Cofinite Topology on the Natural Numbers Continuous Function

Finitely Correlated States

R.F. Werner , in Encyclopedia of Mathematical Physics, 2006

Density of Finitely Correlated Pure States

The natural topology in which to consider the approximation between states on the chain is the weak topology. A sequence ω _n converges weakly to ω if for all local A the expectations converge, that is, ω _n (A) → ω (A).

Let us start from an arbitrary translationally invariant state ω, and see how we can approximate it. First, we can split the chains into intervals of length L, and replace ω by the tensor product of the restrictions of ω to each of these intervals. This state is not translationally invariant, so we average it over the L translations, and call the resulting state ω _L . Consider a local observable A, whose localization region has length R. Then for L − R out of the L translates contributing to ω _L the expectation will be the same as for ω, and we get

${\omega }_L\left(A\right)=\left(1-\frac{R}{L}\right)\omega \left(A\right)+\frac{R}{L}\tilde{\omega }\left(A\right),$

where the error term $\tilde{\omega }$ is again a state. Hence, ω _L converges weakly to ω as L → ∞. One can show easily that ω _L is finitely correlated, with an algebra $\mathcal{B}$ essentially equal to ${\mathcal{A}}^{\otimes L}$ . Hence, the finitely correlated states are weakly dense in the set of translationally invariant states.

We can make the approximating states purer by a very simple trick. In the previous construction we always take two intervals together, and replace the tensor product of the two restrictions by a purification, that is, by a pure state on an interval of length 2L, whose restrictions to the two length-L subintervals coincide with ω. We average this over 2L translates, and call the result η _L . The estimates showing that η _L → ω weakly are exactly the same as before. Moreover, one can show (Fannes et al. 1992a) that η _L is purely generated.

Being defined as a convex combination of other states, η _L is not pure, and the peripheral spectrum of $\widehat{\mathbb{E}}$ will contain all the 2Lth roots of unity. However, we can use that such a rich peripheral spectrum is not generic for $\widehat{\mathbb{E}}$ constructed from an isometry V. Therefore, if we choose an isometry V _ε close to the isometry V generating η _L , we obtain a purely generated state ${\eta }_L^{\epsilon }$ with trivial peripheral spectrum. Since the expression for expectations of such states depends continuously on the generating isometry, we have that ${\eta }_L^{\epsilon }\to {\eta }_L$ as ε → 0. But we know from the previous section that such states are pure. Hence, the pure finitely correlated states are weakly dense in the set of all translationally invariant states (Fannes et al. 1992a).

This has implications for the geometry of the compact convex set of translationally invariant states, which are rather counter-intuitive for the intuitions trained on finite-dimensional convex bodies. To begin with, the extreme points (the ergodic states) are dense in the whole body. This is not such a rare occurrence in infinite-dimensional convex sets, and is shared, for example, by the set of operators F with $0\le F\le \mathbb{1}$ on an infinite-dimensional Hilbert space (Davies 1976). Together with the property that the translationally invariant states form a simplex, it actually fixes the structure of this compact convex set to be the so-called Poulsen simplex. This was known also without looking at finitely correlated states. The rather surprising result of the above density argument is that even the small subclass of states which are extremal, not only in the translationally invariant subset but even in the whole state space, is still dense.

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Topological structure on quotient space

In From Dimension-Free Matrix Theory to Cross-Dimensional Dynamic Systems, 2019

5.1.1 Product topology on quotient subset

First, we consider the partition (4.32), it is natural to assume that each ${\mathcal{M}}_{\mu }$ is a clopen subset in $\mathcal{M}$ , because distinct μ's correspond to distinct shapes of matrices. Now inside each ${\mathcal{M}}_{\mu }$ we assume ${\mu }_y,{\mu }_x\in \mathbb{N}$ are co-prime and ${\mu }_y/{\mu }_x=\mu$ . Then

${\mathcal{M}}_{\mu }=\bigcup _{i=1}^{\infty }{\mathcal{M}}_{\mu }^i,$

where

${\mathcal{M}}_{\mu }^i={\mathcal{M}}_{i{\mu }_y\times i{\mu }_x},i=1,2,\cdots .$

Because of the similar reason, we also assume each ${\mathcal{M}}_{\mu }^i$ is clopen.

Overall, we have a set structure on $\mathcal{M}$ as

(5.1) $\mathcal{M}=\bigcup _{\mu \in {\mathbb{Q}}_{+}}\bigcup _{i=1}^{\infty }{\mathcal{M}}_{\mu }^i.$

It is standard to identify each ${\mathcal{M}}_{\mu }^i$ with ${\mathbb{R}}^{i^2{\mu }_y{\mu }_x}$ . Hence the overall topology of $\mathcal{M}$ can be described as follows:

Definition 5.1

A natural topology on $\mathcal{M}$ , denoted by ${\mathcal{T}}_{\mathcal{M}}$ , consists of

1.

a partition of countable clopen subsets ${\mathcal{M}}_{\mu }^i$ , $\mu \in {\mathbb{Q}}_{+}$ , $i\in \mathbb{N}$ ;

2.

the conventional Euclidean ${\mathbb{R}}^{i^2{\mu }_y{\mu }_x}$ topology for ${\mathcal{M}}_{\mu }^i$ .

Next, we consider the quotient spaces

${\Sigma }_M:=\mathcal{M}/\sim ;{\Sigma }_{\mu }={\mathcal{M}}_{\mu }/\sim .$

It is clear that

(5.2) $\Sigma =\bigcup _{\mu \in {\mathbb{Q}}_{+}}{\Sigma }_{\mu }.$

Note that (5.2) is also a partition. Moreover, if $A\in {\mathcal{M}}_{{\mu }_1}$ , $B\in {\mathcal{M}}_{{\mu }_2}$ , and ${\mu }_1\ne {\mu }_2$ , then $A\nsim B$ , i.e., they are not equivalent. Hence each ${\Sigma }_{\mu }$ can be considered as a clopen subset in ${\Sigma }_M$ . We are, therefore, interested only in constructing a topology on each ${\Sigma }_{\mu }$ .

Definition 5.2

1.

Consider ${\mathcal{M}}_{\mu }^i$ as an Euclidean space ${\mathbb{R}}^{i^2{\mu }_y{\mu }_x}$ with conventional Euclidean topology. Assume $o_i\ne \varnothing$ is an open set. Define a subset $〈o_i〉\subset {\Sigma }_{\mu }$ as follows:

(5.3) $〈A〉\in 〈o_i〉\iff 〈A〉\cap o_i\ne \varnothing .$

2.

Let

$O_i=\{o_i|o_i\text{is an open ball in}{\mathcal{M}}_{\mu }^i\text{ with rational center and rational radius}\}.$

3.

Using $O_i$ , we construct a set of subsets $S_i\subset 2^{{\Sigma }_{\mu }}$ as

$S_i:=〈O_i〉,i=1,2,\cdots .$

Taking $S={\cup }_{i=1}^{\infty }{S}_i$ as a topological subbase, the topology generated by S is denoted by ${\mathcal{T}}_P$ , which makes

$({\Sigma }_{\mu },{\mathcal{T}}_P)$

a topological space. (We refer to the appendix or [7] for a topology generated by a subbase.)

Note that the topological basis consists of the sets of finite intersections of $s_i=〈o_i〉\in S_i$ .

Remark 5.1

1.

It is clear that ${\mathcal{T}}_P$ makes $({\Sigma }_{\mu },{\mathcal{T}}_P)$ a topological space.

2.

The topological basis is

(5.4) $\mathcal{B}:=\{〈o_{i_1}〉\cap 〈o_{i_2}〉\cap \cdots \cap 〈o_{i_r}〉|〈o_{i_j}〉\in S_{i_j}j=1,\cdots ,r;r<\infty \}.$

3.

Fig. 5.1 depicts an element in the topological basis. Here $o_1\in {\mathcal{M}}_{\mu }^i$ , $o_2\in {{\mathcal{M}}_{\mu }}^j$ are two open discs with rational center and rational radius. Then $s_1=〈o_1〉$ and $s_2=〈o_2〉$ are two elements in the subbase, and

$s_1\cap s_2=\left\{〈A〉\right|〈A〉\cap o_i\ne \varnothing ,i=1,2\}$

is an element in the basis.

Figure 5.1. An element in topological basis

Theorem 5.1

The topological space $({\Sigma }_{\mu },{\mathcal{T}}_P)$ is a second countable, Hausdorff (or $T_2$ ) space.

Proof

To see $({\Sigma }_{\mu },\mathcal{T})$ is second countable, it is easy to see that $O_i$ is countable. Then $\{O_i|i=1,2,\cdots \}$ , as a countable union of countable set, is countable. Finally, $\mathcal{B}$ , as the finite subset of a countable set, is countable.

Next, consider $〈A〉\ne 〈B〉\in {\Sigma }_{\mu }$ . Let $A_1\in 〈A〉$ and $B_1\in 〈B〉$ be their irreducible elements respectively. If $A_1,B_1\in {\mathcal{M}}_{\mu }^i$ for the same i, then we can find two open sets $\varnothing \ne o_a,o_b\subset {\mathcal{M}}_{\mu }^i$ , $o_a\cap o_b=\varnothing$ , such that $A_1\in o_a$ and $B_1\in o_b$ . Then by definition, $s_a(o_a)\cap s_b(o_b)=\varnothing$ and $〈A〉\in s_a$ , $〈B〉\in s_b$ .

Finally, assume $A_1\in {\mathcal{M}}_{\mu }^i$ , $B_1\in {\mathcal{M}}_{\mu }^j$ and $i\ne j$ . Let $t=\mathrm{lcm}(i,j)$ . Then

$A_{t/i}=A_1\otimes I_{t/i}\in {\mathcal{M}}_{\mu }^t,B_{t/j}=B_1\otimes I_{t/j}\in {\mathcal{M}}_{\mu }^t.$

Since $A_{t/i}\ne B_{t/j}$ , we can find $o_a,o_b\subset {\mathcal{M}}_{\mu }^t$ , $o_a\cap o_b=\varnothing$ , and $A_{t/i}\in o_a$ and $B_{t/j}\in o_b$ . That is, $s_a(o_a)$ and $s_b(o_b)$ separate $〈A〉$ and $〈B〉$ .   □

Since $s_i(o_i)=〈o_i〉\subset {\Sigma }_{\mu }$ is completely determined by $o_i$ , we can simply identify $s_i(o_i)$ with $O_i$ . Then it is clear that the topology ${\mathcal{T}}_P$ is completely determined by its sub-basis

$\{o_i|o_i\in {\mathcal{T}}_i,i=1,2,\cdots \},$

where ${\mathcal{T}}_i$ is the classical ${\mathbb{R}}^{i^2{\mu }_y{\mu }_x}$ topology. Then it is clear that ${\mathcal{T}}_P$ is the product topology of the product of ${\mathcal{M}}_{\mu }^i$ , $i=1,2,\cdots$ . Hence ${\mathcal{T}}_P$ is called the product topology on ${\Sigma }_{\mu }$ . Then each ${\Sigma }_{\mu }$ , $\mu \in {\mathbb{Q}}_{+}$ is considered as a clopen set of Σ. We still simply call such a topology on Σ the product topology and still use ${\mathcal{T}}_P$ to denote it.

Recall the natural projection $\mathrm{Pr}:\mathcal{M}\to \Sigma$ as

(5.5) $\mathrm{Pr}(A):=〈A〉,A\in \mathcal{M}.$

Particularly, we are interested in each clopen subsets ${\mathcal{M}}_{\mu }$ . That is,

(5.6) $\mathrm{Pr}{|}_{{\mathcal{M}}_{\mu }}(A):=〈A〉\in {\Sigma }_{\mu },A\in {\mathcal{M}}_{\mu }.$

Then the product topology may be described by Fig. 5.2, where a parallel structure of the projection is presented.

Figure 5.2. Parallel structure of projection

Proposition 5.1

$\mathit{Pr}:{\mathcal{M}}_{\mu }\to ({\Sigma }_{\mu },{\mathcal{T}}_P)$ is not continuous. Hence, ${\mathcal{T}}_P$ is not the quotient topology.

Proof

Let O be an open set in ${\mathcal{M}}_{\mu }^1$ . Define

$S(O):=\{〈A〉\in \Sigma |〈A〉\cap O\ne \varnothing \}.$

Then by definition $S(O)$ is open. Now consider

$H_k:=\mathrm{Pr}{}^{-1}{|}_{{\mathcal{M}}_{\mu }^k}(S(O)),k>1.$

It is obvious that $H_k$ is not open, because $H_k$ is a subset of a subspace of ${\mathcal{M}}_{\mu }^k$ . Hence, ${\mathcal{T}}_P$ is not the quotient topology.   □

Consider the natural mapping (5.6) (in general, (5.5)). Since Pr is surjective, we can construct the quotient topology as: $O\in \Sigma$ is open, if and only if, ${\mathrm{Pr}}^{-1}(O)\in {\Sigma }_{\mu }$ is open. This quotient topology is denoted by ${\mathcal{T}}_Q$ . It has a sequential structure of projection, which is described by Fig. 5.3. Similarly to product topology, quotient topology can naturally be extended to Σ.

Figure 5.3. Sequential structure of projection

Both the product topology ${\mathcal{T}}_P$ and the quotient topology ${\mathcal{T}}_Q$ are important in further study. Roughly speaking, the product topology is very useful in studying the geometric structure of Σ while the quotient topology is suitable for investigating algebraic structure of Σ.

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Density Topologies

Wladyslaw Wilczyński , in Handbook of Measure Theory, 2002

THEOREM 3.17 (Filipczak and Filipczak (1988))

T_op is a topology essentially stronger than a natural topology, essentially weaker than a density topology, essentially stronger than a.e. topology and incomparable with the r topology .

An interesting example of topology has been given by Wojdowski (1989). He defined an operator $\Phi$ _a, on the class of B_a sets having the Baire property in the following way: ${\Phi }_a\left(A\right)=\Phi \left(G\left(A\right)\right),$ where G(A) is a regular open part of A. Recall that the set A ∈ B_a if and only if A = GΔP, where G is open (in the natural topology) and P is of the first category. Among all possible representations of this type there is one for which the set G is the greatest and then it is regular open set, which we have denoted above by G(A). The topology of Wojdowski equals the Hashimoto topology generated by the a.e. topology and the σ-ideal of sets of the first category (that is the topology of the form (A P: A ∈ a.e. and P is of the first category).

Sarkhel and De (1981) while studying integrals of the Perron type have introduced a notion of sparse set. A set $E\subset ℝ$ is said to be sparse at a point $x\in ℝ$ on the right if and only if for each ε > 0 there exists k > 0 such that each interval $\left|a,b\right|\subset ]x,x+k[\text{\hspace{0.17em}with}\text{a-x<k}\text{.}\left(b-x\right)$ contains at least one point y such that ${\lambda }^{*}\left(E\bigcap \left|x,y\right|\right)<\epsilon \left(y-x\right).$ Sparseness on the left is defined similarly. A set E is said to be sparse at x if and only if it is sparse at x on the right and on the left.

We shall apply this notion to measurable sets only. Let $S_p\left(E\right)=\{x\in ℝ:ℝ\backslash E$ is sparse at x}.

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Bitopological Spaces: Theory, Relations with Generalized Algebraic Structures, and Applications

In North-Holland Mathematics Studies, 2005

Example 1.4.4

Suppose τ₁ is the antidiscrete topology on R, τ₂ = ω is the natural topology on R and a subset A ⊂ R is the same as in Example 1.4.3. Then A ₁ ⁱ = ∅, A ₁ ^d = R, A ₂ ⁱ = A, and A ₂ ^d = {0}. Hence A ₁ ⁱ = ∅ implies that A ∈ 1-DI(R) = p-DI(R), but A $\overline{\in }$ 2-DI(R) since A ₂ ⁱ ≠ ∅.

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Relation Algebras by Games

Robin Hirsch , Ian Hodkinson , in Studies in Logic and the Foundations of Mathematics, 2002

2.3.6 Canonical extensions

Looking back at the proof of theorem 2.10 , observe that the construction suggests a natural topology on the space of all ultrafilters from ℬ This allows us to pick out the range of the representation h constructed in Stone's theorem.

DEFINITION 2.11

Let ℬ be a boolean algebra and let $X=\mathrm{U}\mathrm{f}\left(\mathrm{ℬ}\right)$ . For each element $b\in \mathrm{ℬ}$ , let h(b)   =   {γ ∈ X: b ∈ γ}. The set $C=\left\{h\left(b\right):b\in \mathrm{ℬ}\right\}$ forms a basis of clopen sets for the Stone space topology $\mathcal{T}$ over X. $\mathcal{T}$ is obtained from C by closing under arbitrary unions.

THEOREM 2.12

For any boolean algebra ℬ, the Stone space topology is compact and totally disconnected.

Proof. For compactness, let Ω be an open cover of $X=\mathrm{U}\mathrm{f}\left(\mathrm{ℬ}\right)$ . Since the open sets are defined to be arbitrary unions of the basis of clopen sets, we can assume that each set in Ω is of the form h(s) for some $s\in \mathrm{ℬ}$ . Thus $X={\displaystyle \cup \left\{h\left(s\right):s\in S\right\}}$ for some set $S\subseteq \mathrm{ℬ}$ .

Now let

$\begin{array}{ccc}U\left(S\right)& =& \left\{b\in \mathrm{ℬ}:b\ge \left(-s_0\right)\cdot \left(-s_1\right)\cdot \dots \cdot \left(-s_{k-1}\right)\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{some}{s}_i\in S,i<k<\mathrm{\omega }\right\}\\ & =& \left\{b\in \mathrm{ℬ}:b+s_0+\dots +s_{k-1}=1\mathrm{some}{s}_i\right\}\text{.}\end{array}$

Then U(S) is a filter containing {− s: s ∈⋅ S}. If U(S) is not a proper filter then 0 ∈ U(S) so there are finitely many elements s ₀,… s _{k  −   1} ∈ S with ${\displaystyle {\sum }_{i<k}{s}_i=1}$ , hence {h(s_i ): i  < k} is a finite open cover of X, as required.

Otherwise U(S) is a proper filter. By BPI (fact 2.8), U(S) extends to an ultrafilter γ. But for any s ∈ S, we know   − s ∈ ∪ (S)   ⊆   γ, so ∉ γ and γ ∉ h(s). But this means that {h(s): s ∈ S} is not an open cover of X, contrary to assumption. This is a contradiction and proves compactness.

To show that the topology is totally disconnected we have to show, for any two distinct ultrafilters γ₁, γ₂ of ℬ, that there is a clopen set that contains one but not the other. This is easy: if γ₁  ≠   γ₂ then without loss of generality there exists b ∈ γ₁ \ γ₂. Thus, γ₁ ∈ h(b) and γ₂ ∉ h(b).

Observe that the range of ℬ under the representation given in the proof of theorem 2.10 consists of the clopen sets in the Stone space topology. These clopen sets form a field of sets isomorphic to ℬ, but they form a subalgebra of another field of sets — the power set over X.

DEFINITION 2.13

Let ℬ be a boolean algebra. $〈\wp \left(\mathrm{U}\mathrm{f}\left(\mathrm{ℬ}\right)\right),Ø,\mathrm{U}\mathrm{f}\left(\mathrm{ℬ}\right),\cup ,\cap ,\backslash 〉$ is called the canonical extension (or sometimes the canonical embedding algebra) of ℬ, denoted ${\mathrm{ℬ}}^{+}$ .

${\mathrm{ℬ}}^{+}$ is complete (arbitrary sums and products exist; see definition 2.23 below) and atomic. Up to isomorphism it is an extension of ℬ, since ℬ embeds in ${\mathrm{ℬ}}^{+}$ via h. We'll see shortly that any complete, atomic extension of ℬ whose set of atoms is compact and totally disconnected under the natural topology must be isomorphic to ${\mathrm{ℬ}}^{+}$ (see theorem 2.73).

These canonical extensions become more significant when we consider generalisations of boolean algebras, such as the BAOs to be seen later, and the relation algebras of the title.

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Generalized dynamic systems

In From Dimension-Free Matrix Theory to Cross-Dimensional Dynamic Systems, 2019

12.2.3 Topology on quotient matrix space

There is a natural projection $\pi :\mathcal{M}\to \Sigma =\mathcal{M}/{\sim }^{\mathrm{\Gamma }}$ as

(12.43) $\pi (A):={〈A〉}^{\mathrm{\Gamma }}.$

(For notational ease, we ignore left or right hereafter. You may consider ${\sim }^{\mathrm{\Gamma }}$ as either ${\sim }_{\ell }^{\mathrm{\Gamma }}$ or ${\sim }_r^{\mathrm{\Gamma }}$ .)

In standard linear algebra matrices of different dimensions have no topological connection. Hence, a natural topology on $\mathcal{M}$ may be determined as follows:

Definition 12.9

The natural topology on $\mathcal{M}$ , denoted by ${\mathcal{T}}_n$ , is defined as follows:

•

For each pair $(m,n)$ the set of matrices ${\mathcal{M}}_{m\times n}$ is considered as a clopen set;

•

Pose on each ${\mathcal{M}}_{m\times n}$ the topology of ${\mathbb{R}}^{mn}$ in a natural way (as conventional one).

Definition 12.10

The quotient topology on Σ, denoted by ${\mathcal{T}}_q$ , is the quotient topology deduced from ${\mathcal{T}}_n$ by using projection π.

To give a precise description of ${\mathcal{T}}_q$ , recall that

${\mathcal{M}}_{\mu }:=\{A\in {\mathcal{M}}_{m\times n}|m/n=\mu \}.$

Then we have a partition as follows:

$\mathcal{M}=\bigcup _{\mu \in {\mathbb{Q}}_{+}}{\mathcal{M}}_{\mu },$

where ${\mathbb{Q}}_{+}$ is the set of positive natural numbers.

Correspondingly, we have quotient spaces as

$\Sigma =\mathcal{M}/{\sim }^{\mathrm{\Gamma }},$

and

${\Sigma }_{\mu }={\mathcal{M}}_{\mu }/{\sim }^{\mathrm{\Gamma }}.$

They yield a partition as follows:

$\Sigma =\bigcup _{\mu \in {\mathbb{Q}}_{+}}{\Sigma }_{\mu }.$

Note that if $A\in {\mathcal{M}}_{{\mu }_1}$ , $B\in {\mathcal{M}}_{{\mu }_2}$ , and ${\mu }_1\ne {\mu }_2$ , then A and B are not equivalent. Then it is obvious that the topology ${\mathcal{T}}_q$ on Σ is determined as follows:

•

Each component ${\Sigma }_{\mu }={\mathcal{M}}_{\mu }/{\sim }^{\mathrm{\Gamma }}$ is considered as a clopen set;

•

Within each component ${\Sigma }_{\mu }$ the quotient topology determined by the projection $\pi :{\mathcal{M}}_{\mu }\to {\Sigma }_{\mu }$ is adopted.

Note that the quotient topology is the most tiny topology which makes π continuous. In other words, $O_{\Sigma }\subset \Sigma$ is open, if and only if,

${\pi }^{-1}(O_{\Sigma }):=\{x|x\in \mathcal{M}\text{and}\pi (x)\in O_{\Sigma }\}$

is open [9].

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Bitopological Spaces: Theory, Relations with Generalized Algebraic Structures, and Applications

In North-Holland Mathematics Studies, 2005

Example 5.1.32

Let us consider the following four topologies on the real line ℝ : τ₁ = ω _c is from Example 5.1.11; γ₁ = ω is the natural topology on ℝ; and τ ₂ = γ₂ is the antidiscrete topology on ℝ. If f : (ℝ, τ₁, τ₂) → (ℝ, γ₁, γ₂) is the identity function, then f ∈ p-AO(ℝ,ℝ) ∩ _p -ACl(ℝ,ℝ), but f $\overline{\in }$ d-O(ℝ,ℝ) ∪ d-Cl(ℝ,ℝ).

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Handbook of Dynamical Systems

Vitaly Bergelson , ... Máté Wierdl , in Handbook of Dynamical Systems, 2006

Sketch of the Proof

Let $\Gamma ={\displaystyle {\cap }_{n=1}^{\infty }\overline{FS\left({\left(x_i\right)}_{i=m}^{\infty }\right)}}$ . (The closures are taken in the natural topology of βℕ.) Clearly, Γ is compact and nonempty. It is not hard to show that Γ is a subsemigroup of (βℕ, +). Being a compact left-topological semigroup, Γ has an idempotent. If p ∈ Γ is an idempotent, then $\overline{\Gamma }=\Gamma \ni p$ which, in particular, implies $FS\left({\left(x_i\right)}_{i=1}^{\infty }\right)\in p$ .

The space βℕ has also another natural semigroup structure, namely, the one inherited from the multiplicative semigroup (ℕ, ·), and is a left topological compact semigroup with respect to this structure too. In particular, there are (many) multiplicative idempotents, namely ultrafilters q with the property

$A\in q\iff \left\{n\in \mathbb{N}:A/n\in q\right\}\in q$

(where $A/n:=\left\{m\in \mathbb{N}:mn\in A\right\}$ ). By complete analogy with the proof of (the additive version of) Hindman's theorem, one can show that any member of a multiplicative idempotent contains a multiplicative IP set, namely a set of finite products of the form

$FP{\left(y_n\right)}_{n=1}^{\infty }=\left\{{\displaystyle \prod _{i\in \alpha }{y}_i:\alpha \subset \mathbb{N},1\le \left|\alpha \right|<\infty }\right\}.$

It follows that for any finite partition $\mathbb{N}={\displaystyle {\cup }_{i=1}^r{C}_i}$ there are i, j ∈ {1, 2, …, r} such that C_i contains an additive IP set and C_j contains a multiplicative IP set. The following theorem due to Hindman shows that one can always have i = j.

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Holomorphic Dynamics

M. Lyubich , in Encyclopedia of Mathematical Physics, 2006

Parameter Phenomena

Spaces of Rational Functions

Let Rat _d stand for the space of rational functions of degree d. As an open subset of the complex projective space $\mathbb{C}{\mathbb{P}}^{2d+1}$ , it is endowed with the natural topology and complex structure.

Hyperbolic Maps and Fatou's Conjecture

Hyperbolic maps form an important and best-understood class of rational maps (compare with Hyperbolic Dynamical Systems). A rational map f is called hyperbolic if one of the following equivalent conditions holds:

•

All critical points of f converge to attracting cycles;

•

The map is expanding on the Julia set:

$|D{f}^n(z)|\ge C{\lambda }^n,z\in J(f)$

where C > 0, λ > 1.

For instance, the maps z ↦ z ² + ε for small ε, z ↦ z ² − 1, and z ↦ z ² + c for $c\in \mathbb{R}\backslash [-2,1/4]$ are hyperbolic. It is easy to see from the first definition that hyperbolicity is a stable property, that is, the set of hyperbolic maps is open in the space Rat _d of rational maps of degree d. One of the central open problems in holomorphic dynamics is to prove that this set is also dense. This problem is known as Fatou's conjecture.

Postcritically Finite Maps and Thurston's Theory

A rational map is called postcritically finite if the orbits of all critical points are finite. In this case, any critical point c is either a superattracting periodic point, or a repelling preperiodic point (i.e., f ⁿ c is a repelling periodic point for some n). If all critical points of f are preperiodic, then $J(f)=\hat{\mathbb{C}}$ .

Important examples of postcritically finite maps with $J(f)=\hat{\mathbb{C}}$ come from the theory of elliptic functions. Namely, let ${\mathcal{P}}_{\tau }:\mathbb{C}/{\Gamma }_{\tau }\to \hat{\mathbb{C}}$ be the Weierstrass $\mathcal{P}$ -function, where Γ _τ is the lattice in $\mathbb{C}$ generated by 1 and τ, Imτ > 0. It satisfies the functional equation ${\mathcal{P}}_{\tau }(\text{<mi mathvariant="italic" is="true">nz</mi>})=f_{\tau ,n}(\mathcal{P}(z))$ , where f _τ,n is a rational function. These functions called Lattés examples possess the desired properties. (For some special lattices, n can be selected complex: the corresponding maps are also called Lattés.)

More generally, one can consider postcritically finite topological branched coverings f: S ² → S ². Two such maps, f and g, are called Thurston combinatorially equivalent if there exist homeomorphisms $h,h\prime :(S^2,O_f)\to (S^2,O_g)$ homotopic rel O _f (and hence coinciding on O _f ) such that h′ ∘ f = h ∘ g.

A combinatorial class is called realizable if it contains a rational function. Thurston (1982) gave a combinatorial criterion for a combinatorial class to be realizable. If it is realizable, then the realization is unique, except for Lattés examples (Thurston's Rigidity Theorem).

Structural Stability and Holomorphic Motions

A map f ∈   Rat _d is called J-stable if for any maps g ∈   Rat _d sufficiently close to f, the maps f ∣ J(f) and g ∣ J(g) are topologically conjugate, and moreover, the conjugacy h _g : J(f) → J(g) is close to id. Thus, the Julia set J(f) moves continuously over the set of J-stable maps. The following result proves a weak version of Fatou's conjecture:

Theorem 4

(Lyubich and Mañé-Sad-Sullivan 1983). The set of J-stable maps is open and dense in Rat _d . Moreover, the set of unstable maps is the closure of maps that have a parabolic periodic point.

A map f ∈   Rat _d is called structurally stable if for any maps g ∈   Rat _d sufficiently close to f, the maps f and g are topologically conjugate on the whole sphere, and moreover, the conjugacy $h_g:\hat{\mathbb{C}}\to \hat{\mathbb{C}}$ is close to id. The set of structurally stable maps is also open and dense in Rat _d (Mañé-Sad-Sullivan).

The proofs make use of the theory of holomorphic motions developed for this purpose but having much broader range of applications in dynamics and analysis. Let X be a subset of $\hat{\mathbb{C}}$ , and let $h_{\lambda }:X\to \hat{\mathbb{C}}$ be a family of injections depending on parameter λ ∈ Λ in some complex manifold with a marked point λ _*. Assume that h _λ* = id and that the functions λ ↦ h _λ (z) are holomorphic in λ for any z ∈ X. Such a family of injections is called a holomorphic motion.

A holomorphic motion of any set X over Λ extends to a holomorphic motion of the whole sphere $\hat{\mathbb{C}}$ over some smaller manifold Λ′ ⊂ Λ (Bers–Royden, Sullivan–Thurston 1986). If h _λ is a holomorphic motion of an open subset of the sphere, then the maps h _λ are quasiconformal (Mañè-Sad-Sullivan). These statements are usually referred to as the λ-lemma.

If $\Lambda =\mathbb{D}$ , then the holomorphic motion of a set $X\subset \hat{\mathbb{C}}$ extends to a holomorphic motion of $\hat{\mathbb{C}}$ over the whole disk $\mathbb{D}$ (Slodkowsky 1991).

Fundamental Conjectures

The above rigidity and stability results led to the following profound conjectures:

QC Rigidity Conjecture

If two rational maps are topologically conjugate, then they are quasiconformally conjugate.

Let us consider the real projective tangent bundle PT over $\hat{\mathbb{C}}$ , with a natural action of the map f. A measurable invariant line field on the Julia set is an invariant measurable section X →   PT over an invariant set X ⊂ J(f) of positive Lebesgue measure. In other words, it is a family of tangent lines L _z ⊂ T _z ,z ∈ X, such that Df(L _z ) = L _fz . Note that such a field can exist only if J(f) has positive Lebesgue measure.

No Invariant Line Fields Conjecture

Let us consider two rational maps, f and $\tilde{f}$ , that are not Lattés examples. If they are quasiconformally conjugate and the conjugacy is conformal on the Fatou set, then they are conjugate by a Möbius transformation. Equivalently, if f is not Lattés, then there are no measurable invariant line fields on J(f).

This conjecture would imply Fatou's conjecture.

Mandelbrot Set

Let us consider the quadratic family f _c : z ↦ z ² + c. (Note that any quadratic polynomial is affinely conjugate to a unique map f _c .) The Mandelbrot set classifies parameters c according to the Basic Dichotomy of the subsection "More properties of the Julia set":

$M=\{c:J(f_c)\text{is}\text{connected}\}=\{c:f_c^n(0)\mapsto \infty \}$

Note that ${\varphi }_n(c)\equiv f_c^n(0)$ is a polynomial in c of degree 2 ^{n − 1}, and these polynomials satisfy a recursive relation ${\varphi }_{n+1}={\varphi }_n^2+c$ . Moreover, $M=\{c:|{\varphi }_n(c)|\le 2,n\in {\mathbb{Z}}_{+}\}$ , which gives an easy way to make a computer image of M (see Figure 5 ).

Figure 5. The Mandelbrot set.

A distinguished curve seen at the picture of M is the main cardioid $\mathcal{C}=\{c={\text{e}}^{2\pi \text{i}\theta }-{\text{e}}^{4\pi \text{i}\theta }/4\},\theta \in \mathbb{R}/\mathbb{Z}$ . For such a $c=c(\theta )\in \mathcal{C}$ , the map f _c has a neutral fixed point α _c with rotation number θ. For c inside the domain H ₀ bounded by $\mathcal{C},f_c$ has an attracting fixed point α _c , and the Julia set J(f _c ) is a Jordan curve (see Figure 1 ).

At the cusp c = 1/4 = c(0) of the main cardioid, the map f _c has a parabolic point with multiplier 1. This point is also called the root of $\mathcal{C}$ . Other parabolic points c = c(q/p) on $\mathcal{C}$ are bifurcation points: if one crosses $\mathcal{C}$ transversally at c, then the fixed point α _c "gives birth" to an attracting cycle of period p. This cycle preserves its "attractiveness" within some component H _q/p of int M attached to $\mathcal{C}$ .

On the boundary of H _q/p , the above attracting cycle becomes neutral, and similar bifurcations happen as one crosses this boundary transversally, etc. In this way we obtain cascades of bifurcations and associated necklaces of components of int M. The most famous one is the cascade of doubling bifurcations that occur along the real slice of M.

Components of int M that occur in these bifurcation cascades give examples of hyperbolic components of int M. More generally, a component H of int M is called hyperbolic of period p if the maps f _c ,c ∈ H, have an attracting cycle of period p. Many other hyperbolic components become visible if one begins to zoom-in into the Mandelbrot set. Some of them are satellite, that is, they are born as above by bifurcation from other hyperbolic components. Others are primitive. They can be easily distinguished geometrically: primitive components have a cusp at their root, while satellite components are bounded by smooth curves.

Given a hyperbolic component H, let us consider the multiplier λ(c),c ∈ H, of the corresponding attracting cycle, as a function of c ∈ H. The function λ univalently maps H onto the unit disk $\mathbb{D}$ (Douady and Hubbard 1982). Thus, there is a single parameter c ₀ ∈ H for which λ(c ₀) = 0, so that f _{c 0} has a superattracting cycle. This parameter is called the center of H.

Nonhyperbolic components of int M are called queer. Conjecturally, there are no queer components. This conjecture is equivalent to Fatou's conjecture for the quadratic family.

The boundary of M coincides with the set of J-unstable quadratic maps (see the subsection "Structural stability and holomorphic motions").

Connectivity and Local Connectivity

Theorem 5

(Douady and Hubbard 1982). The Mandelbrot set is connected.

The proof provides an explicit uniformization $R_M:\mathbb{C}\backslash M\to C\backslash \overline{\mathbb{D}}$ . Namely, let $B_c:{\Omega }_c\to \mathbb{C}\backslash {\mathbb{D}}_{R_c},c\in \mathbb{C}\backslash M$ , be the Böttcher coordinate near ∞. Then R _M (c) = B _c (c). This remarkable formula explains the phase-parameter similarity between the Mandelbrot set near a parameter c ∈ M and the corresponding Julia set J(f _c ) near the critical value c.

The following is the most prominent open problem in holomorphic dynamics:

MLC Conjecture

The Mandelbrot set is locally connected.

If this is the case, then the inverse map $R_M^{-1}$ extends to the unit circle $\mathbb{T}$ , and the Mandelbrot set can be represented as the quotient of $\mathbb{T}$ modulo certain equivalence relation that can be explicitly described. Thus, we would have an explicit topological model for the Mandelbrot set (Douady and Hubbard, Thurston).

The MLC conjecture is equivalent to the following conjecture:

Combinatorial Rigidity Conjecture

If two quadratic maps f _c and f _c′ with all periodic points repelling are combinatorially equivalent, then c = c′.

In turn, this conjecture would imply, in the quadratic case, the above fundamental conjectures. For a progress towards the MLC conjecture (see Universality in Mathematical Physics).

Parabolic Implosion

Parabolic maps $f_{c_0}:z\mapsto z^2+c_0$ are unstable in a dramatic way. In particular, the Julia set J(f _c ) does not depend continuously on c near c ₀. Instead, J(f _c ) tends to fill in a good part of int J(f _{c 0} ). This phenomenon called parabolic implosion has been explored by Douady, Lavaurs, Shishikura, and many others.

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Path Integrals in Noncommutative Geometry

R. Léandre , in Encyclopedia of Mathematical Physics, 2006

Introduction

Let us recall that there are basically two algebraic infinite-dimensional distribution theories:

•

The first one is white-noise analysis (Hida et al. 1993, Berezansky and Kondratiev 1995), and uses Fock spaces and the algebra of creation and annihilation operators.

•

The second one is the noncommutative differential geometry of Connes (1988) and uses the entire cyclic complex.

If we disregard the differential operations, these two distribution theories are very similar. Let us recall quickly their background on geometrical examples. Let V be a compact Riemannian manifold and E a Hermitian bundle on it. We consider an elliptic Laplacian Δ _E acting on sections ω of this bundle. We consider the Sobolev space H _k , k > 0, of sections ω of E such that:

[1] ${\displaystyle {\int }_V〈\left({\text{Δ}}_E^k+1\right)\omega ,\omega 〉\text{d}{m}_V<\infty }$

where dm _V is the Riemannian measure on V and 〈,〉 the Hermitian structure on V. H _{k + 1} is included in H _k and the intersection of all H _k is nothing other than the space of smooth sections of the bundle E, by the Sobolev embedding theorem.

Let us quickly recall Connes' distribution theory: let α(n) be a sequence of real strictly positive numbers. Let

[2] $\sigma ={\displaystyle \sum {\sigma }_n}$

where σ _n belongs to $H_k^{\otimes n}$ with the Hilbert structure naturally inherited from the Hilbert structure of H _k . We put, for C > 0,

[3] ${\Vert \sigma \Vert }_{1,C,k}={\displaystyle \sum C^n\alpha \left(n\right){\Vert {\sigma }_n\Vert }_{H_k^{\otimes n}}}$

The set of σ such that ‖σ‖_1,C,k < ∞ is a Banach space called Co _C,k . The space of Connes functionals Co_∞− is the intersection of these Banach spaces for C > 0 and k > 0 endowed with its natural topology. Its topological dual Co _−∞ is the space of distributions in Connes' sense.

Remark

We do not give the original version of the space of Connes where tensor products of Banach algebras appear but we use here the presentation of Jones and Léandre (1991).

Let us now quickly recall the theory of distributions in the white-noise sense. The main tools are Fock spaces. We consider interacting Fock spaces (Accardi and Boźejko (1998)) constituted of σ written as in [2] such that

[4] ${\Vert \sigma \Vert }_{2,C,k}^2={\displaystyle \sum C^n\alpha {\left(n\right)}^2{\Vert {\sigma }_n\Vert }_{H_k^{\otimes n}}^2<\infty }$

The space of white-noise functionals WN_∞− is the intersection of these interacting Fock spaces Λ _k,C for C > 0, k > 0. Its topological dual WN_−∞ is called the space of white-noise distributions.

Traditionally, in white-noise analysis, one considers in [2] the case where σ _n belongs to the symmetric tensor product of H _k endowed with its natural Hilbert structure. We get a symmetric Fock space ${\text{Λ}}_{C,k}^{\text{s}}$ and another space of white-noise distributions WN_s,−∞. The interest in considering symmetric Fock spaces, instead of interacting Fock spaces, arises from the characterization theorem of Potthoff–Streit. For the sake of simplicity, let us consider the case where α(n) = 1. If ω if a smooth section of E, we can consider its exponential $\text{exp}\left[\omega \right]=\sum {n!}^{-1}{\omega }^{\otimes n}$ . If we consider an element Φ of ${\text{WN}}_{\text{s,}-\infty },〈\text{Φ},\text{exp}\left[\omega \right]〉$ satisfies two natural conditions:

1.

$\left|〈\text{Φ},\text{exp}[\omega ]〉\right|\le C\text{exp}[C{\Vert \omega \Vert }_{H_k}^2]$ for some k > 0.

2.

$z\to 〈\text{Φ},\text{exp}[{\omega }_1+z{\omega }_2]〉$ is entire.

The Potthoff–Streit theorem states the opposite: a functional which sends a smooth section of V into a Hilbert space and which satisfies the two previous requirements defines an element of WN_s,−∞ with values in this Hilbert space. Moreover, if the functional depends holomorphically on a complex parameter, then the distribution depends holomorphically on this complex parameter as well.

The Potthoff–Streit theorem allows us to define flat Feynman path integrals as distributions. It is the opposite point of view, from the traditional point of view of physicists, where generally path integrals are defined by convergence of the finite-dimensional lattice approximations. Hida–Streit have proposed replacing the approach of physicists by defining path integrals as infinite-dimensional distributions, and by using Wiener chaos. Getzler was the first who thought of replacing Wiener chaos by other functionals on path spaces, that is, Chen iterated integrals. In this article, we review the recent developments of path integrals in this framework. We will mention the following topics:

•

infinite-dimensional volume element

•

Feynman path integral on a manifold

•

Bismut–Chern character and path integrals

•

fermionic Brownian motion

The reader who is interested in various rigorous approaches to path integrals should consult the review of Albeverio (1996).

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