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<!DOCTYPE html><html lang="zh-Hans"><head hexo-theme="https://github.com/volantis-x/hexo-theme-volantis/#6.0.3"><meta name="generator" content="Hexo 8.1.2"><meta name="Volantis" content="6.0.3"><meta charset="utf-8"><meta name="robots" content="index,follow,max-image-preview:large"><link rel="canonical" href="https://blog.mhuig.top/notes/person/19"><meta http-equiv="x-dns-prefetch-control" content="on"><link rel="dns-prefetch" href="https://static.mhuig.top"><link rel="preconnect" href="https://static.mhuig.top" crossorigin=""><meta name="renderer" content="webkit"><meta name="force-rendering" content="webkit"><meta http-equiv="X-UA-Compatible" content="IE=Edge,chrome=1"><meta name="HandheldFriendly" content="True"><meta name="mobile-web-app-capable" content="yes"><meta name="viewport" content="width=device-width,initial-scale=1,maximum-scale=5"><meta content="black-translucent" name="apple-mobile-web-app-status-bar-style"><meta content="telephone=no" 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type="application/xml" title="MHuiG Blog Site Map" href="https://blog.mhuig.top/sitemap.xml"><link rel="author" href="https://mhuig.top"><meta name="author" content="MHuiG"><meta name="creator" content="MHuiG"><link rel="archives" href="https://blog.mhuig.top/archives/"><link rel="preload" href="/css/style.css" as="style"><link rel="preload" href="https://static.mhuig.top/npm/volantis-static@0.0.1660614606622/media/fonts/VarelaRound/VarelaRound-Regular.ttf" as="font" type="font/ttf" crossorigin="anonymous"><link rel="preload" href="https://static.mhuig.top/npm/volantis-static@0.0.1660614606622/media/fonts/VarelaRound/VarelaRound-Regular.ttf" as="font" type="font/ttf" crossorigin="anonymous"><link rel="alternate" href="/atom.xml" title="Magicland" type="application/atom+xml"><link rel="alternate" href="/rss2.xml" title="Magicland" type="application/rss+xml"><title>History of mathematics: Hedlund符号动力学与拓扑动力学的奠基人古斯塔夫·赫德伦德 - Magicland</title><meta name="keywords" content="历史,Person, History,人物,赫德伦德,符号动力学,拓扑动力学,测地流遍历性,MHuiG, @MHuiG, Blog, 博客, Magicland, 魔法世界"><meta desc="" name="description" content="古斯塔夫·赫德伦德是符号动力学和拓扑动力学的创始人之一。他揭示了圆周变换的旋转数与莫尔斯类型数的深刻联系，证明了常负曲率曲面上测地流的遍历性，并与莫尔斯共同创立了符号动力学。其研究为动力系统理论奠定了坚实基础。 - MHuiG - Magicland"><meta property="og:type" content="website"><meta property="og:title" content="Magicland"><meta property="og:url" content="https://blog.mhuig.top/notes/person/19"><meta property="og:site_name" content="Magicland"><meta property="og:description" content="古斯塔夫·赫德伦德是符号动力学和拓扑动力学的创始人之一。他揭示了圆周变换的旋转数与莫尔斯类型数的深刻联系，证明了常负曲率曲面上测地流的遍历性，并与莫尔斯共同创立了符号动力学。其研究为动力系统理论奠定了坚实基础。"><meta property="og:locale"><meta property="og:image" content="https://blog.mhuig.top/lib/favicon/android-chrome-192x192.png"><meta property="article:published_time" content="2026-05-16T00:07:00.000Z"><meta property="article:modified_time" content="2026-05-16T00:10:00.000Z"><meta property="article:author" content="MHuiG"><meta property="article:tag" content="History"><meta 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itemscope="" itemtype="http://schema.org/CreativeWork"><meta itemprop="name" content="Hedlund符号动力学与拓扑动力学的奠基人古斯塔夫·赫德伦德"><meta itemprop="description" content="古斯塔夫·赫德伦德是符号动力学和拓扑动力学的创始人之一。他揭示了圆周变换的旋转数与莫尔斯类型数的深刻联系，证明了常负曲率曲面上测地流的遍历性，并与莫尔斯共同创立了符号动力学。其研究为动力系统理论奠定了坚实基础。"></span><span hidden=""><meta itemprop="image" content="/lib/favicon/android-chrome-192x192.png"></span><div class="article-meta" id="top"><span hidden="" itemprop="name headline"></span></div><div id="layoutHelper-page-plugins"></div><div id="post-body" itemprop="articleBody"><p> <span class="p logo center large">Hedlund</span></p><br><h1 hidden="">Hedlund</h1><div class="story post-story"><h2 id="生平"><a href="#生平" class="headerlink" title="生平"></a>生平</h2><p>古斯塔夫・阿诺德・赫德伦德（Gustav Arnold Hedlund, 1904-1993）是符号动力学（symbolic dynamics）和拓扑动力学（topological dynamics）的创始人之一。</p><p>他于 1904 年 5 月 7 日出生于美国 Massachusetts 州 Middlesex 县的 Somerville。<br>他的父母均是来自瑞典的移民。</p><p>父亲莫里茨・特奥多尔・赫德伦德（Mauritz Teodor Hedlund，1867-1929）曾担任 Hanover 的 Dartmouth College 建筑与场地助理主管；母亲特克拉・玛丽亚・斯塔克（Tekla Maria Starck，1872-1946）是一位家庭主妇。在 20 世纪初工业蓬勃发展的新英格兰，Hedlund 与兄弟威廉・西奥多・赫德伦德（Wilhelm Theodore Hedlund，1899-1944）一起在 Somerville 这个人口密集的且具有浓厚学术氛围的 Boston 郊区长大。</p><p>1925 年，他在 Harvard 大学获得了本科学位，在校期间强大的师资力量为他打下了极其扎实的纯数学分析与拓扑学基础。</p><p>随后他前往 Columbia 大学攻读硕士学位（约于 1927 年完成），并同时在亨特学院（Hunter College，现在是 New York 市立大学下的一所四年制学院）担任讲师。</p><p>此后，他返回 Harvard 大学攻读博士学位，师从 20 世纪大范围变分法（calculus of variations in the large）与大范围拓扑学（topology in the large）的绝对先驱莫尔斯（Harold Calvin Marston Morse，1892-1977）。</p><p>Morse 是伯克霍夫（George David Birkhoff，1884-1944）的学生。</p><p>在当时，“大范围（in the large）” 是一个极其前沿的数学理念，意味着跳出微积分的局部视角，将几何空间作为一个整体来研究其全局拓扑性质，这演变成了后来的 Morse 理论（Morse theory）。</p><p>Morse 理论的一些想法可追溯到凯莱（Arthur Cayley，1821-1895）和麦克斯韦（James Clerk Maxwell，1831-1879）所研究的地形学（topography）。</p><p>1930 年，Hedlund 凭借题为《I. 具有周期系数的二维 Riemannian 流形上的测地线；II. Poincaré 旋转数与 Morse 类型数》（I. Geodesics on a two-dimensional Riemannian manifold with periodic coefficients; II. Poincaré's rotation number and Morse's type number）顺利获得博士学位，这项工作确立了他一生将几何学与动力系统紧密结合的研究基调。</p><p>Hedlund 的正式学术生涯始于 1930 年，他在布林莫尔学院（Bryn Mawr College，距离费城不远）担任教职直至 1939 年。在此期间，他不仅教授解析几何、微积分等基础课，还讲授了变分法、位置分析（analysis situs，即拓扑学的早期名称）、微分方程、力学以及复变函数论等高级课程。</p><p>他在这里指导了他的首批博士生，包括 1937 年毕业的两位女学生安娜・格兰特（Anna Grant）和安妮塔・图勒（Annita Tuller，1910-1994）。</p><p>1939 年至 1948 年，他转任 Virginia 大学。</p><p>在这段时期，他指导了沃尔特・赫尔比格・戈特沙尔克（Walter Helbig Gottschalk，1918-2004）（1944 年毕业）和 W・罗伊（温菲尔德）乌茨（W. Roy (Winfield) Utz, Jr.，1919-2003）（1948 年毕业）等学生，这些学者后来推动了拓扑动力学（topological dynamic）的发展。</p><p>1948 年，Hedlund 转入 Yale 大学任教。他被授予 Philip Schuyler Beebe 数学讲席教授头衔直至 1972 年退休。在 1949 年至 1959 年长达十年的时间里，他担任数学系主任，并在 Yale 大学培养了一大批日后卓有建树的学者，包括</p><ul><li><p>莱昂・威廉・格林（Leon William Green，1926-2009）：1948 年 Harvard 本科毕业，1952 年 Yale 博士毕业，1953 年起，任教于 Minnesota。</p></li><li><p>罗伯特・尤金・布莱恩（Robert Eugene Bryan）：1960 年毕业</p></li><li><p>约翰・杜安・弗格森（John Duane Ferguson）：1962 年毕业</p></li><li><p>纳尔逊・格罗・马克利（Nelson Groh Markley）：1966 年毕业，以及</p></li><li><p>伊桑・M・科文（Ethan M. Coven）：1967 年毕业等。</p></li></ul><p>Hedlund 曾三次（1933–1934, 1938–1939, 1953–1954）前往 Princeton 的 Institute for Advanced Study（IAS）担任访问学者。</p><p>在 1962 年至 1963 年冷战期间，他更是跨界出任了 Institute for Defense Analyses（IDA）通信研究部的主任，参与了通信与国防相关的应用数学研究。</p><p>1943 年，他当选为科学界荣誉学会 Sigma Xi（Sigma Xi, The Scientific Research Honor Society）的成员。</p><p>退休后，Hedlund 仍前往 Wesleyan 大学担任客座教授，直至 1993 年 3 月 15 日辞世。</p></div><div class="story post-story"><h2 id="贡献"><a href="#贡献" class="headerlink" title="贡献"></a>贡献</h2><h3 id="一、建立-Poincare-旋转数（Poincare-rotation-number）与-Morse-类型数（Morse-type-number）的联系"><a href="#一、建立-Poincare-旋转数（Poincare-rotation-number）与-Morse-类型数（Morse-type-number）的联系" class="headerlink" title="一、建立 Poincaré 旋转数（Poincaré rotation number）与 Morse 类型数（Morse type number）的联系"></a>一、建立 Poincaré 旋转数（Poincaré rotation number）与 Morse 类型数（Morse type number）的联系</h3><p>在 1932 年发表的一篇基于其博士论文的重要文章中，Hedlund 探索了流形上的周期行为。</p><p>对<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.027ex" xmlns="http://www.w3.org/2000/svg" width="2.243ex" height="1.914ex" role="img" focusable="false" viewBox="0 -833.9 991.6 845.9"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\mathbb{S}^1"><g data-mml-node="msup" data-latex="\mathbb{S}^1 "><g data-mml-node="TeXAtom" data-latex="\mathbb{S}" data-mjx-texclass="ORD"><g data-mml-node="mi" data-latex="S"><path data-c="1D54A" d="M258 661C296 661 334 644 371 612 408 580 431 545 440 507 444 489 451 480 462 480 476 480 483 492 483 517L483 667C483 692 476 704 462 704 458 704 452 701 444 695 431 682 416 676 399 676 384 676 362 681 332 690 302 699 276 704 256 704 120 704 29 635 29 502 29 464 42 429 67 396 85 373 114 350 155 329 162 326 196 310 259 281 270 276 285 267 305 256 338 236 361 212 376 185 384 170 388 154 388 137 388 58 346 31 264 31 218 31 176 52 137 93 98 134 79 178 79 224 79 238 72 245 59 245 41 245 37 231 37 209L37 25C37 0 44-12 59-12 62-12 70-7 81 2 92 11 106 16 121 16 138 16 156 12 177 3 202-7 231-12 263-12 334-12 395 4 444 36 501 74 530 127 530 194 530 297 466 374 337 425 286 444 219 474 192 498 167 519 154 542 154 565 154 624 199 661 258 661M440 640 440 600C433 609 423 620 410 634 423 635 433 637 440 640M125 626C116 607 112 587 112 565 112 521 135 484 180 454 211 434 255 413 311 390 370 365 412 341 435 317 470 279 488 238 488 194 488 140 465 99 420 71 427 90 431 112 431 137 431 188 410 231 367 264 334 289 295 312 249 332 194 355 153 377 126 397 89 424 71 459 71 502 71 559 89 601 125 626M79 96C86 85 97 73 112 58 99 57 88 55 79 51Z"></path></g></g><g data-mml-node="mn" transform="translate(588,363) scale(0.707)" data-latex="1"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47 195 42 170 39 130 39L95 39 95 0C120 2 174 3 257 3 340 3 394 2 419 0L419 39 384 39C343 39 318 42 310 47 302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"></path></g></g></g></g></svg></mjx-container> 上的连续保序变换（continuous, order‑preserving transformations of the circle），系统渐近的宏观平均旋转行为被称为 Poincaré 旋转数（Poincaré rotation number）。</p><p>Poincaré 旋转数是 Henri Poincaré 在 1885 年的论文《关于由微分方程定义的曲线（III）》（Sur les courbes définies par les équations différentielles (III)）里引入的。</p><p>假设<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="1.249ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 552 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="f:\mathbb{S}^1 \to \mathbb{S}^1"><g data-mml-node="mi" data-latex="f"><path data-c="1D453" d="M552 633C552 677 509 705 462 705 400 705 357 665 334 586 329 568 318 517 302 433L237 433C215 433 204 432 204 411 204 400 214 395 235 395L295 395 222 8C211-49 201-91 192-119 180-156 163-175 141-175 126-175 114-171 103-164 135-159 151-140 151-108 151-82 138-69 111-69 77-69 53-99 53-133 53-177 94-205 141-205 166-205 189-195 208-174 240-141 265-94 283-31 294 8 304 46 311 84L369 395 451 395C474 395 484 396 484 419 484 428 474 433 454 433L377 433C383 474 411 625 420 644 430 665 444 675 462 675 477 675 490 671 501 664 470 657 454 639 454 608 454 582 467 569 494 569 528 569 552 599 552 633Z"></path></g></g></g></svg><mjx-break size="4"></mjx-break><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="3.501ex" height="2.452ex" role="img" focusable="false" viewBox="0 -833.9 1547.3 1083.9"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="f:\mathbb{S}^1 \to \mathbb{S}^1"><g data-mml-node="mo" data-latex=":"><path data-c="3A" d="M192 375C192 405 169 431 139 431 109 431 86 405 86 375 86 345 109 319 139 319 169 319 192 345 192 375M192 56C192 86 169 112 139 112 109 112 86 86 86 56 86 26 109 0 139 0 169 0 192 26 192 56Z"></path></g><g data-mml-node="msup" data-latex="\mathbb{S}^1" transform="translate(555.8,0)"><g data-mml-node="TeXAtom" data-latex="\mathbb{S}" data-mjx-texclass="ORD"><g data-mml-node="mi" data-latex="S"><path data-c="1D54A" d="M258 661C296 661 334 644 371 612 408 580 431 545 440 507 444 489 451 480 462 480 476 480 483 492 483 517L483 667C483 692 476 704 462 704 458 704 452 701 444 695 431 682 416 676 399 676 384 676 362 681 332 690 302 699 276 704 256 704 120 704 29 635 29 502 29 464 42 429 67 396 85 373 114 350 155 329 162 326 196 310 259 281 270 276 285 267 305 256 338 236 361 212 376 185 384 170 388 154 388 137 388 58 346 31 264 31 218 31 176 52 137 93 98 134 79 178 79 224 79 238 72 245 59 245 41 245 37 231 37 209L37 25C37 0 44-12 59-12 62-12 70-7 81 2 92 11 106 16 121 16 138 16 156 12 177 3 202-7 231-12 263-12 334-12 395 4 444 36 501 74 530 127 530 194 530 297 466 374 337 425 286 444 219 474 192 498 167 519 154 542 154 565 154 624 199 661 258 661M440 640 440 600C433 609 423 620 410 634 423 635 433 637 440 640M125 626C116 607 112 587 112 565 112 521 135 484 180 454 211 434 255 413 311 390 370 365 412 341 435 317 470 279 488 238 488 194 488 140 465 99 420 71 427 90 431 112 431 137 431 188 410 231 367 264 334 289 295 312 249 332 194 355 153 377 126 397 89 424 71 459 71 502 71 559 89 601 125 626M79 96C86 85 97 73 112 58 99 57 88 55 79 51Z"></path></g></g><g data-mml-node="mn" transform="translate(588,363) scale(0.707)" data-latex="1"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47 195 42 170 39 130 39L95 39 95 0C120 2 174 3 257 3 340 3 394 2 419 0L419 39 384 39C343 39 318 42 310 47 302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"></path></g></g></g></g></svg><mjx-break size="4"></mjx-break><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="5.134ex" height="2.452ex" role="img" focusable="false" viewBox="0 -833.9 2269.3 1083.9"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="f:\mathbb{S}^1 \to \mathbb{S}^1"><g data-mml-node="mo" data-latex="\to"><path data-c="2192" d="M932 234C939 237 943 243 943 250 943 257 939 263 932 266 884 282 839 316 797 368 769 403 750 444 741 491 738 504 730 510 717 510 701 510 693 502 693 485L694 483 694 482C711 395 755 326 828 274L82 274C66 274 58 266 58 250 58 234 66 226 82 226L828 226C755 174 711 105 694 18L694 17 693 15C693-2 701-10 717-10 730-10 738-4 741 9 750 56 769 97 797 132 839 184 884 218 932 234Z"></path></g><g data-mml-node="msup" data-latex="\mathbb{S}^1" transform="translate(1277.8,0)"><g data-mml-node="TeXAtom" data-latex="\mathbb{S}" data-mjx-texclass="ORD"><g data-mml-node="mi" data-latex="S"><path data-c="1D54A" d="M258 661C296 661 334 644 371 612 408 580 431 545 440 507 444 489 451 480 462 480 476 480 483 492 483 517L483 667C483 692 476 704 462 704 458 704 452 701 444 695 431 682 416 676 399 676 384 676 362 681 332 690 302 699 276 704 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data-latex="1"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47 195 42 170 39 130 39L95 39 95 0C120 2 174 3 257 3 340 3 394 2 419 0L419 39 384 39C343 39 318 42 310 47 302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"></path></g></g></g></g></svg></mjx-container> 是<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.027ex" xmlns="http://www.w3.org/2000/svg" width="2.243ex" height="1.914ex" role="img" focusable="false" viewBox="0 -833.9 991.6 845.9"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\mathbb{S}^1"><g data-mml-node="msup" data-latex="\mathbb{S}^1 "><g data-mml-node="TeXAtom" data-latex="\mathbb{S}" data-mjx-texclass="ORD"><g data-mml-node="mi" data-latex="S"><path data-c="1D54A" d="M258 661C296 661 334 644 371 612 408 580 431 545 440 507 444 489 451 480 462 480 476 480 483 492 483 517L483 667C483 692 476 704 462 704 458 704 452 701 444 695 431 682 416 676 399 676 384 676 362 681 332 690 302 699 276 704 256 704 120 704 29 635 29 502 29 464 42 429 67 396 85 373 114 350 155 329 162 326 196 310 259 281 270 276 285 267 305 256 338 236 361 212 376 185 384 170 388 154 388 137 388 58 346 31 264 31 218 31 176 52 137 93 98 134 79 178 79 224 79 238 72 245 59 245 41 245 37 231 37 209L37 25C37 0 44-12 59-12 62-12 70-7 81 2 92 11 106 16 121 16 138 16 156 12 177 3 202-7 231-12 263-12 334-12 395 4 444 36 501 74 530 127 530 194 530 297 466 374 337 425 286 444 219 474 192 498 167 519 154 542 154 565 154 624 199 661 258 661M440 640 440 600C433 609 423 620 410 634 423 635 433 637 440 640M125 626C116 607 112 587 112 565 112 521 135 484 180 454 211 434 255 413 311 390 370 365 412 341 435 317 470 279 488 238 488 194 488 140 465 99 420 71 427 90 431 112 431 137 431 188 410 231 367 264 334 289 295 312 249 332 194 355 153 377 126 397 89 424 71 459 71 502 71 559 89 601 125 626M79 96C86 85 97 73 112 58 99 57 88 55 79 51Z"></path></g></g><g data-mml-node="mn" transform="translate(588,363) scale(0.707)" data-latex="1"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47 195 42 170 39 130 39L95 39 95 0C120 2 174 3 257 3 340 3 394 2 419 0L419 39 384 39C343 39 318 42 310 47 302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"></path></g></g></g></g></svg></mjx-container> 上的保定向的同胚映射（orientation‑preserving homeomorphism），那么<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.464ex" xmlns="http://www.w3.org/2000/svg" width="1.249ex" height="2.059ex" role="img" focusable="false" viewBox="0 -705 552 910"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="f"><g data-mml-node="mi" data-latex="f"><path data-c="1D453" d="M552 633C552 677 509 705 462 705 400 705 357 665 334 586 329 568 318 517 302 433L237 433C215 433 204 432 204 411 204 400 214 395 235 395L295 395 222 8C211-49 201-91 192-119 180-156 163-175 141-175 126-175 114-171 103-164 135-159 151-140 151-108 151-82 138-69 111-69 77-69 53-99 53-133 53-177 94-205 141-205 166-205 189-195 208-174 240-141 265-94 283-31 294 8 304 46 311 84L369 395 451 395C474 395 484 396 484 419 484 428 474 433 454 433L377 433C383 474 411 625 420 644 430 665 444 675 462 675 477 675 490 671 501 664 470 657 454 639 454 608 454 582 467 569 494 569 528 569 552 599 552 633Z"></path></g></g></g></svg></mjx-container> 可提升（lifted）到同胚映射<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="1.699ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 751 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="F:\mathbb{R} \to \mathbb{R}"><g data-mml-node="mi" data-latex="F"><path data-c="1D439" d="M199 656C199 646 210 641 231 641 271 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368 485 396 494 438 494 495 494 552 483 596 462 629 487 622 512 612 535 597M452 495C452 433 441 394 419 378 397 362 348 353 273 353L273 593C273 630 287 642 330 642 425 642 452 595 452 495M413 316C496 184 561 93 610 43L525 43 352 311C357 312 361 312 364 312 378 312 394 313 413 316M140 642 240 642C234 631 231 616 231 596L231 93C231 72 233 55 237 43L140 43C144 55 146 72 146 93L146 592C146 613 144 630 140 642Z"></path></g></g></g></g></svg><mjx-break size="4"></mjx-break><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="4.524ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 1999.8 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="F:\mathbb{R} \to \mathbb{R}"><g data-mml-node="mo" data-latex="\to"><path data-c="2192" d="M932 234C939 237 943 243 943 250 943 257 939 263 932 266 884 282 839 316 797 368 769 403 750 444 741 491 738 504 730 510 717 510 701 510 693 502 693 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410 469 405 442 396 429 378 429 351 429 325 443 312 470 312 502 312 527 341 527 373Z"></path></g></g><g data-mml-node="mi" data-latex="n" transform="translate(2149.1,-686)"><path data-c="1D45B" d="M537 137C514 58 481 18 440 18 427 18 420 28 420 47 420 61 426 84 438 115 478 224 498 296 498 333 498 403 451 442 381 442 322 442 271 416 230 363 222 407 187 442 136 442 80 442 58 390 44 345 34 313 29 294 29 287 29 278 34 273 45 273 50 273 53 274 56 276 61 285 64 292 65 299 83 375 106 413 133 413 151 413 160 399 160 371 160 358 155 331 144 290L87 63C84 50 78 24 78 19 78-1 89-11 111-11 130-11 144-1 151 19 153 24 159 49 170 92L191 181 221 295C232 318 249 341 270 365 299 397 335 413 378 413 411 413 427 391 427 348 427 310 406 234 363 120 356 102 353 87 353 74 353 25 390-11 438-11 482-11 517 14 542 64 561 104 571 131 571 144 571 153 566 158 555 158 552 158 537 149 537 137Z"></path></g><rect width="4658.1" height="60" x="120" y="220"></rect></g></g></g></g></svg><svg data-labels="true" preserveAspectRatio="xMaxYMid" viewBox="1278 -1331.9 1 2163.8"><g data-labels="true" transform="matrix(1 0 0 -1 0 0)"><g data-mml-node="mtd" id="mjx-eqn:2" transform="translate(0,623.9)"><text data-id-align="true"></text><g data-idbox="true" transform="translate(0,-748)"><g data-mml-node="mtext" data-latex="\text{(2)}"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path><path data-c="32" d="M237 666C186 666 143 648 106 612 69 576 50 534 50 483 50 449 75 424 106 424 136 424 161 450 161 480 161 513 137 536 105 536 102 536 100 536 98 535 117 584 161 627 224 627 306 627 352 556 352 470 352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0 450 180 417 180C409 129 402 100 396 91 391 86 361 84 306 84L139 84 236 179C304 243 390 312 419 365 439 400 449 435 449 470 449 588 357 666 237 666Z" transform="translate(389,0)"></path><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z" transform="translate(889,0)"></path></g></g></g></g></svg></g></g></g></g></svg></mjx-container></p><p>Henri Poincaré 证明了上面极限存在，且与所选取的起点<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.025ex" xmlns="http://www.w3.org/2000/svg" width="1.294ex" height="1.025ex" role="img" focusable="false" viewBox="0 -442 572 453"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="x"><g data-mml-node="mi" data-latex="x"><path data-c="1D465" d="M527 373C527 419 482 442 432 442 389 442 355 419 329 373 308 419 273 442 222 442 173 442 133 419 101 374 74 335 60 306 60 287 60 278 65 273 75 273 84 273 90 278 92 287 111 345 153 413 220 413 253 413 269 392 269 351 269 330 251 252 216 118 199 51 169 18 126 18 112 18 99 21 88 26 114 36 127 54 127 80 127 106 114 119 87 119 54 119 29 91 29 58 29 12 76-11 125-11 167-11 201 12 228 58 247 12 283-11 335-11 383-11 423 12 455 57 482 96 496 125 496 144 496 153 491 158 481 158 472 158 467 153 464 144 447 87 402 18 337 18 304 18 287 38 287 79 287 92 292 120 303 165L337 300C356 375 387 413 431 413 445 413 458 410 469 405 442 396 429 378 429 351 429 325 443 312 470 312 502 312 527 341 527 373Z"></path></g></g></g></svg></mjx-container> 无关。因为不同的提升<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.699ex" height="1.538ex" role="img" focusable="false" viewBox="0 -680 751 680"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="F"><g data-mml-node="mi" data-latex="F"><path data-c="1D439" d="M199 656C199 646 210 641 231 641 271 641 291 637 291 628 291 625 289 618 286 605L156 82C150 60 140 47 126 42 119 40 101 39 70 39 48 39 38 37 38 16 38 5 44 0 57 0L187 3 335 0C351 0 359 8 359 23 359 40 348 39 322 39 278 39 253 41 247 46 244 47 243 51 243 56L307 321 400 321C446 321 478 319 478 281 478 268 476 252 471 233 470 230 469 226 468 222 468 211 473 206 484 206 489 207 496 214 503 229L557 444C559 452 560 458 560 461 557 470 551 475 544 475 536 475 530 467 526 452 516 414 502 390 486 378 470 366 442 360 402 360L317 360 379 606C387 640 387 641 429 641L559 641C656 641 700 626 700 537 700 517 699 499 697 484 696 473 695 467 695 466 695 455 700 450 710 450 720 450 726 459 728 478L748 649C751 677 744 680 717 680L233 680C210 680 199 679 199 656Z"></path></g></g></g></svg></mjx-container> 只差一个整数，所以旋转数是<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="2.243ex" height="2.452ex" role="img" focusable="false" viewBox="0 -833.9 991.6 1083.9"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\mathbb{S}^1=\mathbb{R}/\mathbb{Z}"><g data-mml-node="msup" data-latex="\mathbb{S}^1"><g data-mml-node="TeXAtom" data-latex="\mathbb{S}" data-mjx-texclass="ORD"><g data-mml-node="mi" data-latex="S"><path data-c="1D54A" d="M258 661C296 661 334 644 371 612 408 580 431 545 440 507 444 489 451 480 462 480 476 480 483 492 483 517L483 667C483 692 476 704 462 704 458 704 452 701 444 695 431 682 416 676 399 676 384 676 362 681 332 690 302 699 276 704 256 704 120 704 29 635 29 502 29 464 42 429 67 396 85 373 114 350 155 329 162 326 196 310 259 281 270 276 285 267 305 256 338 236 361 212 376 185 384 170 388 154 388 137 388 58 346 31 264 31 218 31 176 52 137 93 98 134 79 178 79 224 79 238 72 245 59 245 41 245 37 231 37 209L37 25C37 0 44-12 59-12 62-12 70-7 81 2 92 11 106 16 121 16 138 16 156 12 177 3 202-7 231-12 263-12 334-12 395 4 444 36 501 74 530 127 530 194 530 297 466 374 337 425 286 444 219 474 192 498 167 519 154 542 154 565 154 624 199 661 258 661M440 640 440 600C433 609 423 620 410 634 423 635 433 637 440 640M125 626C116 607 112 587 112 565 112 521 135 484 180 454 211 434 255 413 311 390 370 365 412 341 435 317 470 279 488 238 488 194 488 140 465 99 420 71 427 90 431 112 431 137 431 188 410 231 367 264 334 289 295 312 249 332 194 355 153 377 126 397 89 424 71 459 71 502 71 559 89 601 125 626M79 96C86 85 97 73 112 58 99 57 88 55 79 51Z"></path></g></g><g data-mml-node="mn" transform="translate(588,363) scale(0.707)" data-latex="1"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47 195 42 170 39 130 39L95 39 95 0C120 2 174 3 257 3 340 3 394 2 419 0L419 39 384 39C343 39 318 42 310 47 302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"></path></g></g></g></g></svg><mjx-break size="4"></mjx-break><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="6.66ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2943.8 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="\mathbb{S}^1=\mathbb{R}/\mathbb{Z}"><g data-mml-node="mo" data-latex="="><path data-c="3D" d="M698 367 80 367C64 367 56 359 56 344 56 329 64 321 80 321L698 321C714 321 722 329 722 344 722 356 711 367 698 367M698 179 80 179C64 179 56 171 56 156 56 141 64 133 80 133L698 133C714 133 722 141 722 156 722 169 711 179 698 179Z"></path></g><g data-mml-node="TeXAtom" data-latex="\mathbb{R}" data-mjx-texclass="ORD" transform="translate(1055.8,0)"><g data-mml-node="mi" data-latex="R"><path data-c="211D" d="M104 590 104 95C104 45 99 43 47 43 26 43 16 36 16 22 16 4 31 0 54 0L324 0C349 0 361 7 361 22 361 36 350 43 329 43 277 43 273 45 273 95L273 310 302 310 450 83C475 43 490 20 494 14 500 5 512 0 530 0L666 0C691 0 703 7 703 22 703 33 697 40 685 42 660 47 625 82 579 146 534 208 494 267 459 322 573 344 630 402 630 496 630 563 600 613 540 644 488 671 429 685 364 685L54 685C29 685 16 678 16 663 16 649 26 642 47 642 99 642 104 640 104 590M535 597C570 576 587 542 587 496 587 427 547 385 468 368 485 396 494 438 494 495 494 552 483 596 462 629 487 622 512 612 535 597M452 495C452 433 441 394 419 378 397 362 348 353 273 353L273 593C273 630 287 642 330 642 425 642 452 595 452 495M413 316C496 184 561 93 610 43L525 43 352 311C357 312 361 312 364 312 378 312 394 313 413 316M140 642 240 642C234 631 231 616 231 596L231 93C231 72 233 55 237 43L140 43C144 55 146 72 146 93L146 592C146 613 144 630 140 642Z"></path></g></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(1777.8,0)"><g data-mml-node="mo" data-latex="/"><path data-c="2F" d="M444 718C445 720 445 723 445 726 445 742 437 750 421 750 410 750 403 745 399 734L57-218C56-220 56-223 56-226 56-242 64-250 80-250 91-250 98-245 102-234Z"></path></g></g><g data-mml-node="TeXAtom" data-latex="\mathbb{Z}" data-mjx-texclass="ORD" transform="translate(2277.8,0)"><g data-mml-node="mi" data-latex="Z"><path data-c="2124" d="M569 685 117 685C103 685 93 684 88 681 83 670 79 661 78 653 73 604 68 554 62 506 62 491 69 483 83 483 94 483 101 489 104 501 119 594 217 642 332 642L405 642 37 42C31 31 28 25 28 22 28 4 43 0 66 0L572 0C601 0 607 5 609 31L636 226C636 241 629 248 615 248 610 248 605 246 602 243 596 231 593 222 592 215 571 117 480 43 362 43L228 43 596 643C602 654 605 660 605 663 605 682 593 685 569 685M156 642C144 635 131 625 116 612L119 642M455 642 546 642 179 43 87 43M517 43C538 56 558 73 575 92L568 43Z"></path></g></g></g></g></svg></mjx-container> 里的元素。</p><p>如果<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.464ex" xmlns="http://www.w3.org/2000/svg" width="1.249ex" height="2.059ex" role="img" focusable="false" viewBox="0 -705 552 910"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="f"><g data-mml-node="mi" data-latex="f"><path data-c="1D453" d="M552 633C552 677 509 705 462 705 400 705 357 665 334 586 329 568 318 517 302 433L237 433C215 433 204 432 204 411 204 400 214 395 235 395L295 395 222 8C211-49 201-91 192-119 180-156 163-175 141-175 126-175 114-171 103-164 135-159 151-140 151-108 151-82 138-69 111-69 77-69 53-99 53-133 53-177 94-205 141-205 166-205 189-195 208-174 240-141 265-94 283-31 294 8 304 46 311 84L369 395 451 395C474 395 484 396 484 419 484 428 474 433 454 433L377 433C383 474 411 625 420 644 430 665 444 675 462 675 477 675 490 671 501 664 470 657 454 639 454 608 454 582 467 569 494 569 528 569 552 599 552 633Z"></path></g></g></g></svg></mjx-container> 是旋转了<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.025ex" xmlns="http://www.w3.org/2000/svg" width="4.414ex" height="1.57ex" role="img" focusable="false" viewBox="0 -683 1951 694"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="2\pi N"><g data-mml-node="mn" data-latex="2"><path data-c="32" d="M237 666C186 666 143 648 106 612 69 576 50 534 50 483 50 449 75 424 106 424 136 424 161 450 161 480 161 513 137 536 105 536 102 536 100 536 98 535 117 584 161 627 224 627 306 627 352 556 352 470 352 403 318 331 250 255L62 43C49 28 50 29 50 0L421 0 450 180 417 180C409 129 402 100 396 91 391 86 361 84 306 84L139 84 236 179C304 243 390 312 419 365 439 400 449 435 449 470 449 588 357 666 237 666Z"></path></g><g data-mml-node="mi" data-latex="\pi" transform="translate(500,0)"><path data-c="1D70B" d="M524 431 194 431C153 431 117 415 88 384 74 369 27 305 27 292 31 285 31 279 43 279 50 279 56 283 62 292 94 341 135 366 184 366L235 366C212 279 170 172 110 45 105 33 102 24 102 19 102-1 113-11 134-11 153-11 167 0 176 21 194 78 207 122 214 152L269 366 372 366C345 247 331 164 331 117 331 68 342-11 379-11 400-11 423 9 423 30 423 35 421 43 417 53 398 100 389 153 389 214 389 261 395 312 406 366L515 366C550 366 567 378 567 403 567 426 549 431 524 431Z"></path></g><g data-mml-node="mi" data-latex="N" transform="translate(1070,0)"><path data-c="1D441" d="M864 683C845 683 783 680 764 680 745 680 682 683 663 683 648 683 641 675 641 659 641 650 648 645 663 644 705 644 726 631 726 605 726 599 725 592 724 585L616 157 401 662C393 681 391 683 366 683L233 683C209 683 200 681 200 659 200 649 211 644 232 644 274 644 295 643 296 640L163 110C154 71 131 49 95 42 69 39 39 43 39 15 39 5 45 0 56 0 74 0 136 3 155 3 174 3 238 0 257 0 272 0 279 8 279 24 279 33 271 38 255 39 214 40 194 53 194 78 194 83 195 90 197 100L326 611 576 21C582 7 590 0 599 0 608 0 614 8 618 24L757 574C770 625 798 643 860 644 874 645 881 653 881 669 878 678 877 683 864 683Z"></path></g></g></g></svg></mjx-container> 的映射，<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="1.993ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 881 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="N\in(0,1)"><g data-mml-node="mi" data-latex="N"><path data-c="1D441" d="M864 683C845 683 783 680 764 680 745 680 682 683 663 683 648 683 641 675 641 659 641 650 648 645 663 644 705 644 726 631 726 605 726 599 725 592 724 585L616 157 401 662C393 681 391 683 366 683L233 683C209 683 200 681 200 659 200 649 211 644 232 644 274 644 295 643 296 640L163 110C154 71 131 49 95 42 69 39 39 43 39 15 39 5 45 0 56 0 74 0 136 3 155 3 174 3 238 0 257 0 272 0 279 8 279 24 279 33 271 38 255 39 214 40 194 53 194 78 194 83 195 90 197 100L326 611 576 21C582 7 590 0 599 0 608 0 614 8 618 24L757 574C770 625 798 643 860 644 874 645 881 653 881 669 878 678 877 683 864 683Z"></path></g></g></g></svg><mjx-break size="4"></mjx-break><svg style="vertical-align:-.566ex" xmlns="http://www.w3.org/2000/svg" width="7.166ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 3167.4 1000"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="N\in(0,1)"><g data-mml-node="mo" data-latex="\in"><path data-c="2208" d="M563 4 373 4C310 4 254 25 208 68 162 111 136 163 130 226L563 226C579 226 587 234 587 250 587 266 579 274 563 274L130 274C136 337 162 389 208 432 254 475 310 496 373 496L563 496C579 496 587 504 587 519 587 535 579 543 563 543L373 543C292 543 223 515 166 458 109 401 81 332 81 250 81 168 109 99 166 42 223-15 292-43 373-43L563-43C579-43 587-35 587-19 587-7 576 4 563 4Z"></path></g><g data-mml-node="mo" data-latex="(" transform="translate(944.8,0)"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path></g><g data-mml-node="mn" data-latex="0" transform="translate(1333.8,0)"><path data-c="30" d="M249-22C390-22 460 92 460 320 460 473 428 575 365 625 330 652 291 666 250 666 109 666 39 551 39 320 39 136 88-22 249-22M361 524C368 489 371 425 371 332 371 240 367 172 360 128 347 48 310 8 249 8 226 8 203 17 182 34 155 57 139 104 132 176 129 201 128 253 128 332 128 419 131 480 136 513 145 568 163 603 191 618 213 630 232 636 249 636 314 636 350 583 361 524Z"></path></g><g data-mml-node="mo" data-latex="," transform="translate(1833.8,0)"><path data-c="2C" d="M139 106C107 106 86 82 86 50 86 20 109-5 139-5 153-5 165-1 174 8L175 0C175-63 154-117 112-160 105-168 101-174 101-178 101-188 105-193 114-193 123-193 135-181 152-158 186-110 203-57 203 0 203 53 185 106 139 106Z"></path></g><g data-mml-node="mn" data-latex="1" transform="translate(2278.4,0)"><path data-c="31" d="M269 666C228 624 168 603 89 603L89 564C141 564 184 572 217 588L217 82C217 64 213 52 204 47 195 42 170 39 130 39L95 39 95 0C120 2 174 3 257 3 340 3 394 2 419 0L419 39 384 39C343 39 318 42 310 47 302 52 297 64 297 82L297 636C297 660 295 666 269 666Z"></path></g><g data-mml-node="mo" data-latex=")" transform="translate(2778.4,0)"><path data-c="29" d="M78-245C138-199 188-131 229-40 268 47 288 129 288 208L288 292C288 371 268 453 229 540 188 631 138 699 78 745 75 747 72 748 71 748 62 748 57 743 57 734 57 730 59 726 62 723 114 683 156 617 187 526 214 447 228 369 228 292L228 208C228 131 214 53 187-26 156-117 114-183 62-223 59-227 57-231 57-234 57-243 62-248 71-248 72-248 75-247 78-245Z"></path></g></g></g></svg></mjx-container> ，那么<mjx-container class="MathJax" jax="SVG" 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467 695 466 695 455 700 450 710 450 720 450 726 459 728 478L748 649C751 677 744 680 717 680L233 680C210 680 199 679 199 656Z"></path></g><g data-mml-node="mo" data-latex="(" transform="translate(751,0)"><path data-c="28" d="M318-248C327-248 332-243 332-234 332-231 330-227 327-223 275-183 233-117 202-26 175 53 161 131 161 208L161 292C161 369 175 447 202 526 233 617 275 683 327 723 330 726 332 730 332 734 332 743 327 748 318 748 317 748 314 747 311 745 251 699 201 631 160 540 121 453 101 371 101 292L101 208C101 129 121 47 160-40 201-131 251-199 311-245 314-247 317-248 318-248Z"></path></g><g data-mml-node="mi" data-latex="x" transform="translate(1140,0)"><path data-c="1D465" d="M527 373C527 419 482 442 432 442 389 442 355 419 329 373 308 419 273 442 222 442 173 442 133 419 101 374 74 335 60 306 60 287 60 278 65 273 75 273 84 273 90 278 92 287 111 345 153 413 220 413 253 413 269 392 269 351 269 330 251 252 216 118 199 51 169 18 126 18 112 18 99 21 88 26 114 36 127 54 127 80 127 106 114 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transform="translate(1000.2,0)"><path data-c="1D441" d="M864 683C845 683 783 680 764 680 745 680 682 683 663 683 648 683 641 675 641 659 641 650 648 645 663 644 705 644 726 631 726 605 726 599 725 592 724 585L616 157 401 662C393 681 391 683 366 683L233 683C209 683 200 681 200 659 200 649 211 644 232 644 274 644 295 643 296 640L163 110C154 71 131 49 95 42 69 39 39 43 39 15 39 5 45 0 56 0 74 0 136 3 155 3 174 3 238 0 257 0 272 0 279 8 279 24 279 33 271 38 255 39 214 40 194 53 194 78 194 83 195 90 197 100L326 611 576 21C582 7 590 0 599 0 608 0 614 8 618 24L757 574C770 625 798 643 860 644 874 645 881 653 881 669 878 678 877 683 864 683Z"></path></g></g></g></svg></mjx-container> ，从而<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:-.464ex" xmlns="http://www.w3.org/2000/svg" width="1.249ex" height="2.059ex" role="img" focusable="false" viewBox="0 -705 552 910"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="f"><g data-mml-node="mi" data-latex="f"><path data-c="1D453" d="M552 633C552 677 509 705 462 705 400 705 357 665 334 586 329 568 318 517 302 433L237 433C215 433 204 432 204 411 204 400 214 395 235 395L295 395 222 8C211-49 201-91 192-119 180-156 163-175 141-175 126-175 114-171 103-164 135-159 151-140 151-108 151-82 138-69 111-69 77-69 53-99 53-133 53-177 94-205 141-205 166-205 189-195 208-174 240-141 265-94 283-31 294 8 304 46 311 84L369 395 451 395C474 395 484 396 484 419 484 428 474 433 454 433L377 433C383 474 411 625 420 644 430 665 444 675 462 675 477 675 490 671 501 664 470 657 454 639 454 608 454 582 467 569 494 569 528 569 552 599 552 633Z"></path></g></g></g></svg></mjx-container> 的旋转数是<mjx-container class="MathJax" jax="SVG" overflow="overflow"><svg style="vertical-align:0" xmlns="http://www.w3.org/2000/svg" width="1.993ex" height="1.545ex" role="img" focusable="false" viewBox="0 -683 881 683"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math" data-latex="N"><g data-mml-node="mi" data-latex="N"><path data-c="1D441" d="M864 683C845 683 783 680 764 680 745 680 682 683 663 683 648 683 641 675 641 659 641 650 648 645 663 644 705 644 726 631 726 605 726 599 725 592 724 585L616 157 401 662C393 681 391 683 366 683L233 683C209 683 200 681 200 659 200 649 211 644 232 644 274 644 295 643 296 640L163 110C154 71 131 49 95 42 69 39 39 43 39 15 39 5 45 0 56 0 74 0 136 3 155 3 174 3 238 0 257 0 272 0 279 8 279 24 279 33 271 38 255 39 214 40 194 53 194 78 194 83 195 90 197 100L326 611 576 21C582 7 590 0 599 0 608 0 614 8 618 24L757 574C770 625 798 643 860 644 874 645 881 653 881 669 878 678 877 683 864 683Z"></path></g></g></g></svg></mjx-container> 。</p><p>同时，在其导师 Marston Morse 创立的变分法理论中有一个核心概念 ——Morse 类型数（Morse type number），用于量化动力系统中极小集（minimal sets）的拓扑复杂性（topological complexity）。</p><p>Hedlund 严格证明了：如果上述圆周变换的旋转数是无理数（irrational number），那么其对应的 Morse 类型数必定是奇数（odd number）。</p><p>这一优美的定理首次提供了一个极其强悍的拓扑不变量。它直接将几何上的旋转行为与变分学性质严密地绑定在一起，为基于极小不变集（minimal invariant sets）对拓扑变换进行分类提供了最早期的系统工具。</p><h3 id="二、证明测地流的遍历性（ergodicity-of-geodesic-flows）与几何动力学"><a href="#二、证明测地流的遍历性（ergodicity-of-geodesic-flows）与几何动力学" class="headerlink" title="二、证明测地流的遍历性（ergodicity of geodesic flows）与几何动力学"></a>二、证明测地流的遍历性（ergodicity of geodesic flows）与几何动力学</h3><p>20 世纪 30 年代，受 John von Neumann 提出的均值遍历定理（mean ergodic theorem）启发，数学界当时希望在具体的连续几何流形上验证这种宏观的统计力学性质。Hedlund 对常负曲率曲面上的测地流、Fuchsian 群作用以及相关轨道结构作出了重要研究，为几何动力系统和遍历理论的发展奠定了基础。</p><p>在这样的几何空间的单位切丛（unit tangent bundle）上，他严格证明了测地流（geodesic flow）关于 Liouville 测度（Liouville measure，即相空间的自然体积测度）是遍历的（ergodic）。</p><p>这意味着几乎所有轨道的时间平均（time averages）都将严格收敛于整个系统的空间平均（space average）。在宏观上表现出极其稠密的轨道（dense orbits）和强混合性质（strong mixing properties）。</p><p>Hedlund 通过严密分析几何流的非游荡集（non-wandering set）和传递性（transitivity），成功将几何约束转化为了严谨的测度论结果。</p><p>此外，他对具有周期测地曲率的 Riemannian 流形（Riemannian manifolds with periodic geodesic curvature）以及 Fuchsian 群的动力学（dynamics of Fuchsian groups）的研究，也为遍历理论（ergodic theory）打下了基础。</p><h3 id="三、符号动力学（symbolic-dynamics）的创立"><a href="#三、符号动力学（symbolic-dynamics）的创立" class="headerlink" title="三、符号动力学（symbolic dynamics）的创立"></a>三、符号动力学（symbolic dynamics）的创立</h3><p>符号动力学研究的对象是定义在离散空间上的动力系统，该离散空间由抽象符号构成的无限序列。</p><p>相关想法可追溯到 Hadamard 在 1898 年的关于负曲率曲面上测地线的论文《曲率相反的曲面及其测地线》（Les surfaces à courbures opposées et leurs lignes géodésiques）。</p><p>1921 年，Marston Morse 将其应用于构造非周期性递归测地线（nonperiodic recurrent geodesic）。</p><p>1938 年，Hedlund 与导师 Morse 发表《符号动力学》（Symbolic dynamics）及其后续《无尽国际象棋、符号动力学与半群中的一个问题》（Unending chess, symbolic dynamics, and a problem in semi-groups），通常被视为符号动力学作为系统理论的起点。</p><p>1948 年，Shannon 在《通信的数学理论》（A mathematical theory of communication）中系统使用符号序列、信源编码和熵的思想，开创了信息论。</p><p>这种离散组合框架（discrete combinatorial framework），使得研究人员能够通过纯符号编码（symbolic encoding）来分析复杂的动力学行为，为后来的混沌理论（chaos theory）提供了重要的理论基础。</p><h3 id="四、拓扑动力学（topological-dynamics）的公理化"><a href="#四、拓扑动力学（topological-dynamics）的公理化" class="headerlink" title="四、拓扑动力学（topological dynamics）的公理化"></a>四、拓扑动力学（topological dynamics）的公理化</h3><p>早期的遍历理论（ergodic theory）高度依赖概率论和特定的测度工具（measure-theoretic tools）。</p><p>然而，Hedlund 指出，许多动力系统的长程演化属性（long-term evolutionary properties）可以剥离微积分，仅仅取决于底层的宏观拓扑结构（topological structure）。</p><p>1955 年，他与得意门生 Walter Gottschalk 共同出版了具有里程碑意义的专著《拓扑动力学》（Topological Dynamics）。</p><p>在这部著作中，他们采用抽象的拓扑半群（topological semigroups），将动力系统重新公理化（axiomatization）为拓扑变换群（topological transformation groups）在紧致空间上的连续作用。</p><p>利用点集拓扑语言，他们系统构建并分析了紧致空间上的同胚、迫近流（proximal flows）、极小集（minimal sets）和殆周期点（almost periodic points）等概念。这一公理体系宣告拓扑动力学正式脱离测度论，确立为一门纯数学分支。</p><p>在他的影响下，其学术后代将这些拓扑思想扩展到了常返变换（recurrent transformations）、区间映射和一般拓扑空间的演化性质，并启发了后来对拓扑熵（topological entropy）的研究。</p><h3 id="五、Curtis-Hedlund-Lyndon-定理（The-Curtis-Hedlund-Lyndon-theorem）"><a href="#五、Curtis-Hedlund-Lyndon-定理（The-Curtis-Hedlund-Lyndon-theorem）" class="headerlink" title="五、Curtis-Hedlund-Lyndon 定理（The Curtis-Hedlund-Lyndon theorem）"></a>五、Curtis-Hedlund-Lyndon 定理（The Curtis-Hedlund-Lyndon theorem）</h3><p>这是 Hedlund 于 1969 年发表的论文《移位动力系统的自同态与自同构》(Endomorphisms and automorphisms of the shift dynamical system) 中提出的著名纯数学定理（学界通常简称为 CHL 定理）。</p><p>该定理最初是作为符号动力学的核心结果提出的，极其优美地给出了离散动力系统演化规则的纯拓扑特征刻画。</p><p>在论文里，Hedlund 将柯蒂斯（Morton Landers Curtis，1921-1989）和林登（Roger Conant Lyndon，1917-1988）列为共同发现者。</p><p>Lyndon 在 1946 年获得博士学位（指导老师是 Saunders MacLane）；他的博士论文产生了 Lyndon-Hochschild-Serre 谱序列（Lyndon-Hochschild-Serre spectral sequence）。</p><p>在原始论文中，Hedlund 研究的是有限字母表上的全移位空间（full shift space over a finite alphabet）。</p><p>CHL 定理指出：一个定义在该空间上的全局映射（global mapping）能够由微观的 “有限局部块编码（sliding block codes）” 生成，当且仅当它同时满足两个宏观拓扑条件（macroscopic topological conditions）：</p><ul><li>它是连续映射；</li><li>它与移位映射可交换（commutes with the shift map）。</li></ul><p>尽管 Hedlund 证明该定理的初衷是为了研究纯数学中的有限型子移位（subshifts of finite type）等代数拓扑结构，并未直接涉足计算机领域。但后来理论计算机科学界发现，CHL 定理所刻画的拓扑映射，在数学本质上完全等价于一维元胞自动机（cellular automata）。</p><p>该定理极具远见地将计算机科学中微观的局部可计算性（local computability）与纯数学中宏观的点集拓扑连续性（point-set topological continuity）等价了起来。</p></div></div><div class="footer"></div><div class="article-meta" id="bottom"><div class="new-meta-box"></div></div><div class="prev-next"><a class="prev" href="/notes/person/18"><p class="title"><i class="fa-solid fa-chevron-left" aria-hidden="true"></i>Shannon克劳德·香农：信息时代的奠基者与科学巨匠</p><p class="content">克劳德·香农是信息论之父，其1948年发表的《通信的数学理论》为现代通信奠定了数学基础。他首次将布尔代数应用于开关电路设计，革新了数字计算机的构建原理。此外，他在密码学、人工智能及自动机领域的早期探索也产生了深远影响</p></a><a class="next" href="/notes/person/20"><p class="title">Gibbs吉布斯：奠基现代物理化学与统计力学的巨匠<i class="fa-solid fa-chevron-right" aria-hidden="true"></i></p><p class="content">美国科学家吉布斯在热力学、化学平衡及统计力学领域做出了奠基性贡献，他创立了化学势、相律及向量分析等核心理论，深刻影响了物理化学与理论物理学的发展</p></a></div><div class="recommended-article"><div class="recommended-article-header"><i class="fa-solid fa-bookmark fa-fw" aria-hidden="true"></i> <span>推荐阅读</span></div><div class="recommended-article-group"> <a class="recommended-article-item" 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Wall的学术生涯与在拓扑学及奇点理论中的奠基性贡献</span></a></div></div></article><article class="post white-box shadow floatable blur" id="comments"><span hidden=""><meta itemprop="discussionUrl" content="/notes/person/19#comments"></span><p ct=""><i class="fa-duotone fa-comments"></i> 留言区</p><div id="layoutHelper-comments"></div></article></div><aside id="l_side" itemscope="" itemtype="http://schema.org/WPSideBar"><section class="widget text desktop mobile pjax"><header><a href="/notes/"><i class="fa-duotone fa-book fa-fw" aria-hidden="true"></i> <span class="name">Notes</span></a></header><div class="content"></div></section><section class="widget list group desktop mobile pjax"><header><i class="fa-duotone fa-p fa-fw" aria-hidden="true"></i> <span class="name">History of mathematics</span></header><div class="content"><ul class="list entry navigation"><li><a class="flat-box" title="/notes/person/" href="/notes/person/" active-action="action-notesperson"><div class="name"> Welcome</div></a></li><li><a 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