1. Complex Dynamics and Crazy Mathematics Dynamics of three very different families of complex functions: 1.Polynomials (z 2 + c) 2. Entire maps ( exp(z)) 3. Rational maps (z n + /z n )

2. We’ll investigate chaotic behavior in the dynamical plane (the Julia sets) z 2 + c exp(z) z 2 + /z 2

3. As well as the structure of the parameter planes. z 2 + c exp(z)z 3 + /z 3 (the Mandelbrot set)

4. A couple of subthemes: 1.Some “crazy” mathematics 2.Great undergrad research topics

5. The Fractal Geometry of the Mandelbrot Set

6. How to count The Fractal Geometry of the Mandelbrot Set

7. The Fractal Geometry of the Mandelbrot Set How to add How to count

8. Many people know the pretty pictures...

9. but few know the even prettier mathematics.

23. Oh, that's nothing but the 3/4 bulb....

24. ...hanging off the period 16 M-set.....

25. ...lying in the 1/7 antenna...

26. ...attached to the 1/3 bulb...

27. ...hanging off the 3/7 bulb...

28. ...on the northwest side of the main cardioid.

29. Oh, that's nothing but the 3/4 bulb, hanging off the period 16 M-set, lying in the 1/7 antenna of the 1/3 bulb attached to the 3/7 bulb on the northwest side of the main cardioid.

30. Start with a function: x + constant 2

31. Start with a function: x + constant 2 and a seed: x 0

32. Then iterate: x = x + constant 10 2

33. Then iterate: x = x + constant 10 2 21 2

34. Then iterate: x = x + constant 10 2 21 2 32 2

35. Then iterate: x = x + constant 10 2 21 2 32 2 43 2

36. Then iterate: x = x + constant 10 2 21 2 32 2 43 2 Orbit of x 0 etc. Goal: understand the fate of orbits.

37. Example: x + 1 Seed 0 2 x = 0 0 x = 1 2 3 4 5 6

38. Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 3 4 5 6

39. Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 3 4 5 6

40. Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 4 5 6

41. Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = 5 6

42. Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = big 5 x = 6

43. Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = big 5 x = BIGGER 6

44. Example: x + 1 Seed 0 2 x = 0 0 x = 1 1 x = 2 2 x = 5 3 x = 26 4 x = big 5 x = BIGGER 6 “Orbit tends to infinity”

45. Example: x + 0 Seed 0 2 x = 0 0 x = 1 2 3 4 5 6

46. Example: x + 0 Seed 0 2 x = 0 0 1 x = 2 3 4 5 6

47. Example: x + 0 Seed 0 2 x = 0 0 1 2 x = 3 4 5 6

48. Example: x + 0 Seed 0 2 x = 0 0 1 2 3 x = 4 5 6

49. Example: x + 0 Seed 0 2 x = 0 0 1 2 3 4 5 6 “A fixed point”

50. Example: x - 1 Seed 0 2 x = 0 0 x = 1 2 3 4 5 6

51. Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 2 3 4 5 6

52. Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 0 2 x = 3 4 5 6

53. Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 0 2 x = -1 3 x = 4 5 6

54. Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 0 2 x = -1 3 x = 0 4 x = 5 6

55. Example: x - 1 Seed 0 2 x = 0 0 x = -1 1 x = 0 2 x = -1 3 x = 0 4 x = -1 5 x = 0 6 “A two- cycle”

56. Example: x - 1.1 Seed 0 2 x = 0 0 x = 1 2 3 4 5 6

57. Example: x - 1.1 Seed 0 2 x = 0 0 x = -1.1 1 x = 2 3 4 5 6

58. Example: x - 1.1 Seed 0 2 x = 0 0 x = -1.1 1 x = 0.11 2 x = 3 4 5 6

59. Example: x - 1.1 Seed 0 2 x = 0 0 x = -1.1 1 x = 0.11 2 x = 3 4 5 6 time for the computer!

60. Observation: For some real values of c, the orbit of 0 goes to infinity, but for other values, the orbit of 0 does not escape.

61. Complex Iteration Iterate z + c 2 complex numbers

62. Example: z + i Seed 0 2 z = 0 0 z = 1 2 3 4 5 6

63. Example: z + i Seed 0 2 z = 0 0 z = i 1 z = 2 3 4 5 6

64. Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = 3 4 5 6

65. Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = -i 3 z = 4 5 6

66. Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = -i 3 z = -1 + i 4 z = 5 6

67. Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = -i 3 z = -1 + i 4 z = -i 5 z = 6

68. Example: z + i Seed 0 2 z = 0 0 z = i 1 z = -1 + i 2 z = -i 3 z = -1 + i 4 z = -i 5 z = -1 + i 6 2-cycle

69. Example: z + i Seed 0 2 1 i -i

70. Example: z + i Seed 0 2 1 i -i

71. Example: z + i Seed 0 2 1 i -i

72. Example: z + i Seed 0 2 -i 1 i

73. Example: z + i Seed 0 2 1 i -i

74. Example: z + i Seed 0 2 -i 1 i

75. Example: z + i Seed 0 2 1 i -i

76. Example: z + i Seed 0 2 -i 1 i

77. Example: z + 2i Seed 0 2 z = 0 0 z = 1 2 3 4 5 6

78. Example: z + 2i Seed 0 2 z = 0 0 z = 2i 1 z = -4 + 2i 2 z = 12 - 14i 3 z = -52 + 336i 4 z = big 5 z = BIGGER 6 Off to infinity

79. Same observation Sometimes orbit of 0 goes to infinity, other times it does not.

80. The Mandelbrot Set: All c-values for which orbit of 0 does NOT go to infinity. Why do we care about the orbit of 0?

81. The Mandelbrot Set: All c-values for which orbit of 0 does NOT go to infinity. As we shall see, the orbit of the critical point determines just about everything for z 2 + c. 0 is the critical point of z 2 + c.

82. Algorithm for computing M Start with a grid of complex numbers

83. Algorithm for computing M Each grid point is a complex c-value.

84. Algorithm for computing M Compute the orbit of 0 for each c. If the orbit of 0 escapes, color that grid point. red = fastest escape

85. Algorithm for computing M Compute the orbit of 0 for each c. If the orbit of 0 escapes, color that grid point. orange = slower

86. Algorithm for computing M Compute the orbit of 0 for each c. If the orbit of 0 escapes, color that grid point. yellow green blue violet

87. Algorithm for computing M Compute the orbit of 0 for each c. If the orbit of 0 does not escape, leave that grid point black.

88. Algorithm for computing M Compute the orbit of 0 for each c. If the orbit of 0 does not escape, leave that grid point black.

89. The eventual orbit of 0

91. 3-cycle

92. The eventual orbit of 0 3-cycle

93. The eventual orbit of 0 3-cycle

94. The eventual orbit of 0 3-cycle

95. The eventual orbit of 0 3-cycle

96. The eventual orbit of 0 3-cycle

97. The eventual orbit of 0 3-cycle

98. The eventual orbit of 0 3-cycle

99. The eventual orbit of 0 3-cycle

100. The eventual orbit of 0

102. 4-cycle

103. The eventual orbit of 0 4-cycle

104. The eventual orbit of 0 4-cycle

105. The eventual orbit of 0 4-cycle

106. The eventual orbit of 0 4-cycle

107. The eventual orbit of 0 4-cycle

108. The eventual orbit of 0 4-cycle

109. The eventual orbit of 0 4-cycle

110. The eventual orbit of 0

112. 5-cycle

113. The eventual orbit of 0 5-cycle

114. The eventual orbit of 0 5-cycle

115. The eventual orbit of 0 5-cycle

116. The eventual orbit of 0 5-cycle

117. The eventual orbit of 0 5-cycle

118. The eventual orbit of 0 5-cycle

119. The eventual orbit of 0 5-cycle

120. The eventual orbit of 0 5-cycle

121. The eventual orbit of 0 5-cycle

122. The eventual orbit of 0 5-cycle

123. The eventual orbit of 0 2-cycle

124. The eventual orbit of 0 2-cycle

125. The eventual orbit of 0 2-cycle

126. The eventual orbit of 0 2-cycle

127. The eventual orbit of 0 2-cycle

128. The eventual orbit of 0 fixed point

129. The eventual orbit of 0 fixed point

130. The eventual orbit of 0 fixed point

131. The eventual orbit of 0 fixed point

132. The eventual orbit of 0 fixed point

133. The eventual orbit of 0 fixed point

134. The eventual orbit of 0 fixed point

135. The eventual orbit of 0 fixed point

136. The eventual orbit of 0 goes to infinity

137. The eventual orbit of 0 goes to infinity

138. The eventual orbit of 0 goes to infinity

139. The eventual orbit of 0 goes to infinity

140. The eventual orbit of 0 goes to infinity

141. The eventual orbit of 0 goes to infinity

142. The eventual orbit of 0 goes to infinity

143. The eventual orbit of 0 goes to infinity

144. The eventual orbit of 0 goes to infinity

145. The eventual orbit of 0 goes to infinity

146. The eventual orbit of 0 goes to infinity

147. The eventual orbit of 0 gone to infinity

148. One reason for the importance of the critical orbit: If there is an attracting cycle for z 2 + c, then the orbit of 0 must tend to it.

149. How understand the of the bulbs? periods

150. How understand the of the bulbs? periods

151. junction point three spokes attached

152. Period 3 bulb junction point three spokes attached

155. Period 4 bulb

158. Period 5 bulb

161. Period 7 bulb

165. Period 13 bulb

166. Filled Julia Set:

167. Fix a c-value. The filled Julia set is all of the complex seeds whose orbits do NOT go to infinity.

168. Example: z 2 Seed: 0 In filled Julia set?

169. Example: z 2 Seed: 0Yes In filled Julia set?

170. Example: z 2 Seed: 0Yes 1 In filled Julia set?

171. Example: z 2 Seed: 0Yes 1 In filled Julia set?

172. Example: z 2 Seed: 0Yes 1 In filled Julia set?

173. Example: z 2 Seed: 0Yes 1 Yes In filled Julia set?

174. Example: z 2 Seed: 0Yes 1 Yes i In filled Julia set?

175. Example: z 2 Seed: 0Yes 1 Yes i In filled Julia set?

176. Example: z 2 Seed: 0Yes 1 Yes i 2i In filled Julia set?

177. Example: z 2 Seed: 0Yes 1 Yes i 2i No In filled Julia set?

178. Example: z 2 Seed: 0Yes 1 Yes i 2i No 5 In filled Julia set?

179. Example: z 2 Seed: 0Yes 1 Yes i 2i No 5No way In filled Julia set?

180. Filled Julia Set for z 2 All seeds on and inside the unit circle. i 1

181. The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic”

182. The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

183. The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

184. The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

185. The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

186. The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

187. The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

188. The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

189. The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

190. The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

191. The Julia Set is the boundary of the filled Julia set That’s where the map is “chaotic” Nearby orbits behave very differently

192. Other filled Julia sets

193. c = 0

194. Other filled Julia sets c = -1

195. Other filled Julia sets c = -1

196. Other filled Julia sets c = -1

197. Other filled Julia sets c = -1

198. Other filled Julia sets c = -1

199. Other filled Julia sets c = -1

200. Other filled Julia sets c = -1

201. Other filled Julia sets c = -1

202. Other filled Julia sets c = -.12+.75i

203. Other filled Julia sets c = -.12+.75i

204. Other filled Julia sets c = -.12+.75i

205. Other filled Julia sets c = -.12+.75i

206. Other filled Julia sets c = -.12+.75i

207. Other filled Julia sets c = -.12+.75i

208. If c is in the Mandelbrot set, then the filled Julia set is always a connected set.

209. Other filled Julia sets But if c is not in the Mandelbrot set, then the filled Julia set is totally disconnected.

210. Other filled Julia sets c =.3

211. Other filled Julia sets c =.3

212. Other filled Julia sets c =.3

213. Other filled Julia sets c =.3

214. Other filled Julia sets c =.3

215. Other filled Julia sets c = -.8+.4i

216. Another reason why we use the orbit of the critical point to plot the M-set: Theorem: (Fatou & Julia) For z 2 + c:

217. Another reason why we use the orbit of the critical point to plot the M-set: Theorem: (Fatou & Julia) For z 2 + c: If the orbit of 0 goes to infinity, the Julia set is a Cantor set (totally disconnected, “fractal dust,” a scatter of uncountably many points.

218. Another reason why we use the orbit of the critical point to plot the M-set: Theorem: (Fatou & Julia) For z 2 + c: But if the orbit of 0 does not go to infinity, the Julia set is connected (just one piece). If the orbit of 0 goes to infinity, the Julia set is a Cantor set (totally disconnected, “fractal dust,” a scatter of uncountably many points.

219. Animations: In and out of M arrangement of the bulbs Saddle node Period doubling Period 4 bifurcation

220. How do we understand the arrangement of the bulbs?

221. How do we understand the arrangement of the bulbs? Assign a fraction p/q to each bulb hanging off the main cardioid. q = period of the bulb

222. Where is the smallest spoke in relation to the “principal spoke”? p/3 bulb principal spoke

223. 1/3 bulb principal spoke The smallest spoke is located 1/3 of a turn in the counterclockwise direction from the principal spoke.

224. 1/3 bulb 1/3

225. 1/3 bulb 1/3

226. 1/3 bulb 1/3

227. 1/3 bulb 1/3

228. 1/3 bulb 1/3

229. 1/3 bulb 1/3

230. 1/3 bulb 1/3

231. 1/3 bulb 1/3

232. 1/3 bulb 1/3

233. 1/3 bulb 1/3

234. ??? bulb 1/3

235. 1/4 bulb 1/3

236. 1/4 bulb 1/3 1/4

237. 1/4 bulb 1/3 1/4

238. 1/4 bulb 1/3 1/4

239. 1/4 bulb 1/3 1/4

240. 1/4 bulb 1/3 1/4

241. 1/4 bulb 1/3 1/4

242. 1/4 bulb 1/3 1/4

243. 1/4 bulb 1/3 1/4

244. 1/4 bulb 1/3 1/4

245. ??? bulb 1/3 1/4

246. 2/5 bulb 1/3 1/4

247. 2/5 bulb 1/3 1/4 2/5

248. 2/5 bulb 1/3 1/4 2/5

249. 2/5 bulb 1/3 1/4 2/5

250. 2/5 bulb 1/3 1/4 2/5

251. 2/5 bulb 1/3 1/4 2/5

252. ??? bulb 1/3 1/4 2/5

253. 3/7 bulb 1/3 1/4 2/5

254. 3/7 bulb 1/3 1/4 2/5 3/7

255. 3/7 bulb 1/3 1/4 2/5 3/7

256. 3/7 bulb 1/3 1/4 2/5 3/7

257. 3/7 bulb 1/3 1/4 2/5 3/7

258. 3/7 bulb 1/3 1/4 2/5 3/7

259. 3/7 bulb 1/3 1/4 2/5 3/7

260. 3/7 bulb 1/3 1/4 3/7 2/5

261. ??? bulb 1/3 1/4 3/7 2/5

262. 1/2 bulb 1/3 1/4 3/7 1/2 2/5

263. 1/2 bulb 1/3 1/4 3/7 1/2 2/5

264. 1/2 bulb 1/3 1/4 3/7 1/2 2/5

265. 1/2 bulb 1/3 1/4 3/7 1/2 2/5

266. ??? bulb 1/3 1/4 3/7 1/2 2/5

267. 2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

268. 2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

269. 2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

270. 2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

271. 2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

272. 2/3 bulb 1/3 1/4 3/7 1/2 2/3 2/5

273. How to count

274. 1/4 How to count

275. 1/3 1/4 How to count

276. 1/3 1/4 2/5 How to count

277. 1/3 1/4 2/5 3/7 How to count

278. 1/3 1/4 2/5 3/7 1/2 How to count

279. 1/3 1/4 2/5 3/7 1/2 2/3 How to count

280. 1/3 1/4 2/5 3/7 1/2 2/3 The bulbs are arranged in the exact order of the rational numbers. How to count

281. 1/3 1/4 2/5 3/7 1/2 2/3 The bulbs are arranged in the exact order of the rational numbers. 1/101 32,123/96,787 How to count

282. Animations: Mandelbulbs Spiralling fingers

283. How to add

284. 1/2

285. How to add 1/2 1/3

286. How to add 1/2 1/3 2/5

287. How to add 1/2 1/3 2/5 3/7

288. += 1/2 + 1/3 = 2/5

289. += 1/2 + 2/5 = 3/7

290. 22 1/2 0/1 Here’s an interesting sequence:

291. 22 1/2 0/1 Watch the denominators 1/3

292. 22 1/2 0/1 Watch the denominators 1/3 2/5

293. 22 1/2 0/1 Watch the denominators 1/3 2/5 3/8

294. 22 1/2 0/1 Watch the denominators 1/3 2/5 3/8 5/13

295. 22 1/2 0/1 What’s next? 1/3 2/5 3/8 5/13

296. 22 1/2 0/1 What’s next? 1/3 2/5 3/8 5/13 8/21

297. 22 1/2 0/1 The Fibonacci sequence 1/3 2/5 3/8 5/13 8/21 13/34

298. The Farey Tree

299. How get the fraction in between with the smallest denominator?

300. The Farey Tree How get the fraction in between with the smallest denominator? Farey addition

301. The Farey Tree

303. .... essentially the golden number

304. Another sequence (denominators only) 1 2

305. Another sequence (denominators only) 1 2 3

306. Another sequence (denominators only) 1 2 3 4

307. Another sequence (denominators only) 1 2 3 4 5

308. Another sequence (denominators only) 1 2 3 4 5 6

309. Another sequence (denominators only) 1 2 3 4 5 6 7

310. sequence 1 2 3 4 5 6 7 Devaney

311. The Dynamical Systems and Technology Project at Boston University website: math.bu.edu/DYSYS: Have fun! Mandelbrot set explorer; Applets for investigating M-set; Applets for other complex functions; Chaos games, orbit diagrams, etc.

312. Farey.qt Farey tree D-sequence Continued fraction expansion Far from rationals Other topics Website

313. Continued fraction expansion Let’s rewrite the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,..... as a continued fraction:

314. Continued fraction expansion 1212 = 1212 the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

315. Continued fraction expansion 1313 = 1212 + 1111 the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

316. Continued fraction expansion 2525 = 1212 + 1111 + 1111 the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

317. Continued fraction expansion 3838 = 1212 + 1111 + 1111 1111 + the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

318. Continued fraction expansion = 1212 + 1111 + 1111 1111 + 1111 + 5 13 the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

319. Continued fraction expansion = 1212 + 1111 + 1111 1111 + 1111 + 8 21 1111 + the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

320. Continued fraction expansion = 1212 + 1111 + 1111 1111 + 1111 + 13 34 1111 + 1111 + the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

321. Continued fraction expansion = 1212 + 1111 + 1111 1111 + 1111 + 13 34 1111 + 1111 + essentially the 1/golden number the sequence: 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34,.....

322. We understand what happens for = 1a1a + 1b1b + 1c1c 1d1d + 1e1e + 1f1f + 1g1g + where all entries in the sequence a, b, c, d,.... are bounded above. But if that sequence grows too quickly, we’re in trouble!!! etc.

323. The real way to prove all this: Need to measure: the size of bulbs the length of spokes the size of the “ears.”

324. There is an external Riemann map : C - D C - M taking the exterior of the unit disk to the exterior of the Mandelbrot set.

325. takes straight rays in C - D to the “external rays” in C - M 01/2 1/3 2/3 external ray of angle 1/3

326. Suppose p/q is periodic of period k under doubling mod 1: period 2 period 3 period 4

327. Suppose p/q is periodic of period k under doubling mod 1: period 2 period 3 period 4 Then the external ray of angle p/q lands at the “root point” of a period k bulb in the Mandelbrot set.

328. 0 1/3 2/3 0 is fixed under angle doubling, so lands at the cusp of the main cardioid.

329. 0 1/3 2/3 1/3 and 2/3 have period 2 under doubling, so and land at the root of the period 2 bulb. 2

330. 0 1/3 2/3 And if lies between 1/3 and 2/3, then lies between and. 2

331. 0 1/3 2/3 So the size of the period 2 bulb is, by definition, the length of the set of rays between the root point rays, i.e., 2/3-1/3=1/3. 2

332. 0 1/3 2/3 1/15 and 2/15 have period 4, and are smaller than 1/7.... 1/7 2/7 3/7 4/7 5/7 6/7 2 3 3 1/15 2/15

333. 0 1/3 2/3 1/15 and 2/15 have period 4, and are smaller than 1/7.... 1/7 2/7 3/7 4/7 5/7 6/7 2 3 3 1/15 2/15

334. 0 1/3 2/3 1/7 2/7 3/7 4/7 5/7 6/7 2 3 3 1/15 2/15 3/15 and 4/15 have period 4, and are between 1/7 and 2/7....

335. 0 1/3 2/3 3/15 and 4/15 have period 4, and are between 1/7 and 2/7.... 1/7 2/7 3/7 4/7 5/7 6/7 2 3 3 1/15 2/15

336. 1/72/7 3/15 and 4/15 have period 4, and are between 1/7 and 2/7....

337. 1/72/7 3/15 and 4/15 have period 4, and are between 1/7 and 2/7.... 3/154/15

338. So what do we know about M? All rational external rays land at a single point in M.

339. So what do we know about M? All rational external rays land at a single point in M. Rays that are periodic under doubling land at root points of a bulb. Non-periodic rational rays land at Misiurewicz points (how we measure length of antennas).

340. So what do we know about M? “Highly irrational” rays also land at unique points, and we understand what goes on here. “Highly irrational" = “far” from rationals, i.e.,

341. So what do we NOT know about M? But we don't know if irrationals that are “close” to rationals land. So we won't understand quadratic functions until we figure this out.

342. MLC Conjecture: The boundary of the M-set is “locally connected” --- if so, all rays land and we are in heaven!. But if not......

343. The Dynamical Systems and Technology Project at Boston University website: math.bu.edu/DYSYS Have fun!

344. A number is far from the rationals if:

346. This happens if the “continued fraction expansion” of has only bounded terms.