Fractals and Invariant Measures

Anna Seitz

The Cantor 3-Set Constructions Fractals and Invariant Measures Cantor Lebesgue Function Associated Measure, µ3 The Cantor Anna Seitz 4-Set Constructions Associated Measure, µ4 Iowa State University A Two- Dimensional Example 5-2-18 Skewed Sierpinski Gasket Associated Measure, ν3 References Outline

Fractals and Invariant Measures 1 The Cantor 3-Set Anna Seitz Constructions The Cantor Cantor Lebesgue Function 3-Set Constructions Associated Measure, µ3 Cantor Lebesgue Function Associated Measure, µ3 2 The Cantor 4-Set The Cantor Constructions 4-Set Constructions Associated Measure, µ4 Associated Measure, µ4 A Two- 3 A Two-Dimensional Example Dimensional Example Skewed Sierpinski Gasket Skewed Sierpinski Gasket Associated Measure, ν3 Associated Measure, ν3 References 4 References The Cantor set is uncountable

Cn+1 ⊂ Cn (descending) ∞ We can say C = ∩n=0Cn or C = limn→∞ Cn

Construction I: Deletion

Fractals and Invariant Measures

Anna Seitz

The Cantor Construct consecutive iterations by deleting middle thirds 3-Set Constructions Cantor Lebesgue Function Associated Measure, µ3 Note: The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References Cn+1 ⊂ Cn (descending) ∞ We can say C = ∩n=0Cn or C = limn→∞ Cn

Construction I: Deletion

Fractals and Invariant Measures

Anna Seitz

The Cantor Construct consecutive iterations by deleting middle thirds 3-Set Constructions Cantor Lebesgue Function Associated Measure, µ3 Note: The Cantor 4-Set Constructions The Cantor set is uncountable Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References ∞ We can say C = ∩n=0Cn or C = limn→∞ Cn

Construction I: Deletion

Fractals and Invariant Measures

Anna Seitz

The Cantor Construct consecutive iterations by deleting middle thirds 3-Set Constructions Cantor Lebesgue Function Associated Measure, µ3 Note: The Cantor 4-Set Constructions The Cantor set is uncountable Associated Measure, µ4 Cn+1 ⊂ Cn (descending) A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References Construction I: Deletion

Fractals and Invariant Measures

Anna Seitz

The Cantor Construct consecutive iterations by deleting middle thirds 3-Set Constructions Cantor Lebesgue Function Associated Measure, µ3 Note: The Cantor 4-Set Constructions The Cantor set is uncountable Associated Measure, µ4 Cn+1 ⊂ Cn (descending) A Two- ∞ Dimensional We can say C = ∩n=0Cn or C = limn→∞ Cn Example Skewed Sierpinski Gasket Associated Measure, ν3 References Construction I: Deletion

Fractals and Invariant Measures

Anna Seitz

The Cantor Construct consecutive iterations by deleting middle thirds 3-Set Constructions Cantor Lebesgue Function Associated Measure, µ3 Note: The Cantor 4-Set Constructions The Cantor set is uncountable Associated Measure, µ4 Cn+1 ⊂ Cn (descending) A Two- ∞ Dimensional We can say C = ∩n=0Cn or C = limn→∞ Cn Example Skewed Sierpinski Gasket Associated Measure, ν3 References What is the Lebesgue measure of C?

Prove it!

2 n m(Cn) = ( ) ⇒ m(C) = lim m(Cn) = 0 3 n→∞

Classic example of uncountable set with Lebesgue measure zero

Construction I: Deletion

Fractals and Invariant Measures

Anna Seitz What is the Lebesgue measure of Cn? The Cantor 3-Set Constructions Cantor Lebesgue Function Associated Measure, µ3 The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References Prove it!

2 n m(Cn) = ( ) ⇒ m(C) = lim m(Cn) = 0 3 n→∞

Classic example of uncountable set with Lebesgue measure zero

Construction I: Deletion

Fractals and Invariant Measures

Anna Seitz What is the Lebesgue measure of Cn? The Cantor 3-Set Constructions What is the Lebesgue measure of C? Cantor Lebesgue Function Associated Measure, µ3 The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References 2 n m(Cn) = ( ) ⇒ m(C) = lim m(Cn) = 0 3 n→∞

Classic example of uncountable set with Lebesgue measure zero

Construction I: Deletion

Fractals and Invariant Measures

Anna Seitz What is the Lebesgue measure of Cn? The Cantor 3-Set Constructions What is the Lebesgue measure of C? Cantor Lebesgue Function Associated Measure, µ3 Prove it! The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References Classic example of uncountable set with Lebesgue measure zero

Construction I: Deletion

Fractals and Invariant Measures

Anna Seitz What is the Lebesgue measure of Cn? The Cantor 3-Set Constructions What is the Lebesgue measure of C? Cantor Lebesgue Function Associated Measure, µ3 Prove it! The Cantor 4-Set Constructions 2 n Associated m(Cn) = ( ) ⇒ m(C) = lim m(Cn) = 0 Measure, µ4 3 n→∞ A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References Construction I: Deletion

Fractals and Invariant Measures

Anna Seitz What is the Lebesgue measure of Cn? The Cantor 3-Set Constructions What is the Lebesgue measure of C? Cantor Lebesgue Function Associated Measure, µ3 Prove it! The Cantor 4-Set Constructions 2 n Associated m(Cn) = ( ) ⇒ m(C) = lim m(Cn) = 0 Measure, µ4 3 n→∞ A Two- Dimensional Example Skewed Sierpinski Gasket Associated Classic example of uncountable set with Lebesgue measure zero Measure, ν3 References Construction II: Iterated Function Systems

Fractals and Invariant Measures

Anna Seitz

The Cantor 3-Set Constructions Cantor Lebesgue Function 1 Associated φ0(x) = x Measure, µ3 3 The Cantor 4-Set 1 2 Constructions φ1(x) = x + Associated 3 3 Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References Contractions

Fractals and Invariant Measures

Anna Seitz

The Cantor 3-Set Definition Constructions Cantor Lebesgue Function For any metric space X with distance function d, F : X → X is Associated Measure, µ3 a contraction if The Cantor 4-Set d(F (x), F (y)) Constructions Associated supx6=y { } < 1. Measure, µ4 d(x, y) A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References Iterated Function Systems

Fractals and Invariant Measures Theorem (Hutchinson) Anna Seitz If {φ , ..., φ } is a finite set of contraction maps on a complete The Cantor 0 N 3-Set metric space X , then there exists a unique compact set K such Constructions SN Cantor Lebesgue that K = φ (K). Also, K is the closure of the set of fixed Function i=0 i Associated Measure, µ3 points of finite compositions of the contraction maps. The Cantor 4-Set Constructions Associated Measure, µ4 C is the unique, compact set that is invariant under the A Two- Dimensional IFS φ0, φ1. I.e. Example Skewed Sierpinski Gasket 1 Associated [ Measure, ν3 C = φi (C). References i=0 Ternary Representation

Fractals and Invariant Measures

Anna Seitz Review: The Cantor 2 1 0 3-Set 14 = 1 · 3 + 1 · 3 + 2 · 3 Constructions Cantor Lebesgue Function So 143 = 112 Associated Measure, µ3 The Cantor We can do this with decimals as well! In base 3... 4-Set Constructions Associated −1 −2 −3 Measure, µ4 .0123 = 0 · 3 + 1 · 3 + 2 · 3 A Two- Dimensional 1 2 5 Example = + = Skewed Sierpinski Gasket 9 27 27 Associated Measure, ν3 References Ternary Representation

Fractals and Invariant Measures

Anna Seitz

The Cantor 3-Set We can write any number in [0, 1] as Constructions Cantor Lebesgue Function ∞ Associated Measure, µ X −i 3 αi 3 , where αi ∈ {0, 1, 2} The Cantor 4-Set i=1 Constructions Associated Measure, µ4 Think of this as a map to any point that you want in the A Two- interval [0, 1]. Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References ∞ X −i C = { αi 3 : αi ∈ {0, 2}} i=1

Useful for proofs such as...

C + C = [0, 2]

Construction III: Ternary Representation

Fractals and Invariant Measures

Anna Seitz

The Cantor For the Cantor Set... 3-Set Constructions Cantor Lebesgue Function Associated Measure, µ3 The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References ∞ X −i = { αi 3 : αi ∈ {0, 2}} i=1

Useful for proofs such as...

C + C = [0, 2]

Construction III: Ternary Representation

Fractals and Invariant Measures

Anna Seitz

The Cantor For the Cantor Set... 3-Set Constructions Cantor Lebesgue Function C Associated Measure, µ3 The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References Useful for proofs such as...

C + C = [0, 2]

Construction III: Ternary Representation

Fractals and Invariant Measures

Anna Seitz

The Cantor For the Cantor Set... 3-Set Constructions ∞ Cantor Lebesgue X −i Function C = { α 3 : α ∈ {0, 2}} Associated i i Measure, µ3 i=1 The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References Construction III: Ternary Representation

Fractals and Invariant Measures

Anna Seitz

The Cantor For the Cantor Set... 3-Set Constructions ∞ Cantor Lebesgue X −i Function C = { α 3 : α ∈ {0, 2}} Associated i i Measure, µ3 i=1 The Cantor 4-Set Constructions Associated Useful for proofs such as... Measure, µ4 A Two- Dimensional C + C = [0, 2] Example Skewed Sierpinski Gasket Associated Measure, ν3 References Draw a continuous function connecting the points (0, 0) and (1, 1) which has a derivative of zero almost everywhere

Cantor Lebesgue

Fractals and Invariant Measures

Anna Seitz

The Cantor 3-Set Constructions Cantor Lebesgue Draw a continuous function connecting the points (0, 0) Function Associated and (1, 1) Measure, µ3 The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References Cantor Lebesgue

Fractals and Invariant Measures

Anna Seitz

The Cantor 3-Set Constructions Cantor Lebesgue Draw a continuous function connecting the points (0, 0) Function Associated and (1, 1) Measure, µ3 The Cantor 4-Set Constructions Draw a continuous function connecting the points (0, 0) Associated Measure, µ4 and (1, 1) which has a derivative of zero almost everywhere A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References 1 2 1 1st Stage: O1 = [ 3 , 3 ] - assign value 2

1 2 1 2 7 8 1 2 3 2nd Stage: O2 = [ 9 , 9 ] ∪ [ 3 , 3 ] ∪ [ 9 , 9 ] - assign values 4 , 4 , 4

1 2 3 4 5 6 7 3rd Stage: O3 - assign values 8 , 8 , 8 , 8 , 8 , 8 , 8

1 4th Stage: O4 - work with 16 s

Continue iteratively!

Cantor Lebesgue

Fractals and Invariant Measures We work with the Cantor Complement: Anna Seitz

The Cantor 3-Set Constructions Cantor Lebesgue Function Associated Measure, µ3 The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References Cantor Lebesgue

Fractals and Invariant Measures We work with the Cantor Complement: Anna Seitz

The Cantor 1 2 1 3-Set 1st Stage: O1 = [ 3 , 3 ] - assign value 2 Constructions Cantor Lebesgue Function Associated 1 2 1 2 7 8 1 2 3 Measure, µ3 2nd Stage: O2 = [ 9 , 9 ] ∪ [ 3 , 3 ] ∪ [ 9 , 9 ] - assign values 4 , 4 , 4 The Cantor 4-Set Constructions Associated 1 2 3 4 5 6 7 Measure, µ4 3rd Stage: O3 - assign values 8 , 8 , 8 , 8 , 8 , 8 , 8 A Two- Dimensional Example 1 Skewed 4th Stage: O - work with s Sierpinski Gasket 4 16 Associated Measure, ν3 References Continue iteratively! Cantor Lebesgue

Fractals and Invariant Measures

Anna Seitz

The Cantor Formula: 3-Set Constructions ( Cantor Lebesgue j n−1 Function 2n , x ∈ On,j , 1 ≤ j ≤ 2 Associated f (x) = Measure, µ3 sup{f (y): y ≤ x}, x ∈ C The Cantor 4-Set Constructions f is continuous Associated Measure, µ4 f has a derivative of zero almost everywhere A Two- Dimensional Example f is increasing Skewed Sierpinski Gasket Associated Measure, ν3 References Cantor Lebesgue

Fractals and Invariant Measures

Anna Seitz

The Cantor 3-Set Constructions Cantor Lebesgue Function Associated Measure, µ3 The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References Invariant Measure

Fractals and Invariant Measures Theorem (Hutchinson) Anna Seitz Let {φ0, . . . , φN } be contractions, and and ρ0, . . . ρN ∈ (0, 1) The Cantor PN 3-Set with i=0 ρi = 1. Then there exists a unique, regular Borel Constructions Cantor Lebesgue measure µ with µ(X ) = 1 such that for any measurable set A, Function Associated Measure, µ3 N The Cantor X −1 4-Set µ(A) = ρi (µ(φi (A)). Constructions i=0 Associated Measure, µ4 A Two- Consequently, for any continuous function f on X , Dimensional Example Skewed Z N Z Sierpinski Gasket X Associated fdµ = (ρi f (φi (x))dµ). Measure, ν3 References i=0 Applied to the Cantor-3 Set

Fractals and Invariant Measures

Anna Seitz There exists a unique, Borel, regular measure µ3 with

The Cantor µ3(C) = 1 such that for any measurable set A, 3-Set Constructions Cantor Lebesgue 1 Function X −1 Associated µ3(A) = ρi (µ3(φ (A)). Measure, µ3 i The Cantor i=0 4-Set Constructions and for any continuous function f on X , Associated Measure, µ4 A Two- Z 1 Z Dimensional X Example fdµ3 = (ρi f (φi (x))dµ3). Skewed Sierpinski Gasket i=0 Associated Measure, ν3 References Interesting Questions 2 What kind of bases can we find for L (C, µ3)? 2 Does L (C, µ3) have an orthonormal basis of exponentials? Does this space have a frame?

Relevence

Fractals and Invariant Measures Now we can talk about: Anna Seitz Z The Cantor 2 2 3-Set L (C, µ3) = {f : |f | dµ3 < ∞} Constructions C Cantor Lebesgue Function Associated which is a Hilbert space! Measure, µ3 The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References Relevence

Fractals and Invariant Measures Now we can talk about: Anna Seitz Z The Cantor 2 2 3-Set L (C, µ3) = {f : |f | dµ3 < ∞} Constructions C Cantor Lebesgue Function Associated which is a Hilbert space! Measure, µ3 The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Interesting Questions Dimensional 2 Example What kind of bases can we find for L (C, µ3)? Skewed Sierpinski Gasket 2 Associated Does L (C, µ3) have an orthonormal basis of exponentials? Measure, ν3 References Does this space have a frame? Cantor-4 Set: Construction

Fractals and Invariant Measures

Anna Seitz

The Cantor 3-Set Constructions Cantor Lebesgue Function Associated Construct consecutive iterations by deleting the second and Measure, µ3 The Cantor fourth fourths 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References Questions on the Cantor-4 Set

Fractals and Invariant Measures

Anna Seitz What is the Lebesgue measure of C˜n? The Cantor 3-Set What is the Lebesgue measure of C˜? Justify! Constructions Cantor Lebesgue ˜ Function What is the base-4 representation of C? Associated Measure, µ3 What is the Iterated Function System under which C˜ is The Cantor 4-Set invariant? What two things must be proven to justify your Constructions Associated answer? Measure, µ4 A Two- There exists a unique Borel measure, say µ4, supported on Dimensional ˜ Example C that is invariant under the IFS stated above. Why? Skewed 2 Sierpinski Gasket Write the definition of the set L (C˜, µ4). Associated Measure, ν3 References 1 1 m(C˜ ) =
n2n =
n. n 4 2

Answers

Fractals and Invariant Measures

Anna Seitz

The Cantor What is the Lebesgue measure of C˜ ? 3-Set n Constructions Cantor Lebesgue Function Associated Measure, µ3 The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References Answers

Fractals and Invariant Measures

Anna Seitz

The Cantor What is the Lebesgue measure of C˜ ? 3-Set n Constructions Cantor Lebesgue Function Associated Measure, µ3 The Cantor 4-Set Constructions Associated Measure, µ4 1
n n 1
n A Two- m(C˜n) = 2 = . Dimensional 4 2 Example Skewed Sierpinski Gasket Associated Measure, ν3 References ∞ \ m(C˜) = m( C˜n) n=0

= lim m(C˜n) n→∞ 1 = lim
n n→∞ 2 = 0

Answers

Fractals and Invariant Measures

Anna Seitz

The Cantor What is the Lebesgue measure of C˜? Justify! 3-Set Constructions Cantor Lebesgue Function Associated Measure, µ3 The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References Answers

Fractals and Invariant Measures

Anna Seitz

The Cantor What is the Lebesgue measure of C˜? Justify! 3-Set Constructions Cantor Lebesgue Function Associated Measure, µ 3 ∞ The Cantor \ 4-Set m(C˜) = m( C˜n) Constructions Associated n=0 Measure, µ4 ˜ A Two- = lim m(Cn) Dimensional n→∞ Example 1 n Skewed
Sierpinski Gasket = lim Associated n→∞ 2 Measure, ν3 = 0 References ∞ X αi C˜ = { : α ∈ {0, 2}} 4i i i=1

Answers

Fractals and Invariant Measures

Anna Seitz

The Cantor 3-Set What is the base-4 representation of C˜? Constructions Cantor Lebesgue Function Associated Measure, µ3 The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References Answers

Fractals and Invariant Measures

Anna Seitz

The Cantor 3-Set What is the base-4 representation of C˜? Constructions Cantor Lebesgue Function Associated Measure, µ3 The Cantor 4-Set ∞ α Constructions ˜ X i Associated C = { i : αi ∈ {0, 2}} Measure, µ4 4 i=1 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References x τ (x) = 0 4 x + 2 τ (x) = 1 4

To justify you must show that: 1 C˜ is compact ˜ S1 ˜ 2 C = i=0 τi (C).

Answers

Fractals and Invariant Measures Anna Seitz What is the Iterated Function System under which C˜ is

The Cantor invariant? Justify your answer. 3-Set Constructions Cantor Lebesgue Function Associated Measure, µ3 The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References Answers

Fractals and Invariant Measures Anna Seitz What is the Iterated Function System under which C˜ is

The Cantor invariant? Justify your answer. 3-Set Constructions Cantor Lebesgue Function Associated Measure, µ3 x τ0(x) = The Cantor 4 4-Set Constructions x + 2 Associated τ1(x) = Measure, µ4 4 A Two- Dimensional Example To justify you must show that: Skewed Sierpinski Gasket Associated 1 C˜ is compact Measure, ν3 ˜ S1 ˜ References 2 C = i=0 τi (C). Hutchinson’s Theorem

Fractals and Invariant Measures

Anna Seitz

The Cantor 3-Set Constructions There exists a unique Borel measure, say µ4, supported on Cantor Lebesgue Function C˜ that is invariant under the IFS stated above. Why? Associated Measure, µ3 The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References Fractals and Invariant Measures

Anna Seitz

The Cantor 3-Set Constructions There exists a unique Borel measure, say µ4, supported on Cantor Lebesgue Function C˜ that is invariant under the IFS stated above. Why? Associated Measure, µ3 The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Hutchinson’s Theorem Skewed Sierpinski Gasket Associated Measure, ν3 References Z 2 2 L (C˜, µ4) = {f : |f | dµ4 < ∞}. C˜

Fractals and Invariant Measures

Anna Seitz

The Cantor 3-Set Write the definition of the set L2(C˜, µ ). Constructions 4 Cantor Lebesgue Function Associated Measure, µ3 The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References Fractals and Invariant Measures

Anna Seitz

The Cantor 3-Set Write the definition of the set L2(C˜, µ ). Constructions 4 Cantor Lebesgue Function Associated Measure, µ3 The Cantor 4-Set Z Constructions 2 ˜ 2 Associated L (C, µ4) = {f : |f | dµ4 < ∞}. Measure, µ4 C˜ A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References A Two-Dimensional Example

Fractals and Invariant Measures

Anna Seitz Apply the IFS The Cantor     3-Set x 1 x Constructions ψ0 = Cantor Lebesgue y 3 y Function Associated Measure, µ     3 x 1 x + 2 The Cantor ψ1 = 4-Set y 3 y Constructions Associated     Measure, µ4 x 1 x ψ2 = A Two- y y + 2 Dimensional 3 Example Skewed to the point (0, 0). Sierpinski Gasket Associated Measure, ν3 References 1 S is compact S2 2 S = i=0 ψi (S) S is known as the Skewed Sierpinski Gasket.

There exists a unique Borel measure, say ν3, supported on S that is invariant under the IFS stated above. Why? 2 Write the definition of the set L (S, ν3).

A Two-Dimensional Example

Fractals and Invariant Measures

Anna Seitz Let S be the closure of the points generated in the The Cantor previous slide. To show that S is the unique compact set 3-Set Constructions invariant under the IFS ψ0, ψ1, ψ2, by Hutchinson, it Cantor Lebesgue Function suffices to show that Associated Measure, µ3 The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References S2 2 S = i=0 ψi (S) S is known as the Skewed Sierpinski Gasket.

There exists a unique Borel measure, say ν3, supported on S that is invariant under the IFS stated above. Why? 2 Write the definition of the set L (S, ν3).

A Two-Dimensional Example

Fractals and Invariant Measures

Anna Seitz Let S be the closure of the points generated in the The Cantor previous slide. To show that S is the unique compact set 3-Set Constructions invariant under the IFS ψ0, ψ1, ψ2, by Hutchinson, it Cantor Lebesgue Function suffices to show that Associated Measure, µ3 1 S is compact The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References A Two-Dimensional Example

Fractals and Invariant Measures

Anna Seitz Let S be the closure of the points generated in the The Cantor previous slide. To show that S is the unique compact set 3-Set Constructions invariant under the IFS ψ0, ψ1, ψ2, by Hutchinson, it Cantor Lebesgue Function suffices to show that Associated Measure, µ3 1 S is compact 2 The Cantor 2 S 4-Set S = i=0 ψi (S) Constructions Associated S is known as the Skewed Sierpinski Gasket. Measure, µ4 A Two- There exists a unique Borel measure, say ν3, supported on Dimensional Example S that is invariant under the IFS stated above. Why? Skewed 2 Sierpinski Gasket Write the definition of the set L (S, ν3). Associated Measure, ν3 References Fractals and Invariant Measures

Anna Seitz

The Cantor 3-Set Constructions Cantor Lebesgue Function Associated Measure, µ3 Thank you! The Cantor 4-Set Constructions Associated Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References References

Fractals and Invariant Measures

Anna Seitz

The Cantor 3-Set Constructions Cantor Lebesgue Function Fitzpatrick and Royden, Real Analysis, 1963 Associated Measure, µ3 Hutchinson, ”Fractals and self-similarity”, 1981 The Cantor 4-Set Hotchkiss, ”Fourier Bases on the Skewed Sierpinski Constructions Associated Gasket”, 2017 Measure, µ4 A Two- Dimensional Example Skewed Sierpinski Gasket Associated Measure, ν3 References