{"id":34401,"date":"2022-05-15T16:43:00","date_gmt":"2022-05-15T08:43:00","guid":{"rendered":"https:\/\/swarma.org\/?p=34401"},"modified":"2022-05-15T16:43:00","modified_gmt":"2022-05-15T08:43:00","slug":"%e8%87%aa%e7%9b%b8%e4%bc%bc%ef%bc%9a%e4%bb%8e%e9%9b%aa%e8%8a%b1%e3%80%81%e6%b5%b7%e5%b2%b8%e7%ba%bf%e5%88%b0%e8%82%a1%e7%a5%a8%e5%b8%82%e5%9c%ba-%e9%9b%86%e6%99%ba%e7%99%be%e7%a7%91","status":"publish","type":"post","link":"https:\/\/swarma.org\/?p=34401","title":{"rendered":"\u81ea\u76f8\u4f3c\uff1a\u4ece\u96ea\u82b1\u3001\u6d77\u5cb8\u7ebf\u5230\u80a1\u7968\u5e02\u573a | \u96c6\u667a\u767e\u79d1"},"content":{"rendered":"
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\u81ea\u76f8\u4f3c\u4e00\u8bcd\u7531<\/span>\u4f2f\u52aa\u74e6\u00b7\u66fc\u5fb7\u5e03\u6d1b\u7279 Benoit Mandelbrot<\/span>\u4e0e1964\u5e74\u5f15\u5165\u3002\u5728\u6570\u5b66\u4e2d\uff0c\u4e00\u4e2a\u81ea\u76f8\u4f3c\u7684\u7269\u4f53\u4e0e\u5b83\u81ea\u8eab\u7684\u67d0\u4e00\u90e8\u5206\u5b8c\u5168\u6216\u8fd1\u4f3c\u5730\u76f8\u4f3c\uff08\u4f8b\u5982\uff1a\u6574\u4f53\u548c\u4e00\u4e2a\u6216\u591a\u4e2a\u90e8\u5206\u5177\u6709\u76f8\u540c\u7684\u5f62\u72b6\uff09\u3002\u73b0\u5b9e\u4e16\u754c\u4e2d\u7684\u8bb8\u591a\u7269\u4f53\uff0c\u4f8b\u5982\u6d77\u5cb8\u7ebf\uff0c\u5728\u7edf\u8ba1\u5b66\u4e0a\u662f\u81ea\u76f8\u4f3c\u7684\uff1a\u5b83\u4eec\u7684\u67d0\u4e9b\u90e8\u5206\u5728\u8bb8\u591a\u4e0d\u540c\u5c3a\u5ea6\u4e0a\u8868\u73b0\u51fa\u76f8\u540c\u7684\u7edf\u8ba1\u7279\u6027\u3002\u81ea\u76f8\u4f3c\u662f<\/span>\u5206\u5f62<\/span>\u7684\u4e00\u4e2a\u5178\u578b\u6027\u8d28\u3002<\/span>\u6807\u5ea6\u4e0d\u53d8\u6027<\/span>\u662f\u81ea\u76f8\u4f3c\u7684\u4e00\u79cd\u7cbe\u786e\u5f62\u5f0f\uff1a\u5728\u4efb\u4f55\u653e\u5927\u500d\u6570\u4e0b\uff0c\u7269\u4f53\u4e2d\u603b\u6709\u66f4\u5c0f\u7684\u90e8\u5206\u4e0e\u6574\u4f53\u76f8\u4f3c\u3002\u4f8b\u5982\uff0c<\/span>\u79d1\u8d6b\u96ea\u82b1<\/span>\u7684\u4e00\u8fb9\u65e2\u5bf9\u79f0\u53c8\u5177\u6709\u6807\u5ea6\u4e0d\u53d8\u6027\uff1b\u5b83\u53ef\u4ee5\u8fde\u7eed\u653e\u59273\u500d\u800c\u4e0d\u6539\u53d8\u5f62\u72b6\u3002\u5206\u5f62\u4e2d\u660e\u663e\u7684\u975e\u5e73\u51e1\u7684\u76f8\u4f3c\u6027\u662f\u901a\u8fc7\u5b83\u4eec\u7684\u7cbe\u7ec6\u7ed3\u6784\u6216\u4efb\u610f\u5c0f\u5c3a\u5ea6\u4e0a\u7684\u7ec6\u8282\u6765\u533a\u5206\u7684\u3002\u5bf9\u6bd4\u4e00\u4e2a\u53cd\u4f8b\u6765\u770b\uff0c\u5c3d\u7ba1\u76f4\u7ebf\u7684\u4efb\u4f55\u90e8\u5206\u90fd\u53ef\u80fd\u7c7b\u4f3c\u4e8e\u6574\u4f53\uff0c\u4f46\u662f\u8fdb\u4e00\u6b65\u653e\u5927\u4e4b\u540e\uff0c\u5374\u6ca1\u6709\u66f4\u591a\u7684\u7ec6\u8282\u663e\u9732\u3002<\/span>
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\u5373\u4f7f\u4ece\u6570\u5b66\u4e0a\u6765\u8bf4\uff0c\u5206\u5f62\u53ef\u4ee5\u5728\u65e0\u9650\u653e\u5927\u7684\u6761\u4ef6\u4e0b\u663e\u793a\u51fa\u81ea\u76f8\u4f3c\u6027\uff0c\u4f46\u662f\u8fd9\u5728\u7269\u7406\u4e0a\u662f\u4e0d\u53ef\u80fd\u5b9e\u73b0\u7684\u3002\u4f69\u7279\u6839\u7b49\u5efa\u8bae\u4f7f\u7528\u8fd1\u4f3c\u65b9\u6cd5\u6765\u7814\u7a76\u81ea\u76f8\u4f3c\u6027\uff1a<\/span><\/p>\n

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\u5728\u6570\u5b66\u4e2d\uff0c\u81ea\u4eff\u5c04\u6027 Self-affinity \u662f\u5206\u5f62\u7684\u7279\u5f81\u4e4b\u4e00\uff0c\u5206\u5f62\u7684\u5404\u90e8\u5206\u5728 x \u65b9\u5411\u548c y \u65b9\u5411\u4e0a\u6309\u4e0d\u540c\u7684\u6bd4\u4f8b\u7f29\u653e\u3002\u8fd9\u610f\u5473\u7740\u8981\u7406\u89e3\u8fd9\u4e9b\u5206\u5f62\u5bf9\u8c61\u7684\u81ea\u76f8\u4f3c\u6027\uff0c\u5fc5\u987b\u4f7f\u7528\u5404\u5411\u5f02\u6027\u4eff\u5c04\u53d8\u6362 Anisotropic Affine Transformation \u8fdb\u884c\u7f29\u653e\u3002<\/span><\/p>\n


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\u4e8c\u5143\u5e7a\u534a\u7fa4\u7684\u81ea\u540c\u6784\u662f\u6a21\u7fa4 Modular Group \uff0c\u81ea\u540c\u6784\u53ef\u4ee5\u63cf\u8ff0\u4e3a\u4e8c\u53c9\u6811\u7684\u53cc\u66f2\u65cb\u8f6c Hyperbolic Rotations \u3002<\/span><\/p>\n


\u6bd4\u81ea\u76f8\u4f3c\u6027\u66f4\u4e00\u822c\u7684\u6982\u5ff5\u662f\u81ea\u4eff\u5c04\u6027\u3002<\/span><\/p>\n


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\u5b9e\u4f8b<\/span><\/strong><\/strong><\/span><\/strong><\/strong><\/p>\n<\/section>\n

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<\/strong><\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n<\/section>\n

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\u66fc\u5fb7\u5e03\u6d1b\u7279\u96c6<\/span>\u5728Misiurewicz \u70b9\u9644\u8fd1\u4e5f\u5177\u6709\u81ea\u76f8\u4f3c\u6027\u3002<\/span><\/p>\n


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\u66fc\u5fb7\u52c3\u7f57\u96c6\u5728\u8d39\u6839\u9c8d\u59c6\u70b9\uff08- 1.401155189… \uff0c0\uff09\u5904\u4e0d\u65ad\u653e\u5927\u663e\u793a\u51fa\u5176\u4e2d\u7684\u81ea\u76f8\u4f3c\u6027\u3002<\/span><\/p>\n

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\u81ea\u76f8\u4f3c\u5bf9\u4e8e\u8ba1\u7b97\u673a\u7f51\u7edc\u7684\u8bbe\u8ba1\u6709\u7740\u91cd\u8981\u7684\u610f\u4e49\uff0c\u56e0\u4e3a\u5178\u578b\u7684\u7f51\u7edc\u6d41\u91cf\u5177\u6709\u81ea\u76f8\u4f3c\u7684\u7279\u6027\u3002\u4f8b\u5982\uff0c\u5728\u7535\u4fe1\u6d41\u91cf\u5de5\u7a0b\u4e2d\uff0c\u5206\u7ec4\u4ea4\u6362\u6570\u636e\u6d41\u91cf\u6a21\u5f0f\u4f3c\u4e4e\u5728\u7edf\u8ba1\u4e0a\u662f\u81ea\u76f8\u4f3c\u7684\u3002\u8fd9\u79cd\u6027\u8d28\u610f\u5473\u7740\u4f7f\u7528Poisson Distribution \u6cca\u677e\u5206\u5e03\u7684\u7b80\u5355\u6a21\u578b\u662f\u4e0d\u51c6\u786e\u7684\uff0c\u800c\u6ca1\u6709\u8003\u8651\u81ea\u76f8\u4f3c\u6027\u7684\u7f51\u7edc\u5f88\u53ef\u80fd\u4ee5\u610f\u60f3\u4e0d\u5230\u7684\u65b9\u5f0f\u8fd0\u884c\u3002<\/span>
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\u7c7b\u4f3c\u5730\uff0c\u4eba\u4eec\u5728\u63cf\u8ff0\u80a1\u7968\u5e02\u573a\u65f6\u8ba4\u4e3a\u5176\u6ce2\u52a8\u5177\u6709\u81ea\u4eff\u5c04\u6027\uff0c\u4e5f\u5c31\u662f\u8bf4\uff0c\u5f53\u6839\u636e\u663e\u793a\u7684\u7ec6\u8282\u7a0b\u5ea6\uff0c\u901a\u8fc7\u9002\u5f53\u7684\u4eff\u5c04\u53d8\u6362\u8fdb\u884c\u8f6c\u6362\u65f6\uff0c\u5b83\u4eec\u663e\u793a\u51fa\u81ea\u76f8\u4f3c\u6027\u3002Andrew Lo\u63cf\u8ff0\u4e86\u8ba1\u91cf\u7ecf\u6d4e\u5b66\u4e2d\u80a1\u7968\u5e02\u573a\u7684\u5bf9\u6570\u56de\u62a5\u81ea\u76f8\u4f3c\u6027\u3002<\/span><\/p>\n


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\u5177\u6709\u4eff\u5c04\u81ea\u76f8\u4f3c\u6027\u7684\u5df4\u6069\u65af\u5229\u8568\u7c7b\u690d\u7269\u7684\u56fe\u50cf<\/span><\/p>\n


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\u6709\u9650\u7ec6\u5206\u89c4\u5219\u662f\u4e00\u79cd\u5efa\u7acb\u81ea\u76f8\u4f3c\u96c6\u7684\u5f3a\u5927\u65b9\u6cd5\uff0c\u5305\u62ec Cantor Set \u5eb7\u6258\u96c6\u548c<\/span>\u8c22\u5c14\u5bbe\u65af\u57fa\u4e09\u89d2<\/span>\u3002<\/span><\/p>\n


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\u63a7\u5236\u8bba\u9886\u57df<\/span><\/strong><\/span><\/h3>\n

\u65af\u5854\u798f\u5fb7 \u00b7 \u6bd4\u5c14<\/span>\u7684\u53ef\u884c\u7cfb\u7edf\u6a21\u578b\u662f\u4e00\u4e2a\u5177\u6709\u4eff\u5c04\u81ea\u76f8\u4f3c\u5c42\u6b21\u7ed3\u6784\u7684\u7ec4\u7ec7\u6a21\u578b\uff0c\u5176\u4e2d\u4e00\u4e2a\u7ed9\u5b9a\u7684\u53ef\u884c\u7cfb\u7edf\u662f\u4e00\u4e2a\u9012\u5f52\u66f4\u9ad8\u4e00\u7ea7\u7684\u53ef\u884c\u7cfb\u7edf\u4e4b\u4e00\u7684\u4e00\u4e2a\u5143\u7d20\uff0c\u5bf9\u4e8e\u8fd9\u4e2a\u7cfb\u7edf\u7684\u5143\u7d20\u662f\u4e00\u4e2a\u9012\u5f52\u5c42\u6b21\u66f4\u4f4e\u7684\u53ef\u884c\u7cfb\u7edf\u3002<\/span><\/p>\n


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\u81ea\u7136\u754c\u4e2d<\/span><\/strong><\/span><\/h3>\n


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\u7f57\u9a6c\u82b1\u6930\u83dc\u7684\u7279\u5199\u955c\u5934\u3002<\/span><\/p>\n


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\u81ea\u7136\u4e2d\u4e5f\u5b58\u5728\u81ea\u76f8\u4f3c\u6027\u3002\u53f3\u8fb9\u662f\u4e00\u4e2a\u6570\u5b66\u751f\u6210\u7684\uff0c\u5b8c\u5168\u81ea\u76f8\u4f3c\u7684\u8568\u7c7b\u56fe\u50cf\uff0c\u4e0e\u81ea\u7136\u8568\u7c7b\u6709\u660e\u663e\u7684\u76f8\u4f3c\u4e4b\u5904\u3002\u5176\u4ed6\u690d\u7269\uff0c\u5982\u7f57\u9a6c\u82b1\u6930\u83dc\uff0c\u8868\u73b0\u51fa\u5f3a\u70c8\u7684\u81ea\u76f8\u4f3c\u6027\u3002<\/span><\/p>\n


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\u97f3\u4e50\u4e16\u754c<\/span><\/strong><\/span><\/h3>\n