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Fractals, Self-Similarity and Hausdorff Dimension Undergraduate Colloquium: Fractals, Self-similarity and Hausdorff Dimension Andrejs Treibergs University of Utah Wednesday, August 31, 2016 2. USAC Lecture on Fractals The URL for these Beamer Slides: \Fractals: self similar fractional dimensional sets" http://www.math.utah.edu/~treiberg/FractalSlides.pdf 3. References Michael Barnsley, \Lecture Notes on Iterated Function Systems," in Robert Devaney and Linda Keen, eds., Chaos and Fractals, Proc. Symp. 39, Amer. Math. Soc., Providence, 1989, 127{144. Gerald Edgar, Classics on Fractals, Westview Press, Studies in Nonlinearity, Boulder, 2004. Jenny Harrison, \An Introduction to Fractals," in Robert Devaney and Linda Keen, eds., Chaos and Fractals, Proc. Symp. 39, Amer. Math. Soc., Providence, 1989, 107{126. Yakov Pesin & Vaughn Climengha, Lectures on Fractal Geometry and Dynamical Systems, American Mathematical Society, Student Mathematical Library 52, Providence, 2009. R. Clark Robinson, An Introduction to Dynamical Systems: Continuous and Discrete, Pearson Prentice Hall, Upper Saddle River, 2004. Shlomo Sternberg, Dynamical Systems, Dover, Mineola, 2010. 4. Outline. Fractals Middle Thirds Cantor Set Example Attractor of Iterated Function System Cantor Set as Attractor of Iterated Function System. Contraction Maps Complete Metric Space of Compact Sets with Hausdorff Distance Hutchnson's Theorem on Attractors of Contracting IFS Examples: Unequal Scaling Cantor Set, Sierpinski Gasket, von Koch Snowflake, Barnsley Fern, Minkowski Curve, Peano Curve, L´evy Dragon Hausdorff Measure and Dimension Dimension of Cantor Set by Covering by Intervals Similarity Dimension Similarity Dimension of Cantor Set Similarity Dimension for IFS of Similarity Transformations Moran's Theorem Similarity Dimensions of Examples Kiesswetter's IFS Construction of Nowhere Differentiable Function 5. Fractal. Cantor Set. A fractal is a set with fractional dimension. A fractal need not be self-similar. In this lecture we construct self-similar sets of fractional dimension. The most basic fractal is the Middle Thirds Cantor Set. One starts from an interval I1 = [0; 1] and at each successive stage, removes the middle third of the intervals remaining in the set. 1 2 I = 0; [ ; 1 2 3 3 1 2 1 2 7 8 I = 0; [ ; [ ; [ ; 1 3 9 9 3 3 9 9 1 2 1 2 7 8 1 I = 0; [ ; [ ; [ ; 4 27 27 9 9 27 27 3 2 19 20 7 8 25 26 [ ; [ ; [ ; [ ; 1 3 27 27 9 27 27 27 ··· T1 Then the Cantor Set is the limit C = n=1 In. 6. Picture of Cantor Sets Figure: The sequence fIng approximating the middle thirds Cantor Set. 7. “Butterfly” Attractor of Lorenz Equations. “Butterfly” ODE limit set is a non self-similar fractal 1 < dimH (A) < 2 8. Cantor Set as the Attractor of an Iterated Function System The Cantor Set may be constructed using Iterated Function Systems. The IFS is given by two maps on the line, F = f`; rg, where x x + 2 `(x) = ; r(x) = : 3 3 ` and r make two shrunken copies of the original interval located at the left and right ends. Define the induced union map taking compact sets A ⊂ R to new compact sets consisting of both shrunken copies F(A) = `(A) [ r(A) where `(A) = f`(x): x 2 Ag. Consider the dynamical system of iterating the maps. We get the Cantor Set as its its attractor (limit) ◦n I2 = F(I1); I3 = F(I2);:::; C = lim F (I1) n!1 where we define F ◦ F(A) = F(F(A)) and n times z }| { F ◦n(A) = F ◦ F ◦ · · · ◦ F(A) 9. Metric Spaces Why does the sequence of sets converge? Let us put the structure of a metric space on the space of compact sets and do a little analysis. n For example, the distance function d on Euclidean Space X = E is v u n uX 2 d(x; y) = kx − yk = t (xi − yi ) : i=1 Euclidean Space has the structure of a metric space, namely, for all x; y; z 2 X we have d(x; x) = 0, d(x; y) = d(y; x), d(x; z) ≤ d(x; y) + d(y; z) triangle inequality (which implies d(x; x) ≥ 0), d(x; y) = 0 implies x = y. 10. Complete Metric Spaces n fxi g ⊂ E is a Cauchy Sequence if for every > 0 there is an N such that d(xi ; xj ) < whenever i; j ≥ N. Euclidean Space is a complete metric space because all Cauchy Sequences n converge. Namely, if fxi g is a Cauchy Sequence, then there is z 2 E such that xi ! z as i ! 1, i.e., for all > 0, there is N > 0 such that d(xi ; z) < whenever i > N. A set K is compact if every sequence fxi g ⊂ K has a subsequence that n converges to a point of K. In Euclidean Space, K 2 E is compact if and only if it is closed and bounded (Heine Borel Theorem). n n Surprisingly, the space K(E ) of all compact sets E and can be endowed with the structure of a complete metric space under the Hausdorff Metric. 11. -Collar of a Set n Let K(E ) denote the nonempty compact subsets. n For any A 2 K(E ) and ≥ 0 define the the -collar of A to be points within of A n A = fx 2 E : d(x; y) ≤ for some y 2 Ag: The distance of a point x to A is d(x; A) = inf d(x; y): y2A It is zero if x 2 A. The -collar may also be given n A = fx 2 E : d(x; A) ≤ g: The infimum is achieved since A is compact. There Figure: -Collar of A. is a y 2 A so that d(x; y) = d(x; A): 12. Hausdorff Distance n Given compact sets A; B 2 K(E ), if we let d(A; B) = max d(x; B): x2A d(A; B) ≤ implies that A ⊂ B. BUT d(A; B) MAY NOT EQUAL d(B; A) so it is not a metric. e.g., 2 A = fx 2 E : jxj ≤ 1g, B = f(2; 0)g then d(B; A) = 1 so B ⊂ A1 but d(A; B) = 3 and A 6⊂ B1. Hausdorff introduced h(A; B) = maxfd(A; B); d(B; A)g = inff ≥ 0 : A ⊂ B and B ⊂ Ag n Theorem (Completeness of K(E )) n K(E ) with Hausdorff Distance h is a complete metric space. n Furthermore, h satisfies for all A; B; C; D 2 K(E ) h(A [ B; C [ D) ≤ maxfh(A; C); h(B; D)g n 13. Proof of the Completeness Theorem for (K(E ); h) Proof. Symmetry (h(A; B) = h(B; A)) and positive definiteness (h(A; B) ≥ 0 with h(A; B) = 0 () A = B) are obvious. To prove the triangle inequality it suffices to show d(A; B) ≤ d(A; C) + d(C; B): This implies the triangle inequality for h: h(A; B) = maxfd(A; B); d(B; A)g ≤ maxfd(A; C) + d(C; B); d(B; C) + d(C; A)g ≤ maxfh(A; C) + h(C; B); h(B; C) + h(C; A)g = h(A; C) + h(C; B): 14. Proof of the Completeness Theorem- Now to show d(A; B) ≤ d(A; C) + d(C; B), d(a; B) = min d(a; b) b2B ≤ min min (d(a; c) + d(c; b)) c2C b2B ≤ min d(a; c) + min min d(c; b) c2C c2C b2B ≤ d(a; C) + min d(c; B) c2C ≤ d(a; C) + min d(C; B) c2C ≤ d(a; C) + d(C; B) Maximizing the right side over a 2 A gives d(a; B) ≤ d(A; C) + d(C; B) Maximizing over a 2 A, d(A; B) ≤ d(A; C) + d(C; B): 15. Proof of the Completeness Theorem- - Sketch of completeness argument: suppose An is a Cauchy Sequence in (K(X ); h). Define A1 to be the set of cluster points of sequences fxng where xn 2 An. Thus x 2 A1 if and only if there is a subsequence of this type such that xkj ! x as j ! 1. Since the sets form a Cauchy Sequence, for every > 0 there is an R() so that h(An; Am) < whenever m; n ≥ R(). In particular, Am ⊂ (An) for all m ≥ n ≥ R() so any sequence xm 2 Am is bounded and thus has a cluster point showing A1 is nonempty. Limits satisfy A1 ⊂ (An) for all n ≥ R(), hence A1 is bounded. A convergent sequence of cluster points is a cluster point, so A1 is closed, thus A1 is compact. To show that An ⊂ (A1) whenever n ≥ R(), pick zn 2 An. For k ≥ R(), h(An; Ak ) < , so there is xk 2 Ak so d(xk ; zn) < . Let z 2 A1 be a cluster point of fxk g. For its converging subsequence d(z; zm) = limj!1 d(xkj ; zm) ≤ so zm 2 (A1). Putting the containments together shows h(Am; A1) ≤ for all m ≥ R(), thus Am converges to A1 in the Hausdorff metric. 16. Contraction . n n A mapping f : E ! E is a λ-contraction if there is a constant 0 ≤ λ < 1 such that n d(f (x); f (y)) ≤ λd(x; y); for all x; y 2 E . Lemma n n n If f : E ! E is a λ-contraction, then the induced map on K(E ) is a contraction in the Hausdorff Metric with the same constant n h(f (A); f (B)) ≤ λh(A; B); for all A; B 2 K(E ) . n Proof. Choose A; B 2 K(E ). d(f (A); f (B)) = max d(f (a); f (B)) ≤ λ max d(a; B) = λd(A; B): a2A a2A Similarly, d(f (B); f (A)) ≤ λd(B; A).
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