Seminar In Topological Dynamics

Size: px
Start display at page:

Download "Seminar In Topological Dynamics"

Transcription

1 Bar Ilan University Department of Mathematics Seminar In Topological Dynamics EXAMPLES AND BASIC PROPERTIES Harari Ronen May,

2 2 1. Basic functions 1.1. Examples. To set the stage, we begin with some standard examples of continuous maps (transformations) which will be used to illustrate different properties. Example 1.1 (Doubling map). Let X denote the unit interval with its endpoints identified, X = R/Z.Define a continuous map T : X X by T (x) = 2x (mod 1). ((mod 1) means drop the integer part), i.e: { 2x, 0 x < 1 T (x) = 2 ; 1 2x 1, 2 x 1. This is usually called the Doubling map, since it doubles distances on X. An equivalent formulation would be if we let X=K:={z C : z = 1} and then define T : X X by T (e 2πiθ ) = e 2πi2θ, where 0 < θ < 1. This is equivalent in the sense that there is a homeomorphism ρ : R/Z K given by ρ(θ + Z) = e 2πiθ which relates the two transformations. (R/Z R/Z K K). Example 1.2 (Rotation on the circle). Let X = R/Z and fix a number α [0, 1).We define a homeomorphism T : X X by T (x) = x + α (mod 1), i.e: T (x) = { x + α, 0 x + α 1; x + α 1, x + α > 1. (An equivalent formulation would be if we let X=K and then define T : X X by T (e 2πiθ ) = e 2πi(θ+α). This is equivalent in the sense that the homeomorphism ρ : R/Z K given by ρ(t + Z) = e 2πit relates the two transformations). Example 1.3 (Shift map). For k 2 let X k = n Z {1, 2,... k}denote the space of all sequences taking values {1, 2,... k} indexed by Z. In order to define a metric we first associate to two sequences x = (x n ) n Z and y = (y n ) n Z an integer N(x, y) :=min{n 0 : x N y N, or x N y N }.We define a metric on X k by: d(x, y) = { ( 1 2 )N(x,y), x y; 0, otherwise. First I ll show that it is a metric: (1) d(x, y) 0 since ( 1 2 )N(x,y) > 0, N(x, y) 0 by definition. (2) d(x, y) = d(y, x) easy. I ll prove the Triangle inequality : d(x, z) = ( 1 2 )N(x,z)

3 3 Define N 0 := N(x, z) := min{n 0 : x N z N or x N z N } Assume that x N0 z N0 so or x N0 y N0 or (x N0 = y N0 ) z N0 y N0 then or N(x, y) N 0 or N(y, z) N 0 so: d(x, z) = ( 1 2 )N0 ( 1 2 )N(x,y) + ( 1 2 )N(y,z) = d(x, y) + d(y, z). Definition 1.4. : Let X be a metric space, and Y is a subset of X. Y is sequentially compact if every sequence in Y contains a subsequence that converges in Y, or equivalently, every sequence in Y has at least one accumulation point in Y. Theorem 1.5. A compact space is sequentially compact. Remark 1.6. (without proof):let M be a metric space,a M.Then for every open set U such that x U,x is an accumulation point if (U {x}) A contains infinite elements. Proof. (of the theorem) Let X be a compact space. If X is not sequentially compact, then there is a sequence in X which does not have any accumulation points. Therefore there is a neighborhood U x for every point x X such that U x contains only a finite number of terms of the sequence. The collection {U x : x X} now forms an open covering of X, hence there must be a finite subcovering because of the compactness of X. However, this finite number of U x s contains at most a finite number of terms of the sequence, which is the contradiction. Theorem 1.7 (Bolzano-Weierstrass). Let (X, d) be a metric space. X is compact if and only if X is sequentially compact. Proof. I proved. Let U be an open cover of X, and δ be the Lebesgue number for the cover U. (Let M be a compact metric space, and U α an open covering of M. Then there exist δ > 0 such that for every x M there is α I such that B(x, δ) U α. Such δ is called the Lebesgue number for the covering). Lemma 1.8. (without proof) Let X be a metric space. If X is sequentially compact then the Lebesgue number is positive for every open covering of X. Then, by this Lemma, we know that δ > 0. For x X, there is member U x in U such that B(x, δ) U x. If no subcovering of U exists, then we are able to find a sequence (x n ) n N in X such that x n / n 1 i=1 U x i (x 1 can be chosen arbitrary). Because X is sequentially compact, there is a convergent subsequence of (x n ) n N in X. On one hand, d(x i, x j ) δ when i j by construction. On the other hand, points in the tail of the convergent subsequence can be chosen arbitrary close. Contradiction. Lemma 1.9. X k is a compact space. Proof. We shall actually show that X k is sequentially compact. Let x m = (x (m) n ) n Z (m = 1, 2,...) be a sequence in X k ; (the sequence looks like: x 1 = (x 1 n,..., x 1 0,..., x 1 n)) and so on... then we need to show that there exist a point x X k and a subsequence x (m l) x (l = 1, 2, 3,...). First observe that the zeroth terms x (m) 0 (m = 1, 2, 3,...) must take some value in {1,2,3,... k} infinitely often. Choose such an x 0 {1, 2, 3,..., k} with x (m) 0 = x 0, for infinitely many m. We continue inductively: For l > 0, choose x l {1, 2, 3,..., k} and x l {1, 2, 3,..., k} such

4 4 that x (m) l = x l,..., x (m) 0 = x 0,..., x (m) l = x l,say, for infinitely many m. Finally, we define x = (x l ) l Z. For each l 0 we choose m l := m such that x (m) l = x l,..., x (m) 0 = x 0,..., x (m) l = x l ; then, d(x (ml), x) 1 and so d(x (ml), x) 0 as 2 l l The Shift Map Definition 2.1. We can define a map σ : X k X k by (σx) n = x n+1, n Z,i.e. σ : (..., x 2, x 1, x 0, x 1, x 2,...) (..., x 1, x 0, x 1, x 2,...) Since this map shifts sequences by one place it is called the shift map. Lemma 2.2. The map σ : X k X k is a homeomorphism. Proof.. To show continuity we observe that if x y and d(x, y) = ( 1 2 )N then we know that x i = y i for N i N. Thus we have that (σx) i = x i+1 = y i+1 = (σy) i for i = (N + 1),..., N 1. This means that d(σx, σy) ( 1 2 )N 1 = 2d(x, y) and we see that σ is continuous. Clearly, σ : X k X k is invertible (since the inverse transformation σ 1 : X k X k simply shifts sequences back one place). Finally, the inverse map σ 1 : X k X k is continuous by the same sort of argument as above. Notation and terminology: Let S be the set: S = {1,..., k} If k N, then a k block or a a block of length k or a word of length k over S is an element of the set S k := S S (k times). Thus, if b S k then b = b 0 b k with b i S for i = 0,..., k. The b i are called the entries of b. The set of all finite words over S will be denoted by S ; so S := {S k : k Z + }. Let Ω := S Z. If x = (x n ) n Z is an element of Ω then we shall often write x = x 2 x 1 x 0 x 1 x 2. If x Ω and j Z then the block x j x j+n is often denoted by x[j; j + n]. An non-empty block b is contained(occurs at) in x Ω (j Z whenever b = x j x j+ b 1 = x[j; j + b 1] where the length b of b is the unique k Z + such that b S k. In this case we say that x contains b (at place j)or that b is a finite sub-block of x. If j = 0, then we also write x = ḃ Definition 2.3. If b is a finite block and j Z, then the cylinder based on b at place j is the set of all elements in Ω in which b occurs(contained in) at place j, that is, the set: C j [b] := {x Ω : x[j; j + b 1] = b} If b = then C j [b] =.

5 5 Definition 2.4. A continuous flow is a pair < X, σ > where X is a topological Hausdorff space and σ : R X X is a mapping such that: (1) σ is continuous. (2) σ(0, x) = x for all x X. (3) σ(t, σ(s, x)) = σ(t + s, x) for all x X and t, s R. A mapping σ : R X X satisfying conditions (2), (3) is called an action of R on X. If condition (1) is fulfilled then it is called a continuous action. Definition 2.5. A subshift is a subflow of (X, σ) defined by a closed non-empty invariant subset. Definition 2.6. For every subset C of S we define: H (C) := {x Ω : no c C occurs in x} H(C)is closed and invariant under σ. So if H (C) then it defines a subshift. 3. Transitivity In this section we shall introduce some basic properties of continuous maps T : X X on compact metric spaces X. Definition 3.1. We say that a homeomorphism T : X X of a compact metric space X is transitive if there exists a point x X such that its orbit {T n x : n Z} = {..., T 2 x, T 1 x, x, T x, T 2 x,..., } is dense in X. We call such a point a transitive point. We say that a continuous map T : X X of a compact metric space X is (forward)transitive if there exists a point x X such that its orbit {T n x : n Z + } = {x, T x, T 2 x,..., } is dense in X. We call such a point x X a (forward) transitive point. We can check each of the examples in section 1.1 for this property. Example 3.2. We shall show that this example is forward transitive when k = 2, other cases being similar. Consider the sequence 1, 2, 11, 12, 21, 22, 111, 112, 121, 122, 221,..., 222, 111,..., 1111,... We can write down x n {1, 2}, n 0, as the nth term in the sequence Finally, consider the point x [0, 1] given by the series x = + (x n 1) n=0 2. We n+1 claim that the point x is a (forward) transitive point. Observe that T x = 2( + (x n 1) n=0 2 )(mod 1)= x n (x n+1 1) n=0 2 (mod1) = + (x n+1 1) n+1 n=0 2. n+1 Similarly, T k x = + x n+k 1 n=0 2 (mod1). n+1 To show that the set {T n x : n 0} is dense it suffices to show that for each interval of the form [ p, p+1 ], which 0 p 2 l 1, we can find N 0 with T N x [ p, p+1 ]. 2 l 2 l 2 l 2 l Given p we can write it in binary form as i 0,..., i N+1, with i 0,..., i n 1 {0, 1}. But for some N we can find x N = i 0, x N+1 = i 1,..., x N+n 1 = i n 1.This means that T N x [ p, p+1 ], as required. 2 l 2 l Proposition 3.3. Let M be a metric space. The following are equivalent: (1) M is compact.

6 6 (2) Every infinite set A M has an accumulation point. (3) Every sequence in M has a sub-sequence which converges. Proof. (1) (2) : assume M is compact.let A M an infinite set. We need to prove that A has an accumulation point.asuume that A does not have an accumulation point,which means that every point x M is not an accumulation point of A.If x A it means that there is an environment x U such that U A = {x}. We shall take this environment and mark it as U x. The family of the sets U x is a cover of M.I claim that this cover does not have a finite sub-cover. Why? because every x A has only one element in the cover that contains him which is U x.therefore every sub-family which doesn t include each of the U x for x A,will not be a cover. Therefore there is no finite sub-cover - contradiction to M being compact. (2) (3) : Assume every A M infinite has an accumulation point. Let {x n } be a sequence in M. We shall prove that {x n } has a sub-sequence which converges.there are two cases: (1) If the same element in M appears infinite number of times in the sequence {x n } so we have a sub-sequence which converges. (2) If all the elements appear a finite number of times so we have an infinite different elements. Let A be the set of values that appear in the sequence {x n } and in this case A is infinite (obvious). So A has an accumulation point, x. We show that {x n } has a sub-sequence converges to x. we shall take the ball B(x, 1).The point x is an accumulation point of A so B(x, 1) A.But A is the set of values of the sequence {x n }, which means x n1 B(x, 1).In B(x, 1 2 ) there are an infinite elements from A so there are elements that appear after the n 1 place.we shall take n 2 > n 1 such that x n2 B(x, 1 2 ) and so on... We shall look at B(x, 1 k ) which has infinite elements from A so there is an element that appear after the n k 1 place. We shall take n k > n k 1 such that x nk B(x, 1 k ) and it is the next element of the sequence. We claim that lim x n k = x k because 0 < d(x nk, x) < 1 k and that s why x n k x. (3) (1): Remark (without proof): If every sequence {x n } in M,has a sub-sequence which converges,then for every open cover{u α } α I of M there exists ɛ-lebesgue number of the cover. Assume that every sequence {x n } has a sub-sequence that converges. We need to prove that M is compact. Let U α be a cover of M. We need to show that it has a finite subcover.let ɛ be the Lebesgue number of the cover. We claim that there are finite elements x 1,..., x n such that {B(x n, ɛ)} is a cover of M.Assume that there is not such a finite set. We take some x 1 M.B(x 1, ɛ) is not covering M.We shall take x 2 / B(x 1, ɛ).b(x 1, ɛ) and B(x 2, ɛ) are not covering M. Let take x 3 / B(x 1, ɛ) B(x 2, ɛ). And so on... after taking x 1,..., x k, {B(x i, ɛ)} k i=1 is not a cover of M and we choose x k+1 / k i=1 B(x i, ɛ).in the sequence {x k } k=1 we

7 7 have that for every k 1 < k 2, d(x k1, x k2 ) ɛ.if the sequence has a sub-sequence that converges, say to x, there were two different elements of the sub-sequence in B(x, ɛ 2 ) and then the distance between them was less than ɛ.contradiction. We got elements {x 1,..., x n } M such that M = n k=1 B(x k, ɛ).for all {1 k n} there is (and we take it)α k I such that B(x k, ɛ) U αk and then M = n k=1 U α k, and therefore M is compact. Example 3.4. There are two different cases, depending on whether or not α is irrational. First assume that α is irrational, then the map T : X X (as in example 1.2)can be shown to be transitive (and even forward transitive) where x = 0,say. It suffices to show that the orbit {T n 0} n Z + is dense.since this is an infinite set in R/Z we can choose x R/Z and a subsequence n i + with T ni 0 = n i α(mod1) x.for any sufficiently small ɛ > 0 we can choose n i > n j with T ni 0 x < ɛ 2 and T nj 0 x < ɛ 2 and thus T ni 0 T nj 0 = T ni nj 0 < ɛ.moreover, T ni 0 T nj 0, since if not this would contradict α being irrational. Thus the points T (ni nj)k 0, k 1 form an ɛ-dense subset of R/Z. Since ɛ can be chosen arbitrary small this completes the proof of transitivity. Now assume that α = p q with p, q Z having no common divisors and q 0. For any x X the orbit {T n x : x Z} would be a finite set {x, x + 1 q,..., x + q 1 q (mod1)}.in particular T is not transitive. Example 3.5. We shall show that this example is forward transitive.the sequence {x n } n N = {1, 2,..., k, 1, 1,..., 1, k, 2, 1,..., 2, k,..., z 0, z 1,..., z N 1...} }{{} All strings appear in X k (in which all finite strings appear once) is a forward transitive point. To see this choose any point z X k (z = (z n ) n Z ) and for any ɛ > 0 choose N > 0 sufficiently large that ( 1 2 )N < ɛ. If we choose r such that x r = z 0,..., x r+n 1 = z N 1 then we see that (σ r x) 0 = x r = z 0,..., (σ r x) N 1 = x r+n 1 = z r+n 1 and so d(σ r x, z) ( 1 2 )N < ɛ. 4. Other characterizations of transitivity The following result gives equivalent conditions for a homeomorphism of a compact metric space to be transitive. Theorem 4.1. The following are equivalent. (1) T : X X is transitive. (2) If U is an open set with T U = U then either U is dense or U =. (3) If U, V are non-empty open sets then for some n Z we have that T n U V. (4) The set {x X : the orbit {T n x} n Z is dense in X} is dense in G δ set. Definition 4.2. A set which can be represented as the intersection of a countable collection of open sets is called a G δ set.

8 8 Proof Assume x X has a dense orbit. Assume that T U = U. We can choose n Z such that T n x U.Moreover, for any m Z we have that T m x T m n U = U. Since the orbit of x is dense (i.e. m Z T m x X is dense)we see that U is dense The T -invariant union n Z T n U is dense in X by assumption 2. Thus n Z T n U V and so n Z with T n U V. 3 4 Consider a dense set {x n } n Z and consider the balls of radius 1 k,k 1, denoted by B(x n, 1 k ). We can identify {x X : {T m x} m Z is dense in X }= + n=0 + k=1 + m= T m B(x n, 1 k ) (i.e. n 0, k 1, m Z with T m x B(x n, 1 k )) This is immediate. 5. Transitivity for subshifts of finite type In section 1.1 we defined the shift transformation σ : X k X k on X k = n Z {1,..., k}.for any closed σ invariant subset X X k (i.e. σ(x) = X) we consider the restriction σ X. We can use the same notation σ : X X. Definition 5.1. A subshift Λ is called a subshift of finite type (abbreviation: ssft) whenever Λ = H (C) for a finite set C of blocks. Definition 5.2. For n N and B S n, we define: (all n blocks are in B). H n (B) := {x Ω : x[i; i + n 1] B for all i Z} H n (B) then it is a subshift. Definition 5.3. Let Λ be a subshift. Than we define: Λ = H 2 (B) := {x Ω : x n x n+1 B for all n Z} for some set B of 2 blocks over S. Consider a finite directed graph Γ with s vertices labeled faithfully, by the symbols 0, 1,..., s 1 of S and let for b 1, b 2 S the pair (b 1, b 2 ) be a directed edge (from b 1 to b 2 ) in Γ iff the 2 block b 1 b 2 belongs to B. Thus the following graph represents the set: {00, 20, 23, 32, 12, 15, 51, 46} of 2 blocks over the symbol set {0, 1, 2, 3, 4, 5, 6}. Let M be the adjacency matrix of the graph Γ, associated with the ssft H 2 (B): M has entries m ij for i, j = 0,..., s 1 and { 1, if ij B,(i,j) an edge in Γ; m ij := 0, if ij/ B, that is (i,j) not an edge in Γ. Clearly, H 2 (B) is determined by M: If x Ω, then x H 2 (B) iff x n x n+1 B for all n Z, iff m xnx n+1 = 1 for all n Z. In the following table a description is given of essentially all possible ssft s of order 2 over the symbol set S := {0, 1}. For each such a subshift the transition matrix M, the corresponding graph Γ and a schematic description of (Λ, σ) are given. The arrows between points in Λ indicate the direction in which a point σ n x runs through its orbit if n increases.

9 9 Lemma 5.4. For every n N and i, j S the following are equivalent: (1) m (n) ij > 0. (2) x Λ : x 0 = i and x n = j. Proof. (1) = (2): By (1) there is a path of length n in Γ form vertex i to vertex j. This path can be extended to a two-sided infinite path in Γ, representing a point x Λ with the desired property. Definition 5.5. Let A be a k k matrix with entries 0 or 1. We call the matrix irreducible if 1 i, j, k, N > 0 such that A N (i, j) > 0. Example 5.6. When k = 3 the matrix A= is irreducible (n = for example. However, the matrix A = is not irreducible. (These properties are readily checked). Definition 5.7. Given a k k matrix A with entries 0 or 1.We define {X A = (x n ) n Z {1,..., k} : A(x n, x n+1 ) = 1, n Z}. n= We define the subshift of finite type σ : X A X A to be the restriction σ XA. The following gives necessary and sufficient conditions for σ : X A X A to be transitive. Theorem 5.8. A subshift of finite type σ : X A X A is transitive if and only if A is irreducible. Proof. Assume that σ is transitive. Consider the sets [i] 0 := {(x n ) n Z X A : x 0 = i} for i = 1,..., k. These sets are open. Given 1 i, j k we know that there exists N > 0 such that σ N [j] 0 [i] 0.Choose (x n ) n Z σ N [j] 0 [i] 0 ;then we know that x 0 = i and x N = j. Notice that k k A N (i, j) = A(i, r 1 )A(r 1, r 2 ) A(r N 2, r N 1 )A(r N 1, j). r 1=1 r N 1 =1 But since A(i, x 1 ) = A(x 1, x 2 ) = = A(x N 1, j) = 1 we see that A N (i, j) 1. (Another way to look at this is by paths: We can write: m (n) ij = s 1 s 1 p 1=0 p 2=0 s 1 p n 1=0 m ip1 m p1p 2 m pn 1j Then m (n) ij is equal to the total number of paths in the directed graph Γ that begin in vertex i, end in vertex j and have length (=number of consecutive edges) n).

10 10 Conversely, assume that for 1 i, j k we have that A N (i, j) 1.Given U, V open sets we can choose (i n ) n Z U and (j n ) n Z V such that for M > 0 sufficiently large U [i M, i M 1,..., i M ] M M := {(x n ) n Z X A : x k = i k, M k M}, V [j M, j M 1,..., j M ] M M := {(x n ) n Z X A : x k = j k, M k M}. By hypothesis we can find N > 0 such that A N (i M, j M ) 1. This means that we can find a string x 1,..., x N 1 such that A(i M, x 1) = A(x 1, x 2) =... = A(x N 1, j M ) = 1 and then define i n, if n M; x n = x n M, if M+1 n M + n 1; j n (2M+N), if M+N n. then we have that x U σ N V i.e U σ N V.

11 11 6. Appendix 6.1. Minimality and the Birkhoff recurrence theorem. In this section I want to present a simple but important recurrence result, called the Birkhoff recurrence theorem. Our starting point is to define the following property. Definition 6.1. A homeomorphism T : X X is minimal if for every point x X the orbit {T n x : n Z} is dense in X. The following is obvious from the definitions. Proposition 6.2. A minimal homeomorphism is necessarily transitive. We can now consider each of the examples from section 1.1 and ask which of these are minimal. Since example 1 is not a homeomorphism we begin with example 2. Example 6.3. Lemma 6.4. When α is irrational then T (x) = x + α is minimal. Proof. It suffices to show that for every x R/Z and every neighbourhood (y ɛ, y + ɛ) (y R/Z, ɛ > 0) we can find n 1 such that T n x (y ɛ, y + ɛ). We already know that T is transitive (i.e. there exists at least one transitive point x 0 R/Z with dense orbit). Fix y R/Z and use the transitivity to choose a subsequence n i with T ni x 0 (y x + x 0 ) as i +. Thus T ni x = n i α + x (mode 1) = n i α + x 0 + (x x 0 ) (mod 1) = T ni (x 0 ) + x x 0 (mod 1) y + (x 0 x) + (x x 0 ) = y (mod 1) Example 6.5. The shift map is not minimal since it contains a fixed point (e.g. x = (..., 1, 1, 1,...)). The following theorem gives equivalent definitions. Theorem 6.6. Let T : X X be a homeomorphism of a compact metric space. The following properties are equivalent. (1) T is minimal. (2) If T E = E is a closed T invariant set, then either E = or E = X. (3) If U is an open set then X = n Z T n U. Proof. (1) (2) Assume that T E = E and choose x E.Hypothesis (1) gives that X = cl({t n x} n Z ) E X. (2) (3) Given a non-empty open set U let E = X ( n Z T n U). By construction T E = E and E X(since U ) and so by hypothesis (2) we have that E =. Thus X = n Z T n U.

12 12 (3) (1)Fix x X and an open neighbourhood U X. Since X T n U for some n Z (by hypothesis (3)) we have that T n x U. This shows that the orbit {T n x} n Z is dense in X. Using property (2) we get the following surprising result that every homeomorphism contains a minimum homeomorphism. Theorem 6.7. Let T : X X be a homeomorphism of a compact metric space X. There exists a non-empty closed set Y X with T Y = Y and T : Y Y is minimal. Proof. This follows from an application of Zorn s Lemma. Let ε denote the family of all closed T -invariant subsets of X with the partial ordering by inclusion, i.e. Z 1 Z 2 iff Z 1 Z 2. Every totally ordered subset (or chain ) {Z α } has a least element Z = α Z α (which is non-empty by compactness of X). Thus by Zorn s lemma, there exists a minimal element Y X(i.e. Y, Y ε with Y Y implies that Y = Y ). By property (2) of theorem 6.6 this can be reinterpreted as saying that T : Y Y is minimal. As a corollary we get the following simple but elegant result. Corollary 6.8. (BIRKHOF F RECURRENCE T HEOREM) Let T : X X be a homeomorphism of a compact metric space X. We can find x X such that T ni x x for a sub-sequence of the integers n i +. Proof. By theorem 6.7 we can choose a T variant subset Y X such that T : Y Y is minimal. For any x Y X we have the required property. Example 6.9. Consider the case X = R/Z and T : X X defined by T x = x + α (mode 1), where α is an irrational number. Let ɛ > 0; then we can find n > 0 (by Birkhoff s theorem) such that αn (mode 1) ɛ, i.e. there exists p N such that ɛ αn p ɛ. Rewriting this, we have that for any irrational α, p, n N such that α p n ɛ n. This is a (marginal) improvement on the most obvious estimate.

13 13 References [1] P.Billingsley, Ergoduc Theory and Information, Addison-Wesley, New York, [2] R. Devaney, An Introduction to the Modern Theory of Dynamical Systems, Addison-Wesley, New York [3] J. de Vries, Elements of Topological Dynamics, Kluwer Academic Publisher, p , [4] A. Katok and B. Hasselblatt, Introduction to Modern Theory of Dynamical Systems, C.U.P., Cambridge, [5] W.Szlenk, An Introduction to the Theory of Smooth Dynamical Systems, Wiley, New York, [6] P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer, Berlin, 1982.

Problem Set 2: Solutions Math 201A: Fall 2016

Problem Set 2: Solutions Math 201A: Fall 2016 Problem Set 2: s Math 201A: Fall 2016 Problem 1. (a) Prove that a closed subset of a complete metric space is complete. (b) Prove that a closed subset of a compact metric space is compact. (c) Prove that

More information

Economics 204 Fall 2012 Problem Set 3 Suggested Solutions

Economics 204 Fall 2012 Problem Set 3 Suggested Solutions Economics 204 Fall 2012 Problem Set 3 Suggested Solutions 1. Give an example of each of the following (and prove that your example indeed works): (a) A complete metric space that is bounded but not compact.

More information

Math 117: Topology of the Real Numbers

Math 117: Topology of the Real Numbers Math 117: Topology of the Real Numbers John Douglas Moore November 10, 2008 The goal of these notes is to highlight the most important topics presented in Chapter 3 of the text [1] and to provide a few

More information

UNIVERSITY OF BRISTOL. Mock exam paper for examination for the Degrees of B.Sc. and M.Sci. (Level 3)

UNIVERSITY OF BRISTOL. Mock exam paper for examination for the Degrees of B.Sc. and M.Sci. (Level 3) UNIVERSITY OF BRISTOL Mock exam paper for examination for the Degrees of B.Sc. and M.Sci. (Level 3) DYNAMICAL SYSTEMS and ERGODIC THEORY MATH 36206 (Paper Code MATH-36206) 2 hours and 30 minutes This paper

More information

1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3

1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3 Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,

More information

7 Complete metric spaces and function spaces

7 Complete metric spaces and function spaces 7 Complete metric spaces and function spaces 7.1 Completeness Let (X, d) be a metric space. Definition 7.1. A sequence (x n ) n N in X is a Cauchy sequence if for any ɛ > 0, there is N N such that n, m

More information

2. Metric Spaces. 2.1 Definitions etc.

2. Metric Spaces. 2.1 Definitions etc. 2. Metric Spaces 2.1 Definitions etc. The procedure in Section for regarding R as a topological space may be generalized to many other sets in which there is some kind of distance (formally, sets with

More information

Course 212: Academic Year Section 1: Metric Spaces

Course 212: Academic Year Section 1: Metric Spaces Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........

More information

MA651 Topology. Lecture 9. Compactness 2.

MA651 Topology. Lecture 9. Compactness 2. MA651 Topology. Lecture 9. Compactness 2. This text is based on the following books: Topology by James Dugundgji Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology

More information

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9

MAT 570 REAL ANALYSIS LECTURE NOTES. Contents. 1. Sets Functions Countability Axiom of choice Equivalence relations 9 MAT 570 REAL ANALYSIS LECTURE NOTES PROFESSOR: JOHN QUIGG SEMESTER: FALL 204 Contents. Sets 2 2. Functions 5 3. Countability 7 4. Axiom of choice 8 5. Equivalence relations 9 6. Real numbers 9 7. Extended

More information

Chapter 2 Metric Spaces

Chapter 2 Metric Spaces Chapter 2 Metric Spaces The purpose of this chapter is to present a summary of some basic properties of metric and topological spaces that play an important role in the main body of the book. 2.1 Metrics

More information

Dynamical Systems and Ergodic Theory PhD Exam Spring Topics: Topological Dynamics Definitions and basic results about the following for maps and

Dynamical Systems and Ergodic Theory PhD Exam Spring Topics: Topological Dynamics Definitions and basic results about the following for maps and Dynamical Systems and Ergodic Theory PhD Exam Spring 2012 Introduction: This is the first year for the Dynamical Systems and Ergodic Theory PhD exam. As such the material on the exam will conform fairly

More information

4. Ergodicity and mixing

4. Ergodicity and mixing 4. Ergodicity and mixing 4. Introduction In the previous lecture we defined what is meant by an invariant measure. In this lecture, we define what is meant by an ergodic measure. The primary motivation

More information

Topological properties

Topological properties CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological

More information

Introduction to Dynamical Systems

Introduction to Dynamical Systems Introduction to Dynamical Systems France-Kosovo Undergraduate Research School of Mathematics March 2017 This introduction to dynamical systems was a course given at the march 2017 edition of the France

More information

Measure and Category. Marianna Csörnyei. ucahmcs

Measure and Category. Marianna Csörnyei.   ucahmcs Measure and Category Marianna Csörnyei mari@math.ucl.ac.uk http:/www.ucl.ac.uk/ ucahmcs 1 / 96 A (very short) Introduction to Cardinals The cardinality of a set A is equal to the cardinality of a set B,

More information

COUNTABLE PRODUCTS ELENA GUREVICH

COUNTABLE PRODUCTS ELENA GUREVICH COUNTABLE PRODUCTS ELENA GUREVICH Abstract. In this paper, we extend our study to countably infinite products of topological spaces.. The Cantor Set Let us constract a very curios (but usefull) set known

More information

Analysis III Theorems, Propositions & Lemmas... Oh My!

Analysis III Theorems, Propositions & Lemmas... Oh My! Analysis III Theorems, Propositions & Lemmas... Oh My! Rob Gibson October 25, 2010 Proposition 1. If x = (x 1, x 2,...), y = (y 1, y 2,...), then is a distance. ( d(x, y) = x k y k p Proposition 2. In

More information

The Heine-Borel and Arzela-Ascoli Theorems

The Heine-Borel and Arzela-Ascoli Theorems The Heine-Borel and Arzela-Ascoli Theorems David Jekel October 29, 2016 This paper explains two important results about compactness, the Heine- Borel theorem and the Arzela-Ascoli theorem. We prove them

More information

MH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then

MH 7500 THEOREMS. (iii) A = A; (iv) A B = A B. Theorem 5. If {A α : α Λ} is any collection of subsets of a space X, then MH 7500 THEOREMS Definition. A topological space is an ordered pair (X, T ), where X is a set and T is a collection of subsets of X such that (i) T and X T ; (ii) U V T whenever U, V T ; (iii) U T whenever

More information

Dynamical Systems 2, MA 761

Dynamical Systems 2, MA 761 Dynamical Systems 2, MA 761 Topological Dynamics This material is based upon work supported by the National Science Foundation under Grant No. 9970363 1 Periodic Points 1 The main objects studied in the

More information

AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES

AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES AN EXPLORATION OF THE METRIZABILITY OF TOPOLOGICAL SPACES DUSTIN HEDMARK Abstract. A study of the conditions under which a topological space is metrizable, concluding with a proof of the Nagata Smirnov

More information

Solutions to Tutorial 8 (Week 9)

Solutions to Tutorial 8 (Week 9) The University of Sydney School of Mathematics and Statistics Solutions to Tutorial 8 (Week 9) MATH3961: Metric Spaces (Advanced) Semester 1, 2018 Web Page: http://www.maths.usyd.edu.au/u/ug/sm/math3961/

More information

Mid Term-1 : Practice problems

Mid Term-1 : Practice problems Mid Term-1 : Practice problems These problems are meant only to provide practice; they do not necessarily reflect the difficulty level of the problems in the exam. The actual exam problems are likely to

More information

Maths 212: Homework Solutions

Maths 212: Homework Solutions Maths 212: Homework Solutions 1. The definition of A ensures that x π for all x A, so π is an upper bound of A. To show it is the least upper bound, suppose x < π and consider two cases. If x < 1, then

More information

HW 4 SOLUTIONS. , x + x x 1 ) 2

HW 4 SOLUTIONS. , x + x x 1 ) 2 HW 4 SOLUTIONS The Way of Analysis p. 98: 1.) Suppose that A is open. Show that A minus a finite set is still open. This follows by induction as long as A minus one point x is still open. To see that A

More information

MASTERS EXAMINATION IN MATHEMATICS SOLUTIONS

MASTERS EXAMINATION IN MATHEMATICS SOLUTIONS MASTERS EXAMINATION IN MATHEMATICS PURE MATHEMATICS OPTION SPRING 010 SOLUTIONS Algebra A1. Let F be a finite field. Prove that F [x] contains infinitely many prime ideals. Solution: The ring F [x] of

More information

16 1 Basic Facts from Functional Analysis and Banach Lattices

16 1 Basic Facts from Functional Analysis and Banach Lattices 16 1 Basic Facts from Functional Analysis and Banach Lattices 1.2.3 Banach Steinhaus Theorem Another fundamental theorem of functional analysis is the Banach Steinhaus theorem, or the Uniform Boundedness

More information

s P = f(ξ n )(x i x i 1 ). i=1

s P = f(ξ n )(x i x i 1 ). i=1 Compactness and total boundedness via nets The aim of this chapter is to define the notion of a net (generalized sequence) and to characterize compactness and total boundedness by this important topological

More information

FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES. 1. Compact Sets

FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES. 1. Compact Sets FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES CHRISTOPHER HEIL 1. Compact Sets Definition 1.1 (Compact and Totally Bounded Sets). Let X be a metric space, and let E X be

More information

Filters in Analysis and Topology

Filters in Analysis and Topology Filters in Analysis and Topology David MacIver July 1, 2004 Abstract The study of filters is a very natural way to talk about convergence in an arbitrary topological space, and carries over nicely into

More information

1 Lecture 4: Set topology on metric spaces, 8/17/2012

1 Lecture 4: Set topology on metric spaces, 8/17/2012 Summer Jump-Start Program for Analysis, 01 Song-Ying Li 1 Lecture : Set topology on metric spaces, 8/17/01 Definition 1.1. Let (X, d) be a metric space; E is a subset of X. Then: (i) x E is an interior

More information

Appendix B Convex analysis

Appendix B Convex analysis This version: 28/02/2014 Appendix B Convex analysis In this appendix we review a few basic notions of convexity and related notions that will be important for us at various times. B.1 The Hausdorff distance

More information

Lecture 16 Symbolic dynamics.

Lecture 16 Symbolic dynamics. Lecture 16 Symbolic dynamics. 1 Symbolic dynamics. The full shift on two letters and the Baker s transformation. 2 Shifts of finite type. 3 Directed multigraphs. 4 The zeta function. 5 Topological entropy.

More information

Rudiments of Ergodic Theory

Rudiments of Ergodic Theory Rudiments of Ergodic Theory Zefeng Chen September 24, 203 Abstract In this note we intend to present basic ergodic theory. We begin with the notion of a measure preserving transformation. We then define

More information

Week 2: Sequences and Series

Week 2: Sequences and Series QF0: Quantitative Finance August 29, 207 Week 2: Sequences and Series Facilitator: Christopher Ting AY 207/208 Mathematicians have tried in vain to this day to discover some order in the sequence of prime

More information

Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi

Real Analysis Math 131AH Rudin, Chapter #1. Dominique Abdi Real Analysis Math 3AH Rudin, Chapter # Dominique Abdi.. If r is rational (r 0) and x is irrational, prove that r + x and rx are irrational. Solution. Assume the contrary, that r+x and rx are rational.

More information

Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University

Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University Lecture Notes in Advanced Calculus 1 (80315) Raz Kupferman Institute of Mathematics The Hebrew University February 7, 2007 2 Contents 1 Metric Spaces 1 1.1 Basic definitions...........................

More information

P-adic Functions - Part 1

P-adic Functions - Part 1 P-adic Functions - Part 1 Nicolae Ciocan 22.11.2011 1 Locally constant functions Motivation: Another big difference between p-adic analysis and real analysis is the existence of nontrivial locally constant

More information

MA651 Topology. Lecture 10. Metric Spaces.

MA651 Topology. Lecture 10. Metric Spaces. MA65 Topology. Lecture 0. Metric Spaces. This text is based on the following books: Topology by James Dugundgji Fundamental concepts of topology by Peter O Neil Linear Algebra and Analysis by Marc Zamansky

More information

Topology. Xiaolong Han. Department of Mathematics, California State University, Northridge, CA 91330, USA address:

Topology. Xiaolong Han. Department of Mathematics, California State University, Northridge, CA 91330, USA  address: Topology Xiaolong Han Department of Mathematics, California State University, Northridge, CA 91330, USA E-mail address: Xiaolong.Han@csun.edu Remark. You are entitled to a reward of 1 point toward a homework

More information

Notes on nets and convergence in topology

Notes on nets and convergence in topology Topology I Humboldt-Universität zu Berlin C. Wendl / F. Schmäschke Summer Semester 2017 Notes on nets and convergence in topology Nets generalize the notion of sequences so that certain familiar results

More information

Measurable Choice Functions

Measurable Choice Functions (January 19, 2013) Measurable Choice Functions Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/fun/choice functions.pdf] This note

More information

g 2 (x) (1/3)M 1 = (1/3)(2/3)M.

g 2 (x) (1/3)M 1 = (1/3)(2/3)M. COMPACTNESS If C R n is closed and bounded, then by B-W it is sequentially compact: any sequence of points in C has a subsequence converging to a point in C Conversely, any sequentially compact C R n is

More information

01. Review of metric spaces and point-set topology. 1. Euclidean spaces

01. Review of metric spaces and point-set topology. 1. Euclidean spaces (October 3, 017) 01. Review of metric spaces and point-set topology Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 017-18/01

More information

Rotation set for maps of degree 1 on sun graphs. Sylvie Ruette. January 6, 2019

Rotation set for maps of degree 1 on sun graphs. Sylvie Ruette. January 6, 2019 Rotation set for maps of degree 1 on sun graphs Sylvie Ruette January 6, 2019 Abstract For a continuous map on a topological graph containing a unique loop S, it is possible to define the degree and, for

More information

NAME: Mathematics 205A, Fall 2008, Final Examination. Answer Key

NAME: Mathematics 205A, Fall 2008, Final Examination. Answer Key NAME: Mathematics 205A, Fall 2008, Final Examination Answer Key 1 1. [25 points] Let X be a set with 2 or more elements. Show that there are topologies U and V on X such that the identity map J : (X, U)

More information

Math 220A Complex Analysis Solutions to Homework #2 Prof: Lei Ni TA: Kevin McGown

Math 220A Complex Analysis Solutions to Homework #2 Prof: Lei Ni TA: Kevin McGown Math 220A Complex Analysis Solutions to Homework #2 Prof: Lei Ni TA: Kevin McGown Conway, Page 14, Problem 11. Parts of what follows are adapted from the text Modular Functions and Dirichlet Series in

More information

Three hours THE UNIVERSITY OF MANCHESTER. 31st May :00 17:00

Three hours THE UNIVERSITY OF MANCHESTER. 31st May :00 17:00 Three hours MATH41112 THE UNIVERSITY OF MANCHESTER ERGODIC THEORY 31st May 2016 14:00 17:00 Answer FOUR of the FIVE questions. If more than four questions are attempted, then credit will be given for the

More information

MATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION. Problem 1

MATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION. Problem 1 MATH 54 - TOPOLOGY SUMMER 2015 FINAL EXAMINATION ELEMENTS OF SOLUTION Problem 1 1. Let X be a Hausdorff space and K 1, K 2 disjoint compact subsets of X. Prove that there exist disjoint open sets U 1 and

More information

Linear distortion of Hausdorff dimension and Cantor s function

Linear distortion of Hausdorff dimension and Cantor s function Collect. Math. 57, 2 (2006), 93 20 c 2006 Universitat de Barcelona Linear distortion of Hausdorff dimension and Cantor s function O. Dovgoshey and V. Ryazanov Institute of Applied Mathematics and Mechanics,

More information

2.31 Definition By an open cover of a set E in a metric space X we mean a collection {G α } of open subsets of X such that E α G α.

2.31 Definition By an open cover of a set E in a metric space X we mean a collection {G α } of open subsets of X such that E α G α. Chapter 2. Basic Topology. 2.3 Compact Sets. 2.31 Definition By an open cover of a set E in a metric space X we mean a collection {G α } of open subsets of X such that E α G α. 2.32 Definition A subset

More information

ORBITAL SHADOWING, INTERNAL CHAIN TRANSITIVITY

ORBITAL SHADOWING, INTERNAL CHAIN TRANSITIVITY ORBITAL SHADOWING, INTERNAL CHAIN TRANSITIVITY AND ω-limit SETS CHRIS GOOD AND JONATHAN MEDDAUGH Abstract. Let f : X X be a continuous map on a compact metric space, let ω f be the collection of ω-limit

More information

Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr.

Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr. Notes for Math 290 using Introduction to Mathematical Proofs by Charles E. Roberts, Jr. Chapter : Logic Topics:. Statements, Negation, and Compound Statements.2 Truth Tables and Logical Equivalences.3

More information

IRRATIONAL ROTATION OF THE CIRCLE AND THE BINARY ODOMETER ARE FINITARILY ORBIT EQUIVALENT

IRRATIONAL ROTATION OF THE CIRCLE AND THE BINARY ODOMETER ARE FINITARILY ORBIT EQUIVALENT IRRATIONAL ROTATION OF THE CIRCLE AND THE BINARY ODOMETER ARE FINITARILY ORBIT EQUIVALENT MRINAL KANTI ROYCHOWDHURY Abstract. Two invertible dynamical systems (X, A, µ, T ) and (Y, B, ν, S) where X, Y

More information

E.7 Alaoglu s Theorem

E.7 Alaoglu s Theorem E.7 Alaoglu s Theorem 359 E.7 Alaoglu s Theorem If X is a normed space, then the closed unit ball in X or X is compact if and only if X is finite-dimensional (Problem A.25). Even so, Alaoglu s Theorem

More information

Sequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1.

Sequences. Chapter 3. n + 1 3n + 2 sin n n. 3. lim (ln(n + 1) ln n) 1. lim. 2. lim. 4. lim (1 + n)1/n. Answers: 1. 1/3; 2. 0; 3. 0; 4. 1. Chapter 3 Sequences Both the main elements of calculus (differentiation and integration) require the notion of a limit. Sequences will play a central role when we work with limits. Definition 3.. A Sequence

More information

Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem

Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem 56 Chapter 7 Locally convex spaces, the hyperplane separation theorem, and the Krein-Milman theorem Recall that C(X) is not a normed linear space when X is not compact. On the other hand we could use semi

More information

Immerse Metric Space Homework

Immerse Metric Space Homework Immerse Metric Space Homework (Exercises -2). In R n, define d(x, y) = x y +... + x n y n. Show that d is a metric that induces the usual topology. Sketch the basis elements when n = 2. Solution: Steps

More information

Analysis Finite and Infinite Sets The Real Numbers The Cantor Set

Analysis Finite and Infinite Sets The Real Numbers The Cantor Set Analysis Finite and Infinite Sets Definition. An initial segment is {n N n n 0 }. Definition. A finite set can be put into one-to-one correspondence with an initial segment. The empty set is also considered

More information

ON SPACE-FILLING CURVES AND THE HAHN-MAZURKIEWICZ THEOREM

ON SPACE-FILLING CURVES AND THE HAHN-MAZURKIEWICZ THEOREM ON SPACE-FILLING CURVES AND THE HAHN-MAZURKIEWICZ THEOREM ALEXANDER KUPERS Abstract. These are notes on space-filling curves, looking at a few examples and proving the Hahn-Mazurkiewicz theorem. This theorem

More information

Lecture Notes on Metric Spaces

Lecture Notes on Metric Spaces Lecture Notes on Metric Spaces Math 117: Summer 2007 John Douglas Moore Our goal of these notes is to explain a few facts regarding metric spaces not included in the first few chapters of the text [1],

More information

A LITTLE REAL ANALYSIS AND TOPOLOGY

A LITTLE REAL ANALYSIS AND TOPOLOGY A LITTLE REAL ANALYSIS AND TOPOLOGY 1. NOTATION Before we begin some notational definitions are useful. (1) Z = {, 3, 2, 1, 0, 1, 2, 3, }is the set of integers. (2) Q = { a b : aεz, bεz {0}} is the set

More information

Exercise Solutions to Functional Analysis

Exercise Solutions to Functional Analysis Exercise Solutions to Functional Analysis Note: References refer to M. Schechter, Principles of Functional Analysis Exersize that. Let φ,..., φ n be an orthonormal set in a Hilbert space H. Show n f n

More information

Analysis III. Exam 1

Analysis III. Exam 1 Analysis III Math 414 Spring 27 Professor Ben Richert Exam 1 Solutions Problem 1 Let X be the set of all continuous real valued functions on [, 1], and let ρ : X X R be the function ρ(f, g) = sup f g (1)

More information

ULTRAFILTER AND HINDMAN S THEOREM

ULTRAFILTER AND HINDMAN S THEOREM ULTRAFILTER AND HINDMAN S THEOREM GUANYU ZHOU Abstract. In this paper, we present various results of Ramsey Theory, including Schur s Theorem and Hindman s Theorem. With the focus on the proof of Hindman

More information

SPECTRAL THEOREM FOR SYMMETRIC OPERATORS WITH COMPACT RESOLVENT

SPECTRAL THEOREM FOR SYMMETRIC OPERATORS WITH COMPACT RESOLVENT SPECTRAL THEOREM FOR SYMMETRIC OPERATORS WITH COMPACT RESOLVENT Abstract. These are the letcure notes prepared for the workshop on Functional Analysis and Operator Algebras to be held at NIT-Karnataka,

More information

MATH 202B - Problem Set 5

MATH 202B - Problem Set 5 MATH 202B - Problem Set 5 Walid Krichene (23265217) March 6, 2013 (5.1) Show that there exists a continuous function F : [0, 1] R which is monotonic on no interval of positive length. proof We know there

More information

MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5

MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5 MATH 722, COMPLEX ANALYSIS, SPRING 2009 PART 5.. The Arzela-Ascoli Theorem.. The Riemann mapping theorem Let X be a metric space, and let F be a family of continuous complex-valued functions on X. We have

More information

Eilenberg-Steenrod properties. (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, )

Eilenberg-Steenrod properties. (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, ) II.3 : Eilenberg-Steenrod properties (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, 8.3 8.5 Definition. Let U be an open subset of R n for some n. The de Rham cohomology groups (U are the cohomology groups

More information

An Introduction to Entropy and Subshifts of. Finite Type

An Introduction to Entropy and Subshifts of. Finite Type An Introduction to Entropy and Subshifts of Finite Type Abby Pekoske Department of Mathematics Oregon State University pekoskea@math.oregonstate.edu August 4, 2015 Abstract This work gives an overview

More information

A Note on the Generalized Shift Map

A Note on the Generalized Shift Map Gen. Math. Notes, Vol. 1, No. 2, December 2010, pp. 159-165 ISSN 2219-7184; Copyright c ICSRS Publication, 2010 www.i-csrs.org Available free online at http://www.geman.in A Note on the Generalized Shift

More information

Part III. 10 Topological Space Basics. Topological Spaces

Part III. 10 Topological Space Basics. Topological Spaces Part III 10 Topological Space Basics Topological Spaces Using the metric space results above as motivation we will axiomatize the notion of being an open set to more general settings. Definition 10.1.

More information

Quick Tour of the Topology of R. Steven Hurder, Dave Marker, & John Wood 1

Quick Tour of the Topology of R. Steven Hurder, Dave Marker, & John Wood 1 Quick Tour of the Topology of R Steven Hurder, Dave Marker, & John Wood 1 1 Department of Mathematics, University of Illinois at Chicago April 17, 2003 Preface i Chapter 1. The Topology of R 1 1. Open

More information

VARIATIONAL PRINCIPLE FOR THE ENTROPY

VARIATIONAL PRINCIPLE FOR THE ENTROPY VARIATIONAL PRINCIPLE FOR THE ENTROPY LUCIAN RADU. Metric entropy Let (X, B, µ a measure space and I a countable family of indices. Definition. We say that ξ = {C i : i I} B is a measurable partition if:

More information

Math 426 Homework 4 Due 3 November 2017

Math 426 Homework 4 Due 3 November 2017 Math 46 Homework 4 Due 3 November 017 1. Given a metric space X,d) and two subsets A,B, we define the distance between them, dista,b), as the infimum inf a A, b B da,b). a) Prove that if A is compact and

More information

Real Analysis Notes. Thomas Goller

Real Analysis Notes. Thomas Goller Real Analysis Notes Thomas Goller September 4, 2011 Contents 1 Abstract Measure Spaces 2 1.1 Basic Definitions........................... 2 1.2 Measurable Functions........................ 2 1.3 Integration..............................

More information

The Structure of C -algebras Associated with Hyperbolic Dynamical Systems

The Structure of C -algebras Associated with Hyperbolic Dynamical Systems The Structure of C -algebras Associated with Hyperbolic Dynamical Systems Ian F. Putnam* and Jack Spielberg** Dedicated to Marc Rieffel on the occasion of his sixtieth birthday. Abstract. We consider the

More information

Week 5 Lectures 13-15

Week 5 Lectures 13-15 Week 5 Lectures 13-15 Lecture 13 Definition 29 Let Y be a subset X. A subset A Y is open in Y if there exists an open set U in X such that A = U Y. It is not difficult to show that the collection of all

More information

Measure Theory on Topological Spaces. Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond

Measure Theory on Topological Spaces. Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond Measure Theory on Topological Spaces Course: Prof. Tony Dorlas 2010 Typset: Cathal Ormond May 22, 2011 Contents 1 Introduction 2 1.1 The Riemann Integral........................................ 2 1.2 Measurable..............................................

More information

F 1 =. Setting F 1 = F i0 we have that. j=1 F i j

F 1 =. Setting F 1 = F i0 we have that. j=1 F i j Topology Exercise Sheet 5 Prof. Dr. Alessandro Sisto Due to 28 March Question 1: Let T be the following topology on the real line R: T ; for each finite set F R, we declare R F T. (a) Check that T is a

More information

THE SET OF RECURRENT POINTS OF A CONTINUOUS SELF-MAP ON AN INTERVAL AND STRONG CHAOS

THE SET OF RECURRENT POINTS OF A CONTINUOUS SELF-MAP ON AN INTERVAL AND STRONG CHAOS J. Appl. Math. & Computing Vol. 4(2004), No. - 2, pp. 277-288 THE SET OF RECURRENT POINTS OF A CONTINUOUS SELF-MAP ON AN INTERVAL AND STRONG CHAOS LIDONG WANG, GONGFU LIAO, ZHENYAN CHU AND XIAODONG DUAN

More information

THE INVERSE FUNCTION THEOREM

THE INVERSE FUNCTION THEOREM THE INVERSE FUNCTION THEOREM W. PATRICK HOOPER The implicit function theorem is the following result: Theorem 1. Let f be a C 1 function from a neighborhood of a point a R n into R n. Suppose A = Df(a)

More information

Ergodic Theorems. Samy Tindel. Purdue University. Probability Theory 2 - MA 539. Taken from Probability: Theory and examples by R.

Ergodic Theorems. Samy Tindel. Purdue University. Probability Theory 2 - MA 539. Taken from Probability: Theory and examples by R. Ergodic Theorems Samy Tindel Purdue University Probability Theory 2 - MA 539 Taken from Probability: Theory and examples by R. Durrett Samy T. Ergodic theorems Probability Theory 1 / 92 Outline 1 Definitions

More information

4 Countability axioms

4 Countability axioms 4 COUNTABILITY AXIOMS 4 Countability axioms Definition 4.1. Let X be a topological space X is said to be first countable if for any x X, there is a countable basis for the neighborhoods of x. X is said

More information

Math 117: Continuity of Functions

Math 117: Continuity of Functions Math 117: Continuity of Functions John Douglas Moore November 21, 2008 We finally get to the topic of ɛ δ proofs, which in some sense is the goal of the course. It may appear somewhat laborious to use

More information

MAT1000 ASSIGNMENT 1. a k 3 k. x =

MAT1000 ASSIGNMENT 1. a k 3 k. x = MAT1000 ASSIGNMENT 1 VITALY KUZNETSOV Question 1 (Exercise 2 on page 37). Tne Cantor set C can also be described in terms of ternary expansions. (a) Every number in [0, 1] has a ternary expansion x = a

More information

Definition 2.1. A metric (or distance function) defined on a non-empty set X is a function d: X X R that satisfies: For all x, y, and z in X :

Definition 2.1. A metric (or distance function) defined on a non-empty set X is a function d: X X R that satisfies: For all x, y, and z in X : MATH 337 Metric Spaces Dr. Neal, WKU Let X be a non-empty set. The elements of X shall be called points. We shall define the general means of determining the distance between two points. Throughout we

More information

MATS113 ADVANCED MEASURE THEORY SPRING 2016

MATS113 ADVANCED MEASURE THEORY SPRING 2016 MATS113 ADVANCED MEASURE THEORY SPRING 2016 Foreword These are the lecture notes for the course Advanced Measure Theory given at the University of Jyväskylä in the Spring of 2016. The lecture notes can

More information

M17 MAT25-21 HOMEWORK 6

M17 MAT25-21 HOMEWORK 6 M17 MAT25-21 HOMEWORK 6 DUE 10:00AM WEDNESDAY SEPTEMBER 13TH 1. To Hand In Double Series. The exercises in this section will guide you to complete the proof of the following theorem: Theorem 1: Absolute

More information

Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra

Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................

More information

Class Notes for MATH 255.

Class Notes for MATH 255. Class Notes for MATH 255. by S. W. Drury Copyright c 2006, by S. W. Drury. Contents 0 LimSup and LimInf Metric Spaces and Analysis in Several Variables 6. Metric Spaces........................... 6.2 Normed

More information

5 Birkhoff s Ergodic Theorem

5 Birkhoff s Ergodic Theorem 5 Birkhoff s Ergodic Theorem Birkhoff s Ergodic Theorem extends the validity of Kolmogorov s strong law to the class of stationary sequences of random variables. Stationary sequences occur naturally even

More information

CHAOTIC BEHAVIOR IN A FORECAST MODEL

CHAOTIC BEHAVIOR IN A FORECAST MODEL CHAOTIC BEHAVIOR IN A FORECAST MODEL MICHAEL BOYLE AND MARK TOMFORDE Abstract. We examine a certain interval map, called the weather map, that has been used by previous authors as a toy model for weather

More information

Measures and Measure Spaces

Measures and Measure Spaces Chapter 2 Measures and Measure Spaces In summarizing the flaws of the Riemann integral we can focus on two main points: 1) Many nice functions are not Riemann integrable. 2) The Riemann integral does not

More information

Existence and Uniqueness

Existence and Uniqueness Chapter 3 Existence and Uniqueness An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect

More information

From now on, we will represent a metric space with (X, d). Here are some examples: i=1 (x i y i ) p ) 1 p, p 1.

From now on, we will represent a metric space with (X, d). Here are some examples: i=1 (x i y i ) p ) 1 p, p 1. Chapter 1 Metric spaces 1.1 Metric and convergence We will begin with some basic concepts. Definition 1.1. (Metric space) Metric space is a set X, with a metric satisfying: 1. d(x, y) 0, d(x, y) = 0 x

More information

CLASS NOTES FOR APRIL 14, 2000

CLASS NOTES FOR APRIL 14, 2000 CLASS NOTES FOR APRIL 14, 2000 Announcement: Section 1.2, Questions 3,5 have been deferred from Assignment 1 to Assignment 2. Section 1.4, Question 5 has been dropped entirely. 1. Review of Wednesday class

More information

Math General Topology Fall 2012 Homework 6 Solutions

Math General Topology Fall 2012 Homework 6 Solutions Math 535 - General Topology Fall 202 Homework 6 Solutions Problem. Let F be the field R or C of real or complex numbers. Let n and denote by F[x, x 2,..., x n ] the set of all polynomials in n variables

More information

Math 118B Solutions. Charles Martin. March 6, d i (x i, y i ) + d i (y i, z i ) = d(x, y) + d(y, z). i=1

Math 118B Solutions. Charles Martin. March 6, d i (x i, y i ) + d i (y i, z i ) = d(x, y) + d(y, z). i=1 Math 8B Solutions Charles Martin March 6, Homework Problems. Let (X i, d i ), i n, be finitely many metric spaces. Construct a metric on the product space X = X X n. Proof. Denote points in X as x = (x,

More information