Timeline for A fractal object at origin but nowhere else: derived from Brownain motion
Current License: CC BY-SA 3.0
10 events
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| Feb 22, 2012 at 16:22 | comment | added | chris in physics | In 3D a random walk taking steps of size 1 with independent and uniform angles is not dense. So let rho be such a path. Can I find a ball B (not circle) which rho does not interesect, and reflect rho inside ball, and then find a fractal dimension 2 for rho at the center of B, where the fractal dimension is defined as above? | |
| Feb 22, 2012 at 8:24 | comment | added | Pablo Shmerkin | A curve of Hausdorff dimension 1 can definitely be dense in the plane. Even the set of points with rational coordinates, which has Hausdorff dimension 0, is dense in the plane | |
| Feb 22, 2012 at 4:25 | comment | added | chris in physics | Hi Douglas, Interesting point. My intuition was that when you smooth the Brownian motion, and you remove its fine detail at short length scales, then you reduce its Hausdorff dimension to 1. Can a curve with Hausdorff dimension 1 fill the plane? Thanks for the comment. | |
| Feb 22, 2012 at 4:19 | comment | added | chris in physics | For a fractal dimension at the center of circle C, I suggest: Let T be an interval of time t. Let rho(T) denote the path of rho(t) evaluated on T. Let ^rho(T) denote rho(T) after it has been smoothed and inverted inside of circle C. Let c denote the center of C. Consider a circle Gamma(R) of radius R also centered on c, where R is smaller than the radius of C. Measure the length of ^rho(T) that is contained in Gamma(R), and denote this by L(R). I claim that in the limit as the interval T becomes large, and R becomes small, L(Ralpha) = alpha^DL(R), where this defines fractal dimension D. | |
| Feb 22, 2012 at 3:52 | comment | added | Douglas Zare | Claim 1 seems wrong. With probability $1$, a random walk taking steps of size $1$ with independent and uniform angles will be dense in the plane. However, I don't think the uniformity of the random walk is important. | |
| Feb 22, 2012 at 0:46 | history | edited | chris in physics | CC BY-SA 3.0 |
changed "origin" to "center of C"
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| Feb 22, 2012 at 0:24 | comment | added | chris in physics | To smooth it out, start with a time t_o and a point r_o = rho(t_o). Find the maximum time t_1 such that there exists a circle of radius R that contains rho(t) for all t in the interval t_o < t < t_1. So now we have a second point r_1 = rho(t_1) on rho. These two points are a distance R apart. Then repeat: Find the maximum time t_2 such that there exists a circle of Radius R that contains rho(t) for all t in the interval t_1 < t < t_2. Repeat for all positive integer i. We can "turn" this procedure "around" to negative integers. Then these points can be joined with a continuous spline. | |
| Feb 21, 2012 at 23:49 | comment | added | ShawnD | I don't think anyone will mind if you take the space to give a rigorous definition of what you mean by "smoothed rho." | |
| Feb 21, 2012 at 23:47 | comment | added | user5810 | What is "point fractal dimension"? $\;$ | |
| Feb 21, 2012 at 23:33 | history | asked | chris in physics | CC BY-SA 3.0 |