The Existence and Structure of Rotational Systems in the Circle

<div>Schematic for the action of <span class="nowrap"><svg height="9.39034pt" id="M343" style="vertical-align:-3.42943pt" version="1.1" viewbox="-0.0498162 -5.96091 7.52435 9.39034" width="7.52435pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M546 430L539 434C529 434 505 438 495 440C473 444 450 448 430 448C352 448 265 412 213 366C145 306 96 203 96 103C96 22 135 -12 160 -12C190 -12 238 14 262 32C310 68 368 120 411 184H413C403 117 396 75 384 21C353 -118 325 -158 291 -184C270 -200 241 -205 208 -205C133 -205 90 -164 74 -110C70 -98 58 -100 49 -107C34 -119 23 -140 23 -155C23 -190 74 -261 166 -261C219 -261 280 -233 314 -208C383 -157 446 -79 470 81C491 223 529 388 546 430ZM456 386C452 357 433 283 420 252C402 216 366 174 325 129C288 88 239 56 212 56C192 56 182 77 182 120C182 165 199 242 226 292C256 348 281 377 311 389C327 395 353 402 375 402C408 402 436 394 456 386Z" id="g113-104"></path></g></svg>.</span> <svg height="8.68572pt" id="M344" style="vertical-align:-0.0498209pt" version="1.1" viewbox="-0.0498162 -8.6359 8.02023 8.68572" width="8.02023pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M534 494L526 650H85L77 494H107C123 564 127 582 189 582H423C485 582 488 567 505 494H534ZM472 234V444H443C433 381 426 374 374 374H231C185 374 179 382 169 444H139V234H168C178 297 184 305 237 305H374C426 305 433 297 443 234H472ZM560 171H532C506 91 498 72 417 72H195C113 72 106 93 80 171H51L62 0H550L560 171Z" id="g113-143"></path></g></svg> is a proper subset of the two intervals marked in blue.</div>

Abstract and Applied Analysis

The Existence and Structure of Rotational Systems in the Circle

Figure 3