Topological Growth Rates and Fractal Dimensions

Transcription

1 Chapter 5 Topological Growth Rates and Fractal Dimensions 5.1 Introduction Throughout this thesis, we obsere close correlations between alues of the topological growth rates and arious other fractal indices. These obserations are based on both analytic deriations and numerical computations of the releant exponents. In this chapter we derie inequalities that relate our topological growth rates to existing scaling indices such as the box-counting dimension and the Besicoitch-Taylor exponent. Such relationships lead to a better understanding of the topological growth rates. The chapter has three sections. We start by giing definitions of box-counting dimension, fat fractal exponents and Besicoitch-Taylor index. These measures of fractal scaling hae close connections with one another, and with the topological growth rates. Sections to examine the disconnectedness and discreteness indices, and for subsets of and. The most detailed results are for compact totally disconnected subsets of the line; these are gien in Section Such sets are defined in terms of countably many complementary open interals. It is well known that the fractal dimension is related to the scaling of the lengths of these deleted interals. We adapt this result to show that and are also related to this scaling. In Section we study subsets of higher-dimensional spaces, and obtain simple inequalities inoling,, and the box-counting dimension. We gie examples in Section to illustrate some of the cases for the inequalities of Sections and consequence of the results in this chapter is that for ero measure Cantor subsets of and proiding the appropriate limits exist. lthough the disconnectedness index takes the same alue as the box-counting dimension under these conditions, we emphasie that this does not imply that is a fractal dimension. ny definition of fractal dimension should extend the classical notion, and therefore an -dimensional manifold must hae dimension, for example. The disconnectedness index, howeer, is ero for any compact, connected manifold. In Section 5.3.4, we take a first step towards relating the growth rate of -dimensional holes to the box-counting dimension. We discuss some of the many open questions in the concluding section of this chapter. The most interesting unproen conjecture concerns strictly self-similar fractals. We hae obsered, for the examples in this thesis, that when a self-similar fractal has a non-ero, then it takes 101

2 the same alue as the similarity dimension. This is not surprising self-similarity is a strong condition and we expect it to dominate any scaling properties. 5.2 Definitions We recall the necessary definitions of box-counting dimension, fat fractal exponents, and the Besicoitch-Taylor index a scaling index that proides a link between our topological growth rates and fractal dimensions Box-counting dimension We discussed the box-counting dimension and its relationship to the Hausdorff dimension briefly in Chapter 1. Here, we restate the definition and gie an equialent formulation in terms of -neighborhoods of a set. Box-counting dimension Recall from Chapter 1 that the box-counting dimension is defined in terms of coers of the fractal by sets of sie. If! is the smallest number of sets with diameters at most needed to coer ", then # %$ &('*)! Of course, this limit may not exist, in which case the $ 0/2143 and $ are used. The corresponding limits are the upper and lower box-counting dimensions, 7 and. The number 8! can be defined in many ways, all of which yield an equialent alue of (see Falconer [23] for details). The definitions of! that we use in Section 5.3 are 1. the smallest number of closed balls of radius that coer " ; and 2. the largest number of disjoint balls of radius with centers in ". Minkowski dimension (5.1) The Minkowski dimension is the scaling rate of the Lebesgue measure of the -neighborhoods of ". We write 9:;" for the Lebesgue measure in, and " & for an -neighborhood of ". The definition of Minkowski dimension is as follows: =< %$ &('*) >; 9:;" (5.2) If the limit does not exist we use the $ 0/C1 3 and the $ This definition of dimension is equialent to box-counting. To see this, let! be the largest number of disjoint balls from definition 2 aboe. If we write D for the olume of the unit ball in i.e., DF BDHG I DHJ LK INMO, etc. then 9:;" & P D! If we triple the radius of the balls, we hae that 9:;" & D O 8! 102

3 } j < ; see [23] for a more detailed proof. The two different formulations of box-counting dimension mean we can derie different types of relationships with the topological growth rates. These bounds imply that # Fat fractal exponents We hae gien a few examples of fat fractals in this thesis. Recall that these sets hae positie Lebesgue measure and therefore integer Hausdorff and box-counting dimensions. It is possible to characterie the irregular structure of a fat fractal by modifying the definition of Minkowski dimension. The measure of the -neighborhoods " & conerges to the measure of " ; the fat fractal exponent characteries the conergence rate as follows: SR T$ 0/C1 3 &('*) > ;9:;" & (5.3) VR The exponent is not a dimension because it gies inconsistent alues for the dimension of the unit -cube, WV, depending on the ambient space. If WX0YZ VR, then [ ; but if WX\Y] ^ with `_ R, then. Umberger et al. [20, 22, 84] gie a finer characteriation of the scaling of 9:;" & by separating out contributions from fattening and filling in holes of the fractal. This inoles considering the fattening of " to its -neighborhood " & and then the unfattening of " & to a set a &. The unfattening operation is achieed by fattening the complement of " &, i.e., a & 0; b" & & a The set & is larger than " any structures of sie less than are filled in or smoothed out. Now define c=! a 9: & U9:;", and d7! : 9:;" & a U9: &. c=! is the measure of filled in holes i.e., the small-scale structure and d! is the measure of fattening caused by large-scale structure. Umberger et al. define scaling rates for both ce! and d7! as gfih, and use these rates as a characteriation of fat fractal structure. In the notation of [22] d7! Note that c=! lk d7! : j %$ ce! &('*) and 9:;" & U9:;" R so The Besicoitch-Taylor index j %$ &('*) U = 5lm This index is deried from a process of packing the complement of a set with regular cells. We start by describing the case for compact totally disconnected subsets of. Let "oyqp r#2s2t be such a set. The complement p r#2s2txu" a is the union of a countable number of open interals,, for w 2x [81]. That is, FF " a p ry2s2tl{ (5.4) We let } a~ 9: a, the Lebesgue measure of, and assume that the sets are ordered by decreasing length, i.e., } P }#G P. Since " Yp ry2s2t a~ and the are disjoint, we hae that FF 9:;" sgur $+.- j:n. 103

4 a a > n U 1 U 2 U 3 U 4 Figure 5.1: Packing the complement of a fractal. The conergence of the series } can be characteried in a number of ways. The original formulation of Besicoitch and Taylor is the index: 546 m ˆ Š } Œ Ž Properties of conergent monotone series can be used to show that the following rates are equialent to the Besicoitch-Taylor index [81]. $ /C143 ' $ /C143 ' (5.5) } (5.6) } } We show in Section that (5.6) has a close connection with our disconnectedness index. Tricot [80] extends the definition of the Besicoitch-Taylor index to subsets " Y ~ by packing a bounded complementary region with regular cells. a For example, we can use -cubes with faces that are parallel to the coordinate axes. Let ) be the smallest closed -cube that contains ", and let } ) a be the side length of ) a. Now let a be the largest cube in ) ]", G be the largest a a cube in ) 0", and so on; see Figure 5.1. If we set } equal to the side length of, then we can define the Besicoitch-Taylor index as in (5.5). The Lebesgue a measure 9: }, so the series } conerges, and therefore. The equialent formula in (5.6) remains the same, but (5.7) becomes $ 0/2143 ' >; } (5.8) We also note that the sets used in the packing can be more general than -cubes; see [80] for details. The cut-out sets described in Falconer [24] inole similar ideas. 104

5 5.2.4 Topological growth rates For ease of reference, we recall definitions from Chapters 2 and 3 for scaling rates in the number of components, sie of components, and persistent Betti numbers. Disconnectedness and discreteness For disconnected sets, the rate of growth in the number of -connected components, =!, is measured by the disconnectedness index,. That is, =! Fš4, and œ%$ =! &;') $+.- (5.9) The sie of the -components is measured by the largest component diameter, žÿ!. If a set is totally disconnected then ž! and %$ ž! &('*) (5.10) is the discreteness index. If the limits do not exist, we use the $ or $ 0/2143 and write # H l H C H or H for the corresponding indices. We also note that the resolution parameter is related to distances between points in the set. Growth rates of Betti numbers j ) In Chapter 3 we introduced the notion of persistent Betti number, ;" &, to count the number of -dimensional holes in a space as a function of resolution. Here, the parameter relates to the j ) -neighborhood, so it is a radius measurement. If ;" & f as gfih, then we characterie the rate of growth by the following index j *%$ ) ;" & &('*) s always, if the limit does not exist, we use the $ 0/C1 3 or $ Recall that for (5.11) h, the Betti number is just the number of connected components, so the definition of ) agrees with that for the disconnectedness index,. In the definition of, we compare the number of holes to their sie. The Besicoitch-Taylor index also compares a number with a sie parameter; this is the reason we expect a link between the two. 5.3 Results In this section, we derie a number of inequalities that relate our topological growth rates to different fractal scaling indices. Sections and examine the disconnectedness and discreteness indices. The first results are for totally disconnected subsets of an interal. The inequalities are straightforward consequences of existing results that relate the Besicoitch-Taylor index and the Minkowski dimension. We then consider, in Section 5.3.2, totally disconnected subsets of higher-dimensional spaces. The examples of Section are mainly Cantor subsets of p h ª t, chosen so as to illustrate arious cases of equality and inequality for the results of the two preceding sections. Finally, in Section 5.3.4, we take a first step towards relating the growth rates of Betti numbers to fractal dimensions. 105

6 5.3.1 Subsets of the line We start with compact totally disconnected subsets of the real line and show that the disconnectedness index,, is closely related to the Besicoitch-Taylor index, which is in turn related to the Minkowski dimension and the fat fractal exponent. We then derie general bounds for the discreteness index,. Disconnectedness Suppose " Y«is compact and totally disconnected, and let " & denote an -neighborhood of ". s in (5.4) the complement of " a~ is a union of open interals,. The number of - connected components! of " is just one more than the number of complementary interals with length } P. This gies us a way to relate the disconnectedness index,, and. Gien, choose so that } } š. Then! ~ and! } } } š } š (5.12) } Following [21] we define This quantity satisfies that $ ' } } š. Taking the limit of each each quantity in (5.12), we hae b (5.13) It is argued in [21] that for physical examples,, and then. In general, howeer, can be arbitrarily large e.g., if ry2sb_, set } r#š (± ², then s. If the limits in (5.12) do not exist, we can obtain similar results to (5.13) by using the $ 0/2143 or $ Both Falconer [24] and Tricot [81] derie inequalities inoling and the Minkowski dimension < when " has ero Lebesgue measure. These results therefore extend to. In summary, the theorem of Section 3.4 in [81] shows that H # =< Slightly different results in Falconer [24] imply the aboe, and also that # < < U < # < (5.14) (5.15) The aboe inequalities tell us that for totally disconnected subsets of with ero measure, the limit =< exists if and only if the limit exists, in which case they are equal. Translating this into an expression for we hae, proiding the limits exist, =< 7< (5.16) 106

7 k n ˆ } } } n } f proof To illustrate the techniques inoled in proing the aboe inequalities, we gie a proof of (5.14) following that in Tricot [81]. We start by obsering that since the lengths } are decreasing, for sufficiently small _0h we can find an integer such that } x. Now consider the measure of the -neighborhood of " 9:;" & ~ 9:;" k x. } š this can be broken down as follows: ³ (5.17) The second term represents the oerlap of the -neighborhood into gaps of length greater than x. and the third term is the length of the gaps that are filled in completely. For ero-measure sets, the first term disappears. The following proof uses critical exponent definitions of and =<, rather than the limit formulations gien in (5.2) and (5.5). Specifically, the Minkowski dimension is < 546 m ˆ Š Œ š 9:;" & f h (5.18) Version (5.6) of the Besicoitch-Taylor index is equialent to 546 m ˆ]Š } Œ fih (5.19) See [81] for a proof that these definitions are equialent to the earlier ones. We now compare critical exponents for the left and right sides of (5.17) proceeding in two stages. The first step shows that # <, the second that 7<. Step 1. # <. Multiplying both sides of (5.17) by Œ š, we hae Œ š 9:;" & ~ x. Œ k Œ š If ˆ]_L =<, then by definition, Œ š 9:;" & fµh, which implies that the right side also tends to ero. Thus, x. Œ f h and since P } M x, we hae that x š Œ } Œ h and this implies ˆ P. Therefore, =<. Step 2. <. Conersely, without loss of generality we can assume that. (This is because if, then step 1 shows that 7< P, but # < from its definition, so we are done.) Now choose ˆ such that. gain we hae that Since } Therefore x. } š, and ˆ Œ š 9:;" & ~ Œ š 9:;" & g x x.œ k Œ š h we hae that Œ š Œ } Œ š 107 k x ³ ³ š Œ } Œ š (} š M x Œ and Œ š (} ³ M x Œ š.

8 j _ } P f } } } t } We want to show that the right side goes to ero. The first term does because ˆ_ means } Œ š We next show that fih. We can rewrite the second term (dropping the x š Œ ) as ³ there is an integer } Œ š ³ } Œ f h as so that for ³ } Œ k ³. š Œ j. Choose such that P, }, i.e. } š ¹ } Œ ³ M w Œ j. Thus which ˆ. Since and ˆ M j _º so the right side tends to ero. Putting all the pieces back together, we hae that Œ š 9:;" & fih as»fih, implying that ˆ P # =< and therefore that =<. Remark. If " is a fat fractal, we can subtract 9:;" from each SR side of (5.17) and obtain results identical to (5.14) and (5.15) for the fat fractal exponent,, instead of 7<. Discreteness For a totally disconnected subset " Yip ry2s2t, the disconnectedness a index,, is independent of the arrangement of the complementary interals,, within p ry2s2t. This is not true of the discreteness index,. In this section, we derie bounds on that are independent of the arrangement of complementary interals. The argument is the same as one we used in Chapter 2 for a Cantor set with h. Let } P }yg P }#J P a~ be the lengths of the. If } FF. }, then the largest -component must be longer than the next interal to be remoed, so žÿ!»p } If is large enough that } and žÿ!, then $+.- ž! }. } Taking the limit on both sides we hae that. On the other hand, the diameter cannot exceed the total length of what remains of the interal p ry2s2t, so ž! s¼ur 9:;" k. We assume again that }. and žÿ! so that žÿ! p 9:;" k. } }. If 9:;" h, then the quantity on the right is related to the Besicoitch-Taylor index ia (5.7). Taking the limit on both sides, we find that P B. If 9:;" _½h, then all we hae is that P h. To summarie, if 9:;" h and the appropriate limits exist, then» We gie examples in Section to illustrate the results obtained here. (5.20) 108

9 5.3.2 Disconnected subsets of ¾ In this section we explore connections between the box-counting dimension 7 and the disconnectedness and discreteness indices and, when " is a compact totally disconnected for which the limits exist, subset of. We start by showing that for any set " (5.21) This follows from comparing the number of -connected components,! with the largest number of disjoint M x -balls with centers in ",! M x (i.e., definition x on page 102). Since any two -components are separated by a distance of at least, any two balls of radius M x with centers in different -components must be disjoint. It follows that If we hae that! $+.- =!»! M x 8! M x! M x $+.-! M x x By taking the limit as Àfµh on each side, it follows that 8. If the limits do not exist, we still hae that H and H # ny connected fractal (e.g., the Sierpinski triangle) has Á, since a connected set with more than one point has Ž h and Á P. More interesting examples for which the inequality is strict are fat Cantor sets in for which, but (see the example in Section 5.3.3). We hae also seen examples of self-similar Cantor sets where equality holds in (5.21). Next, we show that if " is totally disconnected and the appropriate limits exist, then 7 (5.22) We again start by considering the -connected components of ". The number of -components is! and the largest -component diameter is ž!. We set  žÿ! M x, and let 8( be the smallest number of  -balls needed to coer " (i.e., definition 1 on page 102). Clearly =! balls with radius  will coer ", so that From this inequality it follows that when 8( 8(Â, =! $+.-! If we multiply the left side by  M  and rearrange we hae  $+.- 8(Â! $+.-  $

10 But  Since " žÿ! M x so ž! x $+.- ( $+.-  =! (5.23) is totally disconnected we know that ž! fih (Lemma 3 in Chapter 2). If we assume that the limit defining exists and is nonero, then the limit as gfih and the limit as Âfih are equialent. We can therefore take the limits on both sides of the inequality and find ~ ] If the limits do not exist then we can use the limsup or liminf instead. We must be a little more careful when deriing the inequalities since for positie functions, ÃlÅÄ\_0h $ pæã~(ç Äl(Ç t P p $ Ã~(Ç tèp $ Äl(Ç t and Taking the $ in (5.23) we hae nd for the $ 0/C143 Since $ /C143 M Ç $ 0/C143 pæã~(ç Ä(Ç t p $ 0/C1 3ÉÃ~(Ç tèp $ 0/2143»Äl(Ç t $ $ 0/C143 ž! $+.- 8(  $ $ 0/C143 (  M $ 0 5 6Ç, it follows that: =! H H and Á $ 546 $ 0/C143 H H $+.-! žÿ! Finally, putting (5.21) and (5.22) together, tells us that when the limits exist and Ê h then (5.24) ll of the aboe inequalities are consistent with the results obtained in the preious section for Cantor subsets of the line. In fact, for totally disconnected subsets of with ero Lebesgue measure, we hae from (5.16) and (5.21) that Ž 7, and if Ê h, then ÀN» xamples We now discuss some examples that illustrate arious cases of the relationships between dimensions and the discreteness and disconnectedness indices. 110

11 G x x G G J J J Middle-third Cantor set This Cantor set is constructed by successiely remoing the middle third of each remaining interal. There are x š complementary interals with lengths Ä =. From the formulas for middle-ˆ Cantor sets in Chapter 2 we hae that Since the set is self-similar, we know that x O and Á =< The conergence rate of the gap lengths is the limit $ Ä Ä $ x O Ë k V To compute we need the total number of gaps with lengths P Ä ; this is just B š ³ Therefore, $ $+.- x $+.- O It follows that equality holds in all the appropriate relationships deried aboe i.e. (5.13), (5.16), (5.22), and (5.24). fat Cantor set We examine the same fat Cantor set as in Chapter 2. Recall that, ÌÍY%p h ª t and there are x š gaps of sie Ä = G š ) for h ª2x. The set has positie Lebesgue measure so < FF. We showed that when the gaps are remoed from the centers of interals, The conergence rate of the gap lengths is again $ Ä Ä œ x $ and x!xx k V!xX0 V k ) k ) For the Besicoitch-Taylor index we hae that the total number of gaps with lengths P Ä is x so We see that $ $+.- Ä $+.- Ä, and M. $ $+.- x!xx70 V x k Fh x, 111

12 ³ k k k k k n k k k Finally, we show that the fat fractal exponent for this set is also G, using the formula (5.17) for the measure of the -neighborhood of Ì. Gien, choose so that Ä x. Ä š. There are a total of x š gaps longer than x. and the length of these gaps is the sum: š ) x š G G š ) 0 G š Since xî š Ä From (5.17) we therefore hae that By our choice of, we hae ), it follows that the total length of all gaps less than x. is 9:(Ì & b9:(ì ~ x.ª!x š lk ) J ) G 9:(Ì & b9:(ì J ) G š SR Using this in the definition of fat fractal exponent (5.3), we find that VR. countable totally disconnected set Finally, we consider the set " mïh ª G J FF G š ) G š. G. Thus, we see that This set is totally disconnected but not perfect. Falconer shows [23] that the Hausdorff and boxcounting dimensions differ for this set the set is countable, so 7Ð h, but Á G. The distance between neighboring points in the set is and Ä so, and G. To compute the disconnectedness and discreteness growth rates, let be any number such that Ä. Ä. Then the points ª M are -isolated and FF the rest belong to a single -component so that =! g. The largest -component is always the tail of the sequence: p h ª M k t, which means žÿ! ~ M. Thus, o$ ' x Žo$ ' x This example shows that it is possible to hae Á but 7 M. We obsered in Chapter 2 that the Cantor set examples with ero Lebesgue measure had. We conjecture that this is the case for all ero-measure Cantor sets. The example of the countable sequence of points described aboe is a totally disconnected set with ero measure, but Ê. It follows that if our conjecture is true, then the proof will hae to make explicit use of the fact that Cantor sets are perfect, i.e., that they hae no isolated points. 112

13 5.3.4 Other subsets of ¾ We now examine fractal subsets of that hae unbounded growth in the number of L - dimensional non-bounding cycles. Suppose j ) that "ÑY[ ~ and that " is a compact, connected fractal with persistent Betti number š! š4 ÓÒÏÔSÕ as foh. Under these conditions we can show that the growth rate š is bounded aboe by the Minkowski dimension =< if 9:;" h More generally, if 9:;"»P h, š VR is bounded by the fat fractal exponent : š R (5.25) We start by defining a type of Besicoitch-Taylor index for the sequence of persistent hole j ) sies. From the definition of persistent Betti number, we know that if š! g Ö_Th, then there are distinct -cycles in the -neighborhood of ". The presence of an - cycle in " & implies a the existence of an -ball with radius in the bounded complement of ". Therefore, if ) is the smallest -ball containing " j ), and š! ¼ µ_[h, then there are disjoint j ) balls! a Y ) ". Now consider the alues of where there is a jump in the alue of š!. These -alues characterie the sie of a newly-created persistent hole since they define the largest possible radius j ) of a ball that fits inside the corresponding hole in ". Let be the sequence of alues where j ) š! is discontinuous, and let be the difference between the left and right limits of š! at, i.e., the number of holes with sie. In order to define a Besicoitch-Taylor index, we list the radii of the persistent holes in decreasing order, with their multiplicity, and obtain a sequence,  P ªG P ªJ P with  fíh. The index is then just FF XØ $ /C143 '  (5.26) This index has identical equialent formulations as for the Besicoitch-Taylor index in Section Despite this similarity, XØ is not the same index as that obtained by packing the complement with cubes; the latter will detect fractal boundaries as well as the growth rate of holes. XØ The index is closely related to š. Gien a sufficiently small B_Th, we can choose so that   j ). It follows that š! ~ and therefore that j )  š!  (5.27) VØ The limit of the quantity on the left is not quite ; the conergence of the sequence  plays a role. s in Section 5.3.1, we introduce the factor $ '   Taking limits of each quantity in (5.27) we find that If (a common case) then XØ We now show that Ù XØ š. VØ VR š XØ 113 (5.28)

14 P The proof is similar to one in [80], where inequalities inoling the Besicoitch-Taylor index and fat fractal exponent are deried. The idea is to relate the sie of sets that fill in the complement of " to the measure of " &. s we remarked earlier, " is compact and connected, so there are balls of each radius  in the bounded complement of ", i.e. u(â. a Y ) Ž". These balls are disjoint, and if Â, then u(â X Y]" & b". It follows that 9:;" & U"»P ³ The integer is the smallest such that  -ball in (as on page 102). From this inequality it follows that ssuming  Á From the definition of VR (5.3) 9:( \( ~ 9:;" & b"»p0 ³ ³ D  and the constant D is the measure of the unit D Â, we hae that h $+.-  so 9:;" & U" $+.- ³ D   > R P $ ³ ÂÎ k $+.- D 0/C1 3 '  VØ The quantity on the RHS is equialent to by (5.8), so SR. It follows from this result that š SR. If " SR has ero Lebesgue measure, then =<, the Minkowski dimension. Thus, we hae that š < when 9:;" Á h. s an example where equality holds, we saw in Chapter 3 that < % OXM x for the Sierpinski triangle. The aboe proof does not apply to with -balls in the complement are disjoint is not alid. 5.4 Conjectures ½ because the assumption that the In this section, we briefly discuss some relationships that we conjecture to hold, based on the examples in this thesis. The first problem concerns the discreteness index of ero-measure Cantor sets. The second conjecture is that self-similar fractals should hae topological growth rates equialent to their similarity dimension. We finish with some questions about additional inequalities inoling the fat fractal exponent and the. Conjecture 1. Cantor sets with ero Lebesgue measure hae. This holds for all the ero measure Cantor set examples that we hae studied in this thesis and we beliee it to hold in generally. In Section 5.3.2, we showed that for any totally disconnected set with ÚÊ h,. Therefore, all that remains is to show P under suitable assumptions on the set ". Since we hae seen examples of a fat Cantor set and a totally disconnected non-perfect set with, the assumptions on " must include that is has ero measure and is perfect. It may also be the case that only holds for a more restricted class of 114

15 sets for example, self-similar Cantor sets. We hae attempted to proe the conjecture under this condition, but hae so far been unsuccessful. The index is defined in terms of the largest -component diameter. It is possible that a different measure of component sie is needed perhaps the smallest -component diameter, since this is related to the property of perfectness. is a self-similar fractal and Ê h, then # Û. Conjecture 2. If " This has been the case for the examples of Chapters 2 and 3. It is a reasonable conjecture because self-similarity is such a strong property that we expect it to dominate any scaling law. proof of this conjecture might use related constructions to those used in proing that the Hausdorff and similarity dimensions are equialent for self-similar sets that satisfy the open set property; see [23], for example. It seems that the easiest place to start is with self-similar Cantor sets that satisfy a closed set condition. That is, " ZÜ Ã ;" with this union disjoint. Not all self-similar Cantor sets hae this property for example, some Cantor set relaties of the Sierpinski triangle do not. s mentioned in Section 5.3.4, it may be possible to derie further inequalities inoling the topological growth rates and the Minkowski dimension or fat fractal exponent. For example, if "ÝY ~, can we show that SR for h w x The results of this chapter hae used Tricot s formulation of fat fractal scaling [80]. We may be able to obtain different results by comparing our indices with the fat fractal exponents of Umberger et al. [22]. This is only a partial list; there are many promising aenues to explore. 115

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17 Chapter 6 Conclusions and Future Work This thesis has considered the problem of extracting topological information about a set from a finite approximation to it. The essence of our approach is to coarse-grain the data at a sequence of resolutions and extrapolate the limiting trend. Our theoretical work and numerical inestigations show that this multiresolution approach can successfully recoer information about the underlying topology when the data approximate a compact subset of a metric space. The extrapolation is always constrained by the finite nature of the finite-precision data; we identify a cutoff resolution to measure this. lthough the examples studied in this thesis are fairly simple, the theory applies in ery general contexts. With faster numerical implementations, we beliee that our approach to computational topology could be a useful tool for analying data from both physical and numerical experiments. In the following sections we summarie the main results of this thesis then outline directions for further research. 6.1 Summary of results The main contribution of this thesis is the multiresolution approach to computational topology deeloped in Chapters 2 and 3. This approach has a number of adantages oer existing single resolution techniques. First, it is applicable to both smooth and fractal sets, the only condition is that they be compact subsets of a metric space. Second, by examining data at a sequence of resolutions we obtain more accurate knowledge of the underlying topology by identifying persistent features. Finally, it leads to a practical method for estimating the cutoff resolution a measure of confidence in the results. t present, the major drawback to computing topological information especially homology at many resolutions is the high time-cost of the computations. In Chapter 2, we considered the problem of distinguishing between connected and disconnected sets. The key step was introducing the functions!, žÿ!, and W#! to count the number of -connected components, the largest -component diameter, and the number of - isolated points respectiely. Results from Section 2.2 show that the behaior of these functions as the resolution parameter tends to ero tells us whether or not a compact space is connected, totally disconnected, and/or perfect. For arbitrary point-set data, =!, žÿ!, and W#! are easily computed from the minimal spanning tree. nother consequence of these ideas is a technique for estimating the inherent accuracy of the data. When the data approximate a perfect set, we estimate the cutoff resolution as the smallest -alue for which there are no -isolated points. Our characteriation of connected components as a function of resolution has many potential 117

18 applications, some of which are discussed in the following section. The topic of Chapter 3 was computational homology - in particular using the Betti numbers to count the number and type of non-bounding cycles in a space. Since the eroth-order Betti number is the number of path-connected components of a space, this forms a natural extension of the work in Chapter 2. The central lesson from this chapter is that it is not enough to examine the Betti numbers as a function of resolution. This is because coarse-graining a set can introduce spurious holes that are cause by the geometry, rather than the topology, of the space. Instead, an inerse system of -neighborhoods is necessary. The inclusion maps from the inerse system identify holes jþ that persist in the limit as tends to ero. We quantify this by the persistent Betti number! which counts the number of holes in the -neighborhood that hae a preimage in a smaller ß -neighborhood. This enables us to detect those holes that are due to the coarsegraining rather than the underlying topological structure. The system of -neighborhoods also allows us to formalie the relationship between the data and the underlying space. In particular, we derie inequalities inoling the persistent Betti numbers of the data and the underlying space. We anticipate that both the persistent and regular Betti numbers of an -neighborhood will be useful in characteriing the structure of data. The persistent Betti numbers reflect the underlying topological structure while the regular Betti numbers of -neighborhoods gie additional information about how the space is embedded. s we discussed in Chapter 3, more efficient numerical implementations are needed before these techniques can be fully applied to real data. In Chapter 4, we applied the techniques from Chapter 2 to study some simple examples from dynamical systems. These examples each hae well understood structure, so they proide a test of our techniques and illustrate the ersatility of our approach. In the first example, we confirmed the Cantor-set structure of cross-sections from the Hénon attractor. We then studied the breakup of inariant circles in an area-presering twist map. The transition from circle to Cantor set is continuous in a metric sense, so the functions! and ž! are not ery sensitie to this transition. Howeer, by adapting our techniques to examine the scaling of the largest gap, we deelop a new criterion for finding the critical parameter alue that compares well with preious results. Many of the examples in this thesis are fractals. By definition, a fractal has structure on arbitrarily fine scales, so it is possible for! j ) or! to go to infinity as goes to ero. In Chapter 5, we derie inequalities that relate the topological growth rates to arious existing measures of fractal scaling. We find that the growth rates of the number of components or holes are closely related to the Minkowski dimension and fat fractal exponents ia the Besicoitch-Taylor index. Our exponents, howeer, distinguish between fractals that hae the same dimension but different topological structure. They are therefore a useful addition to the collection of tools for characteriing fractal structure. 6.2 Directions for future work number of open problems were discussed in the body of the thesis as they arose. These ranged from easy extensions of the work presented in this thesis, to potential applications, to general questions about whether we can use similar techniques to compute other topological properties, such as branching structure or local connectedness, from finite data. The most interesting problems from each chapter are reisited below. We start by describing extensions of our work in Chapter 2 on connected components and minimal spanning trees. The first two items are simple generaliations that may be of interest in 118

19 applications. The last question concerns the distribution of edge lengths in the minimal spanning tree. 1. We could use different measures of the sie of an -component. xamples include the relatie number of points in an -component or the -dimensional olume of space occupied by a component. Such measures are often used in applications of percolation theory. Recall that we only examined scaling in the largest -component diameter, since this was our test for total disconnectedness. It is likely that the entire distribution of -component sies will gie interesting information in applications. This would require only a slight modification of our algorithms. 2. We obsered in Chapter 2 that the cutoff resolution for nonuniformly distributed data is larger than that for a uniform coering of the underlying set. large cutoff resolution leads to low confidence in the extrapolated underlying topology. It may be possible to reduce the cutoff resolution for nonuniform data by weighting the MST edges by the nearest neighbor distance for each point. This idea is appealing heuristically but needs some formal justification. 3. The function =! is essentially the cumulatie distribution of edge lengths in the minimal spanning tree. For finite data this distribution has two parts. When À_ à the distribution carries information about the topology of the underlying set the focus of this thesis. We conjecture that for à the distribution of MST edge-lengths is related to the distribution of the data points, i.e., a measure associated with the underlying set. It is possible that formal results about this already exist in statistics. In [78] there is a result that relates the total length of a MST to the underlying point distribution. For subsets of, the relationship between distributions of points and corresponding MST edge lengths should reduce to a problem in order statistics [10]. Our work on computational homology in Chapter 3 focussed on the mathematical foundations rather than the implementations, and there is a significant amount of work to be done on the latter. 1. The alpha shape algorithm we described in Section is a subcomplex approach to generating simplicial complexes at multiple resolutions. We argued in Section that a more efficient approach is to use subdiisions of cubical complexes. This requires a slight adjustment of the theory and a substantial amount of work on the implementations. 2. We deried a formula for computing the persistent Betti numbers in Section This is certainly not the only way to compute them. lgorithms for computing regular Betti numbers hae exploited many different results from algebraic topology. It may be possible to adapt some of these to our problem. fficient implementations will also be highly dependent on the type of cell complexes used. 3. In terms of theory, we need a more complete understanding of the continuity of the per- j Þ sistent Betti numbers,!, as ß and tend to ero. This is related to continuity and tautness results for Čech homology. We gae some detailed descriptions of potential applications in dynamical systems at the end of Chapter 4. The most challenging of these is the break-up of inariant tori in fourdimensional symplectic twist maps. In general, we anticipate that our computational techniques will be particularly useful in such higher-dimensional settings where isualiation is difficult. 119

20 More theoretical questions that are related to the study of dynamical systems include the following. 1. Just as the dimension can ary at different points of a multifractal, the scaling of components or holes with resolution may differ for subsets of a fractal. Is it possible to localie our theory to quantify this 2. Newhouse defined the thickness of Cantor subsets of to analye the existence of homoclinic tangencies of stable and unstable manifolds [62]. The definition is gien in terms of ratios of diameters and deleted interals. It may be possible to generalie this notion to Cantor subsets of ~ using techniques from Chapter 2. s we emphasied in Chapter 5 there is ample room for many more results relating our topological growth rates to fractal dimensions. See Section 5.4 for details. From the number of open problems in this short list, it should be clear that computational topology is a rich, interesting, and rapidly eoling discipline. The work in this thesis suggests that further research in this field is likely to be fruitful. 120