From: "K. P. Hart" Subject: Re: Hausdorff distance and fractal dimension Date: Mon, 17 Jan 2000 12:09:41 +0100 Newsgroups: sci.math.research Summary: [missing] I seriously doubt this as there are nowhere differentiable functions arbitrarily close to analytic ones; think of Weierstrass' example. Lower bounds for the Hausdorff dimensions of their graphs were given by Mauldin & Williams, these seem to indicate that d(e)>=1 for all e. see MR 88c:28006 http://www.ams.org/msnmain?fn=105&id=88c_28006&r=42&fmt=doc&l=1000&op3=OR&pg3=IID&s3=80769&v3=Mauldin%2C%20R%2E%20Daniel Adam Bryant wrote > Let A and B be compact subsets of Euclidean space. Let h(A,B) denote > the Hausdorff distance between A and B. Let D(A) and D(B) denote the > fractal dimensions of A and B, respectively. If h(A,B) < e, for > positive e, then can we place an upper bound on |D(A)-D(B)|? That is, > if h(A,B) < e, then |D(A)-D(B)| < d(e), where d is a function of e. > Moreover, is it true that d(e)->0 as e->0? Any references would be very > helpful. -- E-MAIL: K.P.Hart@its.tudelft.nl PAPER: Department of Pure Mathematics PHONE: +31-15-2784572 TU Delft FAX: +31-15-2787245 Postbus 5031 URL: http://aw.twi.tudelft.nl/~hart 2600 GA Delft the Netherlands ============================================================================== From: dlrenfro@gateway.net (Dave L. Renfro) Subject: Re: Hausdorff distance and fractal dimension Date: 19 Jan 2000 00:13:51 -0500 Newsgroups: sci.math.research K. P. Hart [sci.math.research Mon, 17 Jan 2000 12:09:41 +0100] >I seriously doubt this as there are nowhere differentiable functions >arbitrarily close to analytic ones; think of Weierstrass' example. >Lower bounds for the Hausdorff dimensions of their graphs were >given by Mauldin & Williams, these seem to indicate that d(e)>=1 >for all e. see MR 88c:28006 To further elaborate on Hart's reply ... << I originally thought Hart doubted the existence of nowhere differentiable functions arbitrarily close to analytic ones. Thinking this, I wrote the comments below. Then, just before sending my post in, I realized that Hart was claiming (correctly) the opposite. In the event that my comments may be useful, I'm posting them anyway. >> There are nowhere differentiable *continuous* functions arbitrarily close to *any* continuous function, analytic or otherwise. Let G be any nowhere differentiable function on a compact interval I. Then the set of all polynomial translates of G is dense in C(I) = the set of continuous functions defined on I with the sup norm [because the polynomials are dense in C(I), the map from C(I) to C(I) given by f |---> f+G is continuous, and continuous maps take dense sets to dense sets], and hence it is also dense in the Hausdorff metric on the set of compact subsets of I x R. [R is the set of real numbers.] In fact, the set of nowhere differentiable continuous functions is not only dense in C(I), it has a first category (i.e. meager) complement in C(I). It is also true that the functions in C(I) whose graph has Hausdorff dimension d is dense in C(I) for each d such that 1 <= d <= 2. If d=1, this set has a first category complement in C(I) [Mauldin/Williams give a 3 sentence proof of this on p. 794; this had been previously proved by A. J. Ostaszewski (1974) and Peter Gruber (1983), which I believe Mauldin/Williams were not aware of] and, for any d such that 1 < d <= 2, this set is first category in C(I). The same statement regarding density is true for each of the fractal dimensions "upper box dimension", "lower box dimension", "upper packing dimension", and "lower packing dimension" [for these terms, see Falconer's 1990 or 1997 books, Edgar's 1998 book, or Mattila's 1995 book], but my comments about the Baire category "superdensity" results will vary, depending on the specific fractal dimension notion being used. For the full picture, with references, see the diagram on the first page of my conference abstract "A porosity description of the typical continuous graph", Real Analysis Exchange 22 (1996-97), 70-73. Dave L. Renfro ============================================================================== From: "Daniel H. Luecking" Subject: Re: Hausdorff distance and fractal dimension Date: Mon, 17 Jan 2000 01:39:26 -0600 Newsgroups: sci.math.research On Sun, 16 Jan 2000, Adam Bryant wrote: > Hello, > > Let A and B be compact subsets of Euclidean space. Let h(A,B) denote > the Hausdorff distance between A and B. Let D(A) and D(B) denote the > fractal dimensions of A and B, respectively. If h(A,B) < e, for > positive e, then can we place an upper bound on |D(A)-D(B)|? Only the dimension of the space. A single point has dimension 0. A ball of radius e centered at that point has dimension N (in R^N). The Hausdorff distance between the point and the ball is e. A set inside that ball can have any fractal dimension between 0 and N, and will also be within distance e of the point and within distance 2e of the ball (and each other). -- Dan Luecking Dept. of Mathematical Sciences luecking@comp.uark.edu University of Arkansas http://comp.uark.edu/~luecking/ Fayetteville, AR 72101 ============================================================================== From: Mark C McClure Subject: Re: Hausdorff distance and fractal dimension Date: Mon, 17 Jan 2000 14:04:02 +0000 (GMT) Newsgroups: sci.math.research Adam Bryant wrote: : Let A and B be compact subsets of Euclidean space. Let h(A,B) denote : the Hausdorff distance between A and B. Let D(A) and D(B) denote the : fractal dimensions of A and B, respectively. If h(A,B) < e, for : positive e, then can we place an upper bound on |D(A)-D(B)|? That is, : if h(A,B) < e, then |D(A)-D(B)| < d(e), where d is a function of e. : Moreover, is it true that d(e)->0 as e->0? Any references would be very : helpful. No. Any compact set, regardless of dimension, may be approximated arbitrarily closely by a finite set, which has dimension zero. -- __/\__ Mark McClure \ / Department of Mathematics __/\__/ \__/\__ UNC - Asheville \ / Asheville, NC 28804 /__ __\ http://www.unca.edu/~mcmcclur/ \ / __/\__ __/ \__ __/\__ \ / \ / \ / __/\__/ \__/\__/ \__/\__/ \__/\__ ============================================================================== From: "G. A. Edgar" Subject: Re: Hausdorff distance and fractal dimension Date: Tue, 18 Jan 2000 09:05:30 -0500 Newsgroups: sci.math.research > > No. Any compact set, regardless of dimension, may be approximated > arbitrarily closely by a finite set, which has dimension zero. And, on the other hand, any nonempty compact set in n-dimensional Euclidean space is approximated arbitrarily closely by its open neighborhoods, which have dimension n. -- Gerald A. Edgar edgar@math.ohio-state.edu ============================================================================== From: Denis WENDUM Subject: Re: Hausdorff distance and fractal dimension Date: 18 Jan 2000 09:56:50 +0100 Newsgroups: sci.math.research abryant@edge.net wrote: >> Let A and B be compact subsets of Euclidian space. ... His question boils down to : Is (a) fractal dimension of A continuous as function of A w.r.t the Hausdorff distance between compact sets. Unfortunately the answer is NO in general. Take for instance A=segment [0,1]x(0) (in R2) Take B=[0,1]x[0,H] H>0. Then : dist(A,B)<=H but dim(A)=1,dim(B)=2.(for reasonable definition of "the" fractal dimension, wich is by no means unique.) Another example migth be: A={0/N,1/N,... 1} B=[0,1] in R. dist(A,B)<=1/(N) while dim(A)=0 dim(B)=1. The two books written by Falconer, published by Cambridge University Press provide a good start for fractal geometry.(Just now I can't remember the precise titles, wich anyhow contain the keyword fractal) *********************************************** *** THIS MESSAGE REFLECTS ONLY MY OWN VIEWS *** ***********************************************