-to-one and onto and which has a continuous inverse) between them. A homotopy theorist is less discriminating than a point set topologist; two spaces are identical in his eyes if they satisfy a much weaker equivalence relation defined as follows.

DEFINITION 2.1.2

Two spaces X and Y are homotopy equivalent if there are continuous maps f: X → Y and g: Y → X such that g f and f g are homotopic to the identity maps on X and Y. The maps f and g are homotopy equivalences. A space that is homotopy equivalent to a single point is contractible. Spaces which are homotopy equivalent have the same homotopy type.

For example, every real vector space is contractible and a solid torus is homotopy equivalent to a circle.

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Topology, General

Taqdir Husain , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

V.G Connected and Locally Connected Spaces

For the definition of these spaces, see Section I.A. In addition to the properties of these spaces given in that section, we have the following.

A continuous map of a topological space into another carries connected sets into connected sets. A product of connected spaces is connected. If A is a connected subset of a topological space and B is any set such that $\text{A}\subset \text{B}\subset \overline{\text{A}}$ , then B is connected. Any interval of $\mathbb{R}$ , including $\mathbb{R}$ itself, is connected. Each convex subset of a real topological vector space (see Section VIII.B) is connected. An interesting characterization of connected spaces is: A topological space (X, T) is connected if and only if every continuous map f of X into a discrete space Y is constant, i.e., f(x)   =   {y}, y  ∈ Y.

A characterization of locally connected spaces is: A topological space is locally connected if and only if the components of its open sets are open. Local connectedness is preserved under closed maps.

A connected, metrizable compactum (X, T) is sometimes called a continuum. Let (X, T) be a continuum and (Y, T′) a Hausdorff space. If f: (X, T)   →   (Y, T′) is a continuous surjective map, then (Y, T′) is also a continuum. Further, if (Y, T′) is locally connected, so is (Y, T′).

A topological space each of whose components consists of a singleton is called totally disconnected. For example, the set of integers is totally disconnected.

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Ważewski Method and Conley Index

Roman Srzednicki , in Handbook of Differential Equations: Ordinary Differential Equations, 2004

Definition 1.1

A continuous map r : X → A such that r (a) = a for every a ∈ A is called a retraction. A is called a retract of X if there exists a retraction X → A. A continuous map r : X → A is called a strong deformation retraction if it is equal to the map $x\mapsto h(x,1)$ , where h : X × [0, 1] → X is a continuous map such that h (x, 0) = x and h (x, 1) ∈ A for every x ∈ X, and h (a, t) = a for every a ∈ A and t ∈ [0, 1]. A is called a strong deformation retract of X if there exists a strong deformation retraction X → A.

Obviously, a strong deformation retract is also a retract and a retract of a connected set is also connected. Moreover, it is well known that the Brouwer fixed point theorem is equivalent to the fact that the unit sphere S^{n −1} in ${ℝ}^n$ is not a retract of the closed unit ball Bⁿ.

In 1947, Ważewski presented the paper [85] in which he proved a theorem on the existence of solutions which are contained in given set for all positive values of time. In the contemporary used mathematical notation the theorem can be described as follows. Assume that v is a continuous vector-field on M, where M is an open subset of ${ℝ}^n$ (it can be a smooth manifold without boundary as well). We refer to M as to the phase-space. It is assumed that through each point x ₀ of M passes a unique saturated solution t ↦ ϕ(x ₀, t) of the Cauchy problem

The map $\varphi :(x,t)\mapsto \varphi (x,t)$ is a called local flow; its purely topological description is provided in Definition 2.1. Let us mention here that usually ϕ (x, t) is not defined for all t and by the trajectory (respectively, the positive semitrajectory, the negative semitrajectory) of x we mean the set of all points ϕ (x, t) (respectively, the set of all points ϕ (x, t) with t ⩾ 0, all points ϕ (x, t) with t ⩽ 0) whenever they are defined.

Let V be an open subset of the phase space. Let x ∈ ∂V. Some types of behavior of the trajectory of x with respect to V are described in the following

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Stable Homotopy and Iterated Loop Spaces

Gunnar Carlsson , R. James Milgram , in Handbook of Algebraic Topology, 1995

2.3 Serre fibrations

A Serre fibration has the same definition as an Hurewicz fibration except the spaces X and Y are restricted to being finite polyhedral complexes. These are particularly useful when we are dealing with mapping spaces X^Y = {f : Y → X | f continuous} which are assumed, as in 2.2.2, to have the compact-open topology.

Given any continuous map f : Y → X we have the associated Serre fibration $E_{Y,\overline{X}}^{\overline{X}}\stackrel{\pi }{\to }X$ where $\overline{X}$ the mapping cone of f, $E_{A,B}^C$ is the space of paths in C that start in A, end in B and $\pi :E_{A,B}^C\to B$ is projection onto the endpoint. The fiber of π over the point x is the subspace $E_{Y,x}^{\overline{X}}$ and we have the commutative diagram

(2.1).

where i includes y ε Y as the constant path at y.

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Analytical Aspects of Liouville-Type Equations with Singular Sources

Gabriella Tarantello , in Handbook of Differential Equations: Stationary Partial Differential Equations, 2004

Theorem 5.2.1

Assume that M admits positive genus, and let W satisfy (5.2.1) with α_j > 0, j = 1, …, m, and V as in (5.2.2). Then, for any λ ∈ (8π, 16π) \ {8π(1 + α _j ), j = 1, …, m}, problem (5.1.7) admits a solution.

Proof

We shall use the variational formulation of problem (5.1.7) and, under the given assumptions, proceed to construct a critical point for the functional

(5.2.3) $\begin{array}{cc}{I}_{\text{λ}}(v)=\frac{1}{2}\left|\right|{\nabla }_{g}v|{|}_2^2-\text{λ}\log {\displaystyle {\int }_{M}W{\text{e}}^v}\text{d}{\tau }_g& \text{in}E.\end{array}$

Note that by our assumption on W, we have log W ∈ L ¹(M). Without loss of generality, also we may assume

(5.2.4) ${\displaystyle {\int }_M\log \left(\left|M\right|W\right)\text{d}{\tau }_g}=0$

since problem (5.1.7) remains unchanged if we replace W with μW, μ > 0.

Note that (5.2.4), together with Jensen's inequality (1.2.5), implies log ∫ _MW e ^v ) ⩾ 0.

Hence, we deduce the following monotonicity property for I_λ

(5.2.5) $\begin{array}{cccc}{\text{λ}}_1⩽{\text{λ}}_2& \text{implies}& I_{{\text{λ}}_1}(v)⩾I_{{\text{λ}}_2}(v)& \forall v\in E.\end{array}$

Let X: M $\mapsto$ ℝ ^N be the embedding map of M into ℝ ^N for some N ⩾ 3, and Γ₁ ⊂ M \ {p ₁, …, p_m } be a regular simple closed curve such that X(Γ₁) links with a closed curve Γ₂ ⊂ ℝ ^N \ X(M).

The existence of Γ₁ and Γ₂, is deduced from the assumption that M admits positive genus.

For any v ∈ E, let

$m(v)=\frac{{\displaystyle {\int }_{M}X{\text{e}}^v}\text{d}{\tau }_g}{{\displaystyle {\int }_M{\text{e}}^v}\text{d}{\tau }_g}\in {ℝ}^N$

define the center of mass of v.

Consider the set of continuous map

$h:B_1(0)\to E$

with the following properties:

(i)

I_λ (h(z)) → −∞, as |z| → 1⁻;

(ii)

m(e ^iθ): = lim_ρ→1−m(h(ρe ^iθ)) defines a continuous map from ∂B ₁(0) into X(Γ₁) with nonzero topological degree.

Denote by D_λ the set of such maps.

Claim 1

If λ > 8π, then D_λ is not empty.

To establish Claim 1, let p ∈ M \ {p ₁, …, p_m }, so that W(p) > 0. In a small neighborhood of p introduce the function U _ε,p that, in an isothermal coordinate system y centered at p, is expressed as follows:

$\begin{array}{cc}{U}_{\epsilon ,p}(y)=\log \frac{\epsilon }{{(\epsilon +W(p)\pi |y{|}^2)}^2},& \epsilon >0,y\in B_{r_0}(0)\end{array}$

for suitable r ₀ > 0 sufficiently small.

Consider v _ε,p ∈ E the unique solution for the problem:

$\{\begin{array}{l}\begin{array}{ll}-{\Delta }_g{v}_{\epsilon ,p}=8\pi \left(\frac{\eta W{\text{e}}^{U_{\epsilon ,p}}}{{\displaystyle {\int }_M\eta W{\text{e}}^{U_{\epsilon ,p}}}}-\frac{1}{\left|M\right|}\right)\hfill & \text{in}M,\hfill \end{array}\hfill \\ {\displaystyle {\int }_M{v}_{\epsilon ,p}}\text{d}{\tau }_g=0,\hfill \end{array}$

where η denotes a standard cut-off function supported in a small neighborhood of p, where U _ε,p is defined.

Clearly v _ε,p depends continuously on ε > 0 and p ∈ M \ {p ₁, …, p_m }. Moreover, by Green's representation formula, it is possible to show that, as ε → 0, we have:

(5.2.6) $\left|\right|{\nabla }_g{v}_{\epsilon ,p}|{|}_2^2=32\pi \log \frac{1}{\epsilon }+\text{O(1),}$

(5.2.7) $\log {\displaystyle {\int }_{M}W{\text{e}}^{v_{\epsilon ,p}}}\text{d}{\tau }_g=2\log \frac{1}{\epsilon }+\text{O(1),}$

(5.2.8) $\begin{array}{cc}\frac{{\text{e}}^{v_{\epsilon ,p}}}{{\displaystyle {\int }_M{\text{e}}^{v_{\epsilon ,p}}}}\to {\delta }_p,& \text{weakly}\text{in}\text{the}\text{sense}\text{of}\text{measure}\end{array}$

(e.g., see the proof of Proposition 3.1 in [NT2] for details). Denote by p = p(θ), θ ∈ [0, 2π) a regular, simple parametrization of Γ₁. In view of (5.2.6)–(5.2.8), it is possible to check that when λ > 8π, the function h(ρe ^iθ) = v _1−ρ,p(θ) with ρ ∈ (0, 1) and θ ∈ [0, 2π) satisfies properties (i) and (ii) above, and therefore belongs to ${\mathcal{D}}_{\text{λ}}$ .

Hence, for λ > 8π, the value

(5.2.9) $c_{\text{λ}}=\underset{h\in {\mathcal{D}}_{\text{λ}}}{\inf }\underset{z\in B_1}{\sup }{I}_{\text{λ}}\left(h(z)\right)$

is well defined.

Claim 2

(5.2.10) $\text{If}\text{λ}\in (8\pi ,16\pi ),\text{then}{c}_{\text{λ}}>-\infty .$

If λ ∈ (8π, 16π), then c_λ > − ∞.

The proof of (5.2.10) relies in an essential way upon the improved version of the Moser–Trudinger inequality (5.1.9), the relative proof of which may be found for instance in [CL3] and [Au].

Lemma 5.2.2

Let S ₁, S ₂ be subsets of M satisfying dist (S ₁, S ₂) ⩾ δ₀ > 0. Given γ₀ ∈ (0, 1/2) and ε > 0, there exists a constant c = c(δ₀, γ₀, ε) such that, for all v ∈ E, verifying

(5.2.11) $\begin{array}{cc}\frac{{\displaystyle {\int }_{S_j}{\text{e}}^v}\text{d}{\tau }_g}{{\displaystyle {\int }_M{\text{e}}^v}\text{d}{\tau }_g}⩾{\gamma }_0,& j=1,2,\end{array}$

we have

${\displaystyle {\int }_M{\text{e}}^u}\text{d}{\tau }_g⩽c{\text{e}}^{\frac{1}{32\pi -\epsilon }\left|\right|\nabla v|{|}_{L^2(M)}^2}.$

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Modern General Topology

In North-Holland Mathematical Library, 1985

Proof

Let f be a closed continuous map from a metric space M onto X. Then by H) there is a closed set F of M such that f(F) = X, and f is irreducible on F. Let G be an arbitrary closed set of X. Then by J)f ^-1(G)∩F has a closure-preserving open outer base $\mathcal{U}$ in F. Hence by I) $\left\{X-f\left(F-U\right)|U\in \mathcal{U}\right\}$ is a closure-preserving open outer base of G in X. Thus by VI.8.I) X is M ₁. It is also easy to see that every subspace of X is M ₁.

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Complex Analysis

Joseph P.S. Kung , Chung-Chun Yang , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

I.C Curves

A curve γ is a continuous map from a real interval [ a, b] to C. The curve γ is said to be closed if γ(a)   =   γ(b); it is said to be open otherwise. A simple closed curve is a closed curve with γ(t ₁)   =   γ(t ₂) if and only if t ₁  = a and t ₂  = b.

An intuitively obvious theorem about curves that turned out to be very difficult to prove is the Jordan curve theorem. This theorem is usually not necessary in complex analysis, but is useful as background.

The Jordan Curve Theorem. The image of a simple closed curve (not assumed to be differentiable) separates the extended complex plane into two regions. One region is bounded (and "inside" the curve) and the other is unbounded.

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Handbook of Dynamical Systems

G. Raugel , in Handbook of Dynamical Systems, 2002

Definition

Let S be a semigroup of continuous maps from X into X. A closed invariant subset A of X is said to be invariantly connected if it cannot be represented as the union of two nonempty, disjoint, closed, positively invariant sets.

The positive orbit of x ∈ X is the set ${\gamma }^{+}\left(x\right)=\left\{S\left(t\right)x|t\in G^{+}\right\}$ . If E ⊂ X, the positive orbit of E is the set

${\gamma }^{+}(E)=\bigcup _{t\in G^{+}}S(t)E=\bigcup _{z\in E}{\gamma }^{+}(z).$

More generally, for τ ∈ G ⁺, we define the orbit after the time τ of E by

${\gamma }_{\tau }^{+}(E)={\gamma }^{+}\left(S(\tau )E\right).$

Let now I be an interval of R and S(t), t ⩾ 0, a semigroup. We recall that a mapping u from I into X is a trajectory (or orbit) of S(t) on I if u(t + s) = S(t)u(s), for any s ∈ I and t ⩾ 0 such that t + s ∈ I. In particular, if I = (−∞, 0] and u(0) = z ∈ X, u is called a negative orbit through z and is often denoted by γ⁻(z). If I = R and u(0) = z, then u is called a complete orbit through z and is often denoted by γ(z). We let Γ⁻(z) be the set of all negative orbits through z. If Γ⁻(z) is not empty, it may contain more than one negative orbit, because we have not assumed the property of backward uniqueness. We also let Γ(z) = Γ⁻(z)∪ γ⁺(z) be the set of all complete orbits through z. In the same way, we define the sets γ⁻(E), Γ⁻(E) and Γ(E), for any subset E of X. For later use, for any z ∈ X, we introduce the following set:

$H(t,z)=\left\{y\in X|\text{there}\text{exists}\text{a}\text{negative}\text{orbit}{u}_z\text{through}z\text{such}\text{that}{u}_z(0)=z\text{and}{u}_z(-t)=y\right\}.$

We remark that ${\Gamma }^{-}\left(z\right)={\bigcup }_{t⩾0}H\left(t,z\right)$ . Likewise, if E ⊂ X, we define the set $H\left(t,E\right)={\bigcup }_{z\in E}H\left(t,z\right)$ and remark that ${\Gamma }^{-}\left(E\right)={\bigcup }_{t⩾0}H\left(t,E\right)$ .

In a similar way, replacing (−∞, 0] (respectively R) by (−∞, 0] ∩ ℤ (respectively ℤ), we define the negative and complete orbits of maps S. In the framework of maps, it is very easy to give examples of non backward uniqueness. Consider the non injective logistic map S:[0, 1] → [0, 1], Sx = λx(1 − x), with 2 < λ ⩽ 4. The point x ₀ = (λ − 1)/λ. is a fixed point of S and the point y = λ⁻¹ satisfies Sy = x ₀. The iterates S⁻ⁿy ∈ (0, x ₀) are well defined and y(y) = {S⁻ⁿy | n = 0, 1, 2....} U {x ₀} is a complete orbit trough x ₀.

The proof of the following lemma is elementary.

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Handbook of Dynamical Systems

Yuri Kifer , Pei-Dong Liu , in Handbook of Dynamical Systems, 2006

Theorem 6.1.5

Let F be a RDS with F_ω being continuous maps of ℝ and assume that ϑⁿ is ergodic for all n ∈ ℕ. Let $\left\{x_1\left(\omega \right),\dots ,x_N\left(\omega \right)\right\}$ be a random periodic cycle of minimal period N, where x ₁ < · · · < x_N ℙ-a.s. Then for any $l⊲N$ , where ⊳ denotes the Sharkovsky ordering, F has a random periodic orbit of minimal period l or 2l. Let ϑ be weakly mixing and assume that F has a random periodic orbit of minimal period three. Then it also has a random periodic orbit of minimal period l or 2l for all l ∈ ℕ.

Periodic orbits are also somewhat involved in the proof of the thermodynamic formalism constructions for β-transformations in [160] and by this reason a corresponding extension to the random β-transformations encounters difficulties and has not be done yet (see the corresponding discussion in Section 4 of [100]). Observe that existence of random absolutely continuous invariant measures for such transformations was derived in [41] and [40] shows that they have an exponential decay of correlations property.

Among other interesting questions which would be important to understand is the meaning of the fiber (relative) spectrum for random transformations. This could be important both for an extension of the classical spectral theory of dynamical systems to the random setup and, for instance, for a characterization of random Anosov diffeomorphisms via an appropriate gap in a random version of Mather's spectrum (see [127]). Observe that (fiber) relative discrete spectrum is easier to define and some results based on it appeared already in the literature (see [114] and references therein).

We mention also the basic problem of representations of Markov chains by appropriate classes of random transformations discussed in Section 1.1 of [82] which asks about conditions on transition probabilities P (x, ·) of a Markov chain X_n which enable us to find a probability measure μ on a nice family of transformations Φ (homeomorphisms, smooth maps, diffeomorphisms, automorphisms, linear transformations, etc.) such that $P\left(x,\Gamma \right)=\mu \left\{F\in \Phi :Fx\in \Gamma \right\}$ . Then X_n can be viewed as a composition of independent identically distributed random maps with the distribution μ. Apart from a theoretical interest of this problem such representations can be useful in proving various results about Markov chains. For instance, an appropriate representation of this sort has been recently employed in [25] in the study of random perturbations of the Hènnon-like maps. In spite of some progress in this problem achieved in [139] and [140] there are no sufficiently general results about such representations by families of maps mentioned above and a further research on this problem is needed.

Discrete time random hyperbolic dynamical systems are still in a relatively good shape in comparison to stochastic flows where in view of noninvariance of the flow direction no substantial theory of hyperbolic RDS has been developed as yet which remains an important standing problem.

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