Self-similar Fractals: Projections, Sections and Percolation

Transcription

1 Self-similar Fractals: Projections, Sections and Percolation University of St Andrews, Scotland, UK

2 Summary Self-similar sets Hausdorff dimension Projections Fractal percolation Sections or slices Projections percolation sections For this talk we will generally work in R 2.

3 Self-similar sets Example: Middle-third Cantor set (in R) Made up of 2 copies of itself scaled by a factor 1 3. In particular, if f 1, f 2 : R R are the similarities f 1 (x) = 1 3 x; f 1(x) = 1 3 x + 2 3, then E = f 1 (E) f 2 (E). (1)

4 Self-similar sets A mapping f : R 2 R 2 with f (x) f (y) = r x y (x, y R 2 ) is called a similarity with ratio r. If 0 < r < 1 then f is contracting.

5 Self-similar sets A mapping f : R 2 R 2 with f (x) f (y) = r x y (x, y R 2 ) is called a similarity with ratio r. If 0 < r < 1 then f is contracting. A family of contracting similarities f 1,..., f m : R 2 R 2 is a (special case of) an iterated function system or IFS. Theorem (Self-similar sets) Given an IFS of contracting similarities there exists a unique non-empty compact set E R 2 such that called a self-similar set. E = m f i (E) i=1

6 Self-similar sets Example: (Right-angled) Sierpiński triangle f 1 (x, y) = ( 1 2 x, 1 2 y); f 2(x, y) = ( 1 2 x + 1 2, 1 2 y); f 3(x, y) = ( 1 2 x, 1 2 y ). E = f 1 (E) f 2 (E) f 3 (E)

7 Iterative construction of a self-similar set Start with a region D and take its images under iterations of the f i. Then E = k=0 1 i 1,...,i k m f i 1 f i2 f ik (D).

8 Iterative construction of a self-similar set Start with a region D and take its images under iterations of the f i. Then E = k=0 1 i 1,...,i k m f i 1 f i2 f ik (D).

9 Iterative construction of a self-similar set Start with a region D and take its images under iterations of the f i. Then E = k=0 1 i 1,...,i k m f i 1 f i2 f ik (D).

10 Iterative construction of a self-similar set Start with a region D and take its images under iterations of the f i. Then E = k=0 1 i 1,...,i k m f i 1 f i2 f ik (D).

11 Iterative construction of a self-similar set Start with a region D and take its images under iterations of the f i. Then E = k=0 1 i 1,...,i k m f i 1 f i2 f ik (D).

12 Iterative construction of a self-similar set Start with a region D and take its images under iterations of the f i. Then E = k=0 1 i 1,...,i k m f i 1 f i2 f ik (D).

13 Self-similar sets Self-similar sets: Sierpiński triangle and carpet, snowflake and spiral

14 More self-similar sets

15 Yet more self-similar sets

16 Hausdorff dimension The Hausdorff dimension of a set E R 2 is { dim H E = inf s : for all ɛ > 0 there is a countable cover {U i } of E such that } (diam U i ) s < ɛ. i

17 Dimension of self-similar sets Let f 1,..., f m : R 2 R 2 be contracting similarities and let E be the associated self-similar set satisfying m E = f i (E). i=1 ( ) Then provided, the union in ( ) is nearly disjoint (i.e. strong-separation condition or open set condition holds), dim H E = s where m i=1 r s i = 1, where r i is the similarity ratio of f i. E.g. Hausdorff dimension of the Sierpiński triangle is given by 3(1/2) s = 1 or s = log 3/ log 2.

18 Von Koch curves of various dimensions

19 Marstrand s projection theorems Theorem (Marstrand 1954) Let E R 2 be a Borel set. Then, for almost all θ [0, π), (i) dim H proj θ E = min{dim H E, 1}, (ii) L(proj θ E) > 0 if dim H E > 1. [proj θ denotes orthogonal projection onto the line L θ, dim H is Hausdorff dimension, L is Lebsegue measure or length.]

20 Bristol 2014: Marstrand, Mattila, Falconer, Davies

21 Exceptional directions Marstrand s theorem tells nothing about which particular directions may have projections with dimension or measure smaller than normal, i.e. when dim H proj θ E < min{dim H E, 1}, or dim H E > 1 and L(proj θ E) = 0. The set shown has dimension log 4/ log(5/2) = 1.51, but with some projections of dimension < 1.

22 Projections of specific self-similar sets How dim H proj θ E varies with θ has been investigated in certain specific cases. For example: Let E be the 1-dimensional Sierpínski triangle, so dim H E = 1. For projections in direction θ: (a) if θ = p/q is rational, and p + q 0 (mod 3) dim H proj θ E < 1; and p + q 0 (mod 3) proj θ E contains an interval, (b) if θ is irrational, dim H proj θ E = 1 but L(proj θ E) = 0. (Kenyon 1997, Hochman 2014) Similar investigations have been done for certain other regular sets.

23 Self-similar sets with rotations Write the similarities on R 2 as f i (x) = r i O i (x) + t i where 0 < r i < 1 is the scale factor, O i is a rotation and t i is a translation. We say that the family {f 1,..., f m } has dense rotations if at least one of the O i is a rotation by an irrational multiple of π, equivalently if group{o 1,..., O m } is dense in SO(2, R).

24 Self-similar sets with rotations Theorem (Peres & Shmerkin 2009, Hochman & Shmerkin 2012) Let E R 2 be a self-similar set defined by a family {f 1,..., f m } of similarities with dense rotations. Then dim H proj θ E = min{dim H E, 1} for all θ.

25 Self-similar sets with rotations Theorem (Peres & Shmerkin 2009, Hochman & Shmerkin 2012) Let E R 2 be a self-similar set defined by a family {f 1,..., f m } of similarities with dense rotations. Then dim H proj θ E = min{dim H E, 1} for all θ. Proof uses CP chains. Depends on the ergodic behaviour of the scenery flow of pairs (x, µ) where x E and µ is a normalised measure, under the transformation (x, µ) (f 1 i (x), (f 1 i µ) ).

26 Measure of projections These methods are unable to show that for a self-similar set E with dim H E > 1 all projections have positive measure. However, this is very nearly so in the plane.

27 Measure of projections These methods are unable to show that for a self-similar set E with dim H E > 1 all projections have positive measure. However, this is very nearly so in the plane. Theorem (Shmerkin & Solomyak 2014) Let E R 2 be the self-similar attractor of an IFS with dense rotations with 1 < dim H E < 2. Then L(proj θ E) > 0 for all θ except for a set of θ of Hausdorff dimension 0. The proof involves a careful analysis of how the Fourier transform of the projections of a natural measure on E varies with θ.

28 Mandelbrot percolation on a square Squares are repeatedly divided into 3 3 subsquares Each square is retained independently with probability p ( 0.6).

29 Mandelbrot percolation on a square Squares are repeatedly divided into 3 3 subsquares Each square is retained independently with probability p ( 0.6).

30 Mandelbrot percolation on a square Squares are repeatedly divided into 3 3 subsquares Each square is retained independently with probability p ( 0.6).

31 Mandelbrot percolation on a square Squares are repeatedly divided into 3 3 subsquares Each square is retained independently with probability p ( 0.6).

32 Mandelbrot percolation on a square Squares are repeatedly divided into 3 3 subsquares Each square is retained independently with probability p ( 0.6).

33 Mandelbrot percolation on a square If p > 1/M 2 then E p with positive probability, conditional on which dim H E p = 2 + log p/ log M almost surely.

34 Projections of Mandelbrot percolation Theorem (Rams & Simon, 2012) Let E p be the Mandelbrot percolation set obtained by dividing sqaures into M M subsquares, each square being retained with probability p > 1/M 2. Conditional on E p : (i) dim H proj θ E p = min{dim H E p, 1},

35 Projections of Mandelbrot percolation Theorem (Rams & Simon, 2012) Let E p be the Mandelbrot percolation set obtained by dividing sqaures into M M subsquares, each square being retained with probability p > 1/M 2. Conditional on E p : (i) dim H proj θ E p = min{dim H E p, 1}, (ii) if p > 1/M then dim H E p > 1, and, for all θ [0, π), proj θ E p contains an interval and in particular L(proj θ E p ) > 0. Proof depends on a geometrical analysis of how lines intersect the grid squares.

36 Percolation on a self-similar set Start with a self similar set. At each stage of the iterated construction, retain each component with probability p.

37 Percolation on a self-similar set Start with a self similar set. At each stage of the iterated construction, retain each component with probability p.

38 Percolation on a self-similar set Start with a self similar set. At each stage of the iterated construction, retain each component with probability p.

39 Percolation on a self-similar set Start with a self similar set. At each stage of the iterated construction, retain each component with probability p.

40 Percolation on a self-similar set Start with a self similar set. At each stage of the iterated construction, retain each component with probability p.

41 Percolation on a self-similar set Start with a self similar set. At each stage of the iterated construction, retain each component with probability p.

42 Percolation on a self-similar set If p > 1/m then E p with positive probability, conditional on which dim H E p = s, where p m i=1 r i s = 1

43 Projection of percolation on a self-similar set with rotations Let E R 2 be a self-similar set defined by a family {f 1,..., f m } of similarities with dense rotations. Perform the percolation process on E with respect to its hierarchical construction, with each similar component retained independently with probability p, to obtain a percolation set E p E.

44 Projection of percolation on a self-similar set with rotations Let E R 2 be a self-similar set defined by a family {f 1,..., f m } of similarities with dense rotations. Perform the percolation process on E with respect to its hierarchical construction, with each similar component retained independently with probability p, to obtain a percolation set E p E. Theorem (Jin & F, 2014) Let E have dense rotations (+OSC) and let p > 1/m. Then E p with positive probability, conditional on which dim H proj θ E p = min{dim H E p, 1} for all θ. The is a special case of a result on projections (and other smooth images) of random cascade measures on self-similar sets. Proof uses CP trees on a space of random measures.

45 Marstrand s section theorem Theorem (Marstrand 1954) Let E R 2 be a Borel set with dim H E > 1. Then, (i) for all θ [0, π) dim H (E proj 1 θ a) dim H E 1 for L-almost all a, (ii) for L-almost all θ [0, π) L { a L θ : dim H (E proj 1 θ a) dim H E 1 } > 0.

46 Marstrand s section theorem Question: When is L { a L θ : dim H (E proj 1 θ a) dim H E 1 } > 0 for all θ rather than almost all θ?

47 Marstrand s section theorem Question: When is L { a L θ : dim H (E proj 1 θ a) dim H E 1 } > 0 for all θ rather than almost all θ? The graph of a function can have dimension as large as 2. However, E proj 1 0 a is a single point for all a L 0, so dim H (E proj 1 0 a) = 0.

48 Sections of self-similar sets Theorem (F & Jin 2014) Let E R 2 be a self-similar set with dense rotations with 1 < dim H E 2. Then, for all ɛ > 0: (i) L { a L θ : dim H (E proj 1 θ a) > dim H E 1 ɛ } > 0 for all θ except for a set of θ of Hausdorff dimension 0.

49 Sections of self-similar sets Theorem (F & Jin 2014) Let E R 2 be a self-similar set with dense rotations with 1 < dim H E 2. Then, for all ɛ > 0: (i) L { a L θ : dim H (E proj 1 θ a) > dim H E 1 ɛ } > 0 for all θ except for a set of θ of Hausdorff dimension 0. (ii) If, in addition, E is connected or proj θ E is an interval for all { θ, then dim H a Lθ : dim B (E proj 1 θ a) > dim H E 1 ɛ } = 1 for all θ.

50 Sections of self-similar sets Theorem (F & Jin 2014) Let E R 2 be a self-similar set with dense rotations with 1 < dim H E 2. Then, for all ɛ > 0: (i) L { a L θ : dim H (E proj 1 θ a) > dim H E 1 ɛ } > 0 for all θ except for a set of θ of Hausdorff dimension 0. (ii) If, in addition, E is connected or proj θ E is an interval for all { θ, then dim H a Lθ : dim B (E proj 1 θ a) > dim H E 1 ɛ } = 1 for all θ. To obtain such results on sections of a self-similar set E we use the results on projections of percolation subsets of E.

51 Percolation to analyse dimensions of deterministic sets To find the Hausdorff dimension of subsets of the self-similar set E (OSC assumed) it is enough to take covers by basic sets of the iterative construction of E, that is sets of the form f i1 f ik (D). Thus, for F E: { dim H F = inf s : for all ɛ > 0 there are basic sets {U i } with F U i and } (diam U i ) s < ɛ. i i

52 Percolation to analyse dimensions of deterministic sets We can use (random) percolation sets to test the dimension of deterministic (i.e. non-random) subsets of self-similar sets. Lemma Let E be a self-similar set (with OSC, say) constructed iteratively with basic sets {U i }. Let E p be obtained by fractal percolation on the basic sets {U i }, and suppose for some α > 0 P{U i survives the percolation process} c(diam U i ) α for all i. If F E and dim H F < α then E p F = almost surely.

53 Percolation to analyse dimensions of deterministic sets We can use (random) percolation sets to test the dimension of deterministic (i.e. non-random) subsets of self-similar sets. Lemma Let E be a self-similar set (with OSC, say) constructed iteratively with basic sets {U i }. Let E p be obtained by fractal percolation on the basic sets {U i }, and suppose for some α > 0 P{U i survives the percolation process} c(diam U i ) α for all i. If F E and dim H F < α then E p F = almost surely. In particular, if F E and E p F with positive probability, then dim H F α.

54 Percolation to analyse dimensions of deterministic sets Lemma Let E p be obtained from the self-similar set E by fractal percolation on the basic sets {U i }, and suppose P{U i survives the percolation process} c(diam U i ) α for all i. If F E and dim H F < α then E p F = almost surely.

55 Percolation to analyse dimensions of deterministic sets Lemma Let E p be obtained from the self-similar set E by fractal percolation on the basic sets {U i }, and suppose P{U i survives the percolation process} c(diam U i ) α for all i. If F E and dim H F < α then E p F = almost surely. Proof Given ɛ > 0 let I be a family of indices such that F U i and (diam U i ) α < ɛ. i I Then E ( #{i I : E p U i } ) i I i I P{U i survives} c i I (diam U i ) α < cɛ, so P ( E p F ) P ( E p i I U i ) < cɛ. This is true for all ɛ > 0, so P ( E p F ) = 0.

56 Using percolation to analyse sections of self-similar sets Proposition Let E be a self-similar set (with OSC, say) constructed iteratively using a hierarchy of basic sets {U i }. Let E p be the random set obtained by some fractal percolation process on the {U i } and suppose that for some α > 0 P{U i survives the percolation process} c(diam U i ) α for all i. For each θ, if then P { L(proj θ E p ) > 0 } > 0, L { a L θ : dim H (E proj 1 θ a) α} > 0.

57 Using percolation to analyse sections of self-similar sets Proof Let S = { a L θ : dim H (E proj 1 θ a) < α}. For each a S, taking F = E proj 1 a in the lemma, E p proj 1 θ θ a = E p E proj 1 θ a = almost surely. In other words, for each a S, a proj θ E p with probability 1.

58 Using percolation to analyse sections of self-similar sets Proof Let S = { a L θ : dim H (E proj 1 θ a) < α}. For each a S, taking F = E proj 1 a in the lemma, E p proj 1 θ θ a = E p E proj 1 θ a = almost surely. In other words, for each a S, a proj θ E p with probability 1. By Fubini s theorem, with probability 1, a proj θ E p for L-almost all a S. Hence, with positive probability, 0 < L(proj θ E p ) = L ( (proj θ E p ) \ S ) L ( (proj θ E) \ S ).

59 Sections of self-similar sets Theorem (F & Jin 2014) Let E R 2 be a self-similar set with dense rotations with 1 < dim H E 2. Then, for all ɛ > 0: (i) L { a L θ : dim H (E proj 1 θ a) > dim H E 1 ɛ } > 0 for all θ except for a set of θ of Hausdorff dimension 0. (ii) If, in addition, E is connected or proj θ E is an interval for all { θ, then dim H a Lθ : dim B (E proj 1 θ a) > dim H E 1 ɛ } = 1 for all θ.

60 Sections of Mandelbrot percolation Theorem (F & Jin 2014) Let E p be the Mandelbrot percolation set obtained by dividing sqaures into M M subsquares, each square being retained with probability p > 1/M 2. Then, for all ɛ > 0, conditional on E p, for all θ. L { a L θ : dim H (E p proj 1 θ a) > dim H E p 1 ɛ } > 0

61 Sections of Mandelbrot percolation Theorem (F & Jin 2014) Let E p be the Mandelbrot percolation set obtained by dividing sqaures into M M subsquares, each square being retained with probability p > 1/M 2. Then, for all ɛ > 0, conditional on E p, for all θ. L { a L θ : dim H (E p proj 1 θ a) > dim H E p 1 ɛ } > 0 Proof. Similar idea, using that the intersection of two independent percolation sets E p E q has the same distribution as the single percolation set E pq.

62 Thank you!