每次提到数学这个词，大家能够想到的就是初等代数，平面几何，组合运算，微积分，线性代数，概率论等方向。但在整个数学领域（Earth of Math）上，还有很多更有意思的领域和研究方向，包括数论，几何，拓扑，分形几何，分析，概率统计，博弈论，代数等诸多方向，每一个方向都有很多优秀的数学家在从事相关研究。
当年在数学系的时候，所研究的方向是分形几何（Fractal Geometry）和复动力系统（Complex Dynamics），位于 Earth of Math 的左侧，称之为分形湖泊（Fractal Lakes）。所谓分形，其实是一个粗糙或者零碎的集合形状，可以分成多个部分，且每个部分放大之后与整体有某种相似性，即具有自相似性的性质。而动力系统则是基于某种固定的规则，描述一个空间内的所有点随时间的变化情况，例如钟摆的晃动，水的流动，湖泊里面鱼类的数量。备注：动力系统并不是指汽车的动力系统和发动机引擎，这两者毫无关系。
针对不同的定义域，函数的迭代有着完全不同的研究方法。当时的研究方向是复动力系统（Complex Dynamical Systems）。复动力系统理论的研究始于 1920 年，当时是由数学家 Fatou 和 Julia 研究的，因此复动力系统中的两个重要的集合就是以 Fatou 和 Julia 来命名的，分别称之为 Fatou set 和 Julia Set。随着计算机技术的演进，在上世纪八十年代这些集合可以通过计算机进行可视化，分形几何和复动力系统理论开始蓬勃发展起来。在与双曲几何、分形几何、现代分析学和混沌学等学科发展相互促进的同时，围绕双曲猜想以及 Mandelbrot 集合的研究工作，成为当今复动力系统的研究热点。
举个例子，函数 （）的 Julia 集合的动图如下：
之前在学校研究动力系统的时候，收集过一些书籍，在此列举给大家，希望对初学者有一定的帮助。One Dimensional Real and Complex Dynamics（实与复动力系统）需要学习的资料如下：
如果是在 NUS 的 IMS（Institute for Mathematical Sciences）举办的学术会议，一般来说只要是在校的研究生都是可以参加的。记得当时参加的第一个学术会议是关于 PDE 的，标题叫做 Hyperbolic Conservation Laws and Kinetic Equations：Theory, Computation, and Applications（1 November – 19 December 2010）。笔者去听这个系列讲座是因为在 2010 年选择了一门 PDE 的研究生课程，而这个讲座则是作为课程的一部分。
笔者参与的另外一个学术会议则是关于动力系统的，标题叫做 Workshop on Non-uniformly Hyperbolic and Neural One-dimensional Dynamics（23 – 27 April 2012），主要是关于非一致双曲动力系统方向的研讨会。笔者记得当时所修的课程应该只有概率论（Probability II）一门课，因此上课的任务不算很重。参会的时间恰好是学期快结束的时候，科研的任务也不算特别繁重。因此，积极参与各种学术会议也算是科研的其中一部分，一来通过参会可以了解当前的学术研究情况，二来可以认识学术界的各种人士，也算是扩大学术交流圈子的好机会。
从数学家族谱（Mathematics Genealogy Project）上面可以看到：Gian-Carlo Rota 的导师是 Jacob T. Schwartz，Rota 于 1956 年在耶鲁大学获得数学博士学位，其博士论文的题目是 Extension Theory of Differential Operators。
在 1997 年，Rota 发表了两篇关于人生经验和忠告的文章，分别是 “Ten Lessons I wish I Had Been Taught” 和 “Ten Lessons for the Survival of a Mathematics Department“。下面就来逐一分享这两篇文章中的一些观点。
Ten Lessons I wish I Had Been Taught
（a）每次讲座都应该只有一个重点。（Every lecture should make only one main point.）
Every lecture should state one main point and repeat it over and over, like a theme with variations. An audience is like a herd of cows, moving slowly in the direction they are being driven towards. If we make one point, we have a good chance that the audience will take the right direction; if we make several points, then the cows will scatter all over the field. The audience will lose interest and everyone will go back to the thoughts they interrupted in order to come to our lecture.
（b）不要超时。（Never run overtime.）
Running overtime is the one unforgivable error a lecturer can make. After fifty minutes (one micro-century as von Neumann used to say) everybody’s attention will turn elsewhere even if we are trying to prove the Riemann hypothesis. One minute overtime can destroy the best of lectures.
（c）提及听众的成果。（Relate to your audience.）
As you enter the lecture hall, try to spot someone in the audience with whose work you have some familiarity. Quickly rearrange your presentation so as to manage to mention some of that person’s work. In this way, you will guarantee that at least one person will follow with rapt attention, and you will make a friend to boot.
Everyone in the audience has come to listen to your lecture with the secret hope of hearing their work mentioned.
（d）给听众一些值得回忆的东西。（Give them something to take home.）
Most of the time they admit that they have forgotten the subject of the course and all the mathematics I thought I had taught them. However, they will gladly recall some joke, some anecdote, some quirk, some side remark, or some mistake I made.
（a）开讲前保持黑板干净（Make sure the blackboard is spotless.）
By starting with a spotless blackboard you will subtly convey the impression that the lecture they are about to hear is equally spotless.
（b）从黑板的左上角开始书写（Start writing on the top left-hand corner.）
What we write on the blackboard should correspond to what we want an attentive listener to take down in his notebook. It is preferable to write slowly and in a large handwriting, with no abbreviations.
When slides are used instead of the blackboard, the speaker should spend some time explaining each slide, preferably by adding sentences that are inessential, repetitive, or superfluous, so as to allow any member of the audience time to copy our slide. We all fall prey to the illusion that a listener will find the time to read the copy of the slides we hand them after the lecture. This is wishful thinking.
多次公布同样的结果（Publish the Same Result Several Times）
The mathematical community is split into small groups, each one with its own customs, notation, and terminology. It may soon be indispensable to present the same result in several versions, each one accessible to a specific group; the price one might have to pay otherwise is to have our work rediscovered by someone who uses a different language and notation and who will rightly claim it as his own.
说明性的工作反而更有可能被记得（You Are More Likely to Be Remembered by Your Expository Work）
When we think of Hilbert, we think of a few of his great theorems, like his basis theorem. But Hilbert’s name is more often remembered for his work in number theory, his Zahlbericht, his book Foundations of Geometry, and for his text on integral equations.
每个数学家只有少数的招数（Every Mathematician Has Only a Few Tricks）
You admire Erdös’s contributions to mathematics as much as I do, and I felt annoyed when the older mathematician flatly and definitively stated that all of Erdös’s work could be “reduced” to a few tricks which Erdös repeatedly relied on in his proofs. What the number theorist did not realize is that other mathematicians, even the very best, also rely on a few tricks which they use over and over. But on reading the proofs of Hilbert’s striking and deep theorems in invariant theory, it was surprising to verify that Hilbert’s proofs relied on the same few tricks. Even Hilbert had only a few tricks!
别害怕犯错（Do Not Worry about Your Mistakes）
There are two kinds of mistakes. There are fatal mistakes that destroy a theory, but there are also contingent ones, which are useful in testing the stability of a theory.
使用费曼的方法（Use the Feynman Method）
You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say, “How did he do it? He must be a genius!”
不要吝啬你的赞美（Give Lavish Acknowledgments）
I have always felt miffed after reading a paper in which I felt I was not being given proper credit, and it is safe to conjecture that the same happens to everyone else.
写好摘要（Write Informative Introductions）
If we wish our paper to be read, we had better provide our prospective readers with strong motivation to do so. A lengthy introduction, summarizing the history of the subject, giving everybody his due, and perhaps enticingly outlining the content of the paper in a discursive manner, will go some of the way towards getting us a couple of readers.
为老年做好心理准备（Be Prepared for Old Age）
You must realize that after reaching a certain age you are no longer viewed as a person. You become an institution, and you are treated the way institutions are treated. You are expected to behave like a piece of period furniture, an architectural landmark, or an incunabulum.
Ten Lessons for the Survival of a Mathematics Department
不要在其他系讲自己系同事的坏话（Never wash your dirty linen in public）
Departments of a university are like sovereign states: there is no such thing as charity towards one another.
别越级打报告（Never go above the head of your department）
Your letter will be viewed as evidence of disunity in the rank and file of mathematicians. Human nature being what it is, such a dean or provost is likely to remember an unsolicited letter at budget time, and not very kindly at that.
不要进行领域评价（Never Compare Fields）
You are not alone in believing that your own field is better and more promising than those of your colleagues. We all believe the same about our own fields. But our beliefs cancel each other out. Better keep your mouth shut rather than make yourself obnoxious. And remember, when talking to outsiders, have nothing but praise for your colleagues in all fields, even for those in combinatorics. All public shows of disunity are ultimately harmful to the well-being of mathematics.
别看不起别人使用的数学（Remember that the grocery bill is a piece of mathematics too）
The grocery bill, a computer program, and class field theory are three instances of mathematics. Your opinion that some instances may be better than others is most effectively verbalized when you are asked to vote on a tenure decision. At other times, a careless statement of relative values is more likely to turn potential friends of mathematics into enemies of our field. Believe me, we are going to need all the friends we can get.
善待擅长教学的老师（Do not look down on good teachers）
Mathematics is the greatest undertaking of mankind. All mathematicians know this. Yet many people do not share this view. Consequently, mathematics is not as self-supporting a profession in our society as the exercise of poetry was in medieval Ireland. Most of our income will have to come from teaching, and the more students we teach, the more of our friends we can appoint to our department. Those few colleagues who are successful at teaching undergraduate courses should earn our thanks as well as our respect. It is counterproductive to turn up our noses at those who bring home the dough.
学会推销自己的数学成果（Write expository papers）
When I was in graduate school, one of my teachers told me, “When you write a research paper, you are afraid that your result might already be known; but when you write an expository paper, you discover that nothing is known.”
It is not enough for you (or anyone) to have a good product to sell; you must package it right and advertise it properly. Otherwise you will go out of business.
When an engineer knocks at your door with a mathematical question, you should not try to get rid of him or her as quickly as possible.
不要把提问者拒之门外（Do not show your questioners to the door）
What the engineer wants is to be treated with respect and consideration, like the human being he is, and most of all to be listened to with rapt attention. If you do this, he will be likely to hit upon a clever new idea as he explains the problem to you, and you will get some of the credit.
Listening to engineers and other scientists is our duty. You may even learn some interesting new mathematics while doing so.
联合阵线（View the mathematical community as a United Front）
Grade school teachers, high school teachers, administrators and lobbyists are as much mathematicians as you or Hilbert. It is not up to us to make invidious distinctions. They contribute to the well-being of mathematics as much as or more than you or other mathematicians. They are right in feeling left out by snobbish research mathematicians who do not know on which side their bread is buttered. It is our best interest, as well as the interest of justice, to treat all who deal with mathematics in whatever way as equals. By being united we will increase the probability of our survival.
Flakiness is nowadays creeping into the sciences like a virus through a computer, and it may be the present threat to our civilization. Mathematics can save the world from the invasion of the flakes by unmasking them and by contributing some hard thinking. You and I know that mathematics is not and will never be flaky, by definition.
This is the biggest chance we have had in a long while to make a lasting contribution to the well-being of Science. Let us not botch it as we did with the few other chances we have had in the past.
善待所有人（Learn when to withdraw）
Let me confess to you something I have told very few others (after all, this message will not get around much): I have written some of the papers I like the most while hiding in a closet. When the going gets rough, we have recourse to a way of salvation that is not available to ordinary mortals: we have that Mighty Fortress that is our Mathematics. This is what makes us mathematicians into very special people. The danger is envy from the rest of the world.
When you meet someone who does not know how to differentiate and integrate, be kind, gentle, understanding. Remember, there are lots of people like that out there, and if we are not careful, they will do away with us, as has happened many times before in history to other Very Special People.
Rota, Gian-Carlo. “Ten lessons I wish I had been taught.” Indiscrete thoughts. Birkhäuser, Boston, MA, 1997. 195-203.
Rota, Gian-Carlo. “Ten Lessons for the Survival of a Mathematics Department.” Indiscrete Thoughts. Birkhäuser, Boston, MA, 1997. 204-208.
从课表上面来看，基本上可以确定几个结论。首先，数学专业作为基础学科，其特点就是理论知识偏多，而学习到的技能偏少，毕竟所学的内容都是理论型，培养的学生都是理论型选手。因此直接导致的结果就是数学系的学生掌握了一堆理论，但是却没有办法把它们直接转化成生产力。在实战中，总不能就靠一门 C++ 来谋求工作吧。其次，既然数学系传授给学生的实用的技能偏少，那么数学系的学生在需要转行的话，就肯定要补充新的技能。在从理论派走向实战派的过程中，不仅要找好自己的前进方向，还需要花费一定的时间和精力去转行。在这里需要澄清一点，转行并不是轻轻松松，而是需要花费时间，勇气和精力的。
In this paper, we obtain the explicit value of the Hausdorff dimension of the graphs of the classical Weierstrass functions, by proving absolute continuity of the SRB measures of the associated solenoidal attractors.
Weierstrass (1872) and Hardy (1916) were interested in because they are concrete examples of continuous but nowhere differentiable functions.
Remark 1The graph of tends to be a “fractal object” because is self-similar in the sense that
We will come back to this point later.
Remark 2 is a -function for all . In fact, for all , we have
whenever , i.e., .
The study of the graphs of as fractal sets started with the work of Besicovitch-Ursell in 1937.
Remark 3The Hausdorff dimension of the graph of a -function is
Indeed, for each , the Hölder continuity condition
leads us to the “natural cover” of by the family of rectangles given by
Nevertheless, a direct calculation with the family does not give us an appropriate bound on . In fact, since for each , we have
for . Because is arbitrary, we deduce that . Of course, this bound is certainly suboptimal for (because we know that anyway).Fortunately, we can refine the covering by taking into account that each rectangle tends to be more vertical than horizontal (i.e., its height is usually larger than its width ). More precisely, we can divide each rectangle into squares, say
such that every square has diameter . In this way, we obtain a covering of such that
for . Since is arbitrary, we conclude the desired bound
A long-standing conjecture about the fractal geometry of is:
Conjecture (Mandelbrot 1977): The Hausdorff dimension of the graph of is
Remark 4In view of remarks 2 and 3, the whole point of Mandelbrot’s conjecture is to establish the lower bound
Remark 5The analog of Mandelbrot conjecture for the box and packing dimensions is known to be true: see, e.g., these papers here and here).
In a recent paper (see here), Shen proved the following result:
Theorem 1 (Shen)For any integer and for all , the Mandelbrot conjecture is true, i.e.,
Remark 6The techniques employed by Shen also allow him to show that given a -periodic, non-constant, function, and given integer, there exists such that
for all .
Remark 7A previous important result towards Mandelbrot’s conjecture was obtained by Barańsky-Barány-Romanowska (in 2014): they proved that for all integer, there exists such that
for all .
The remainder of this post is dedicated to give some ideas of Shen’s proof of Theorem1 by discussing the particular case when and is large.
2. Ledrappier’s dynamical approach
If is an integer, then the self-similar function (cf. Remark 1) is also -periodic, i.e., for all . In particular, if is an integer, then is an invariant repeller for the endomorphism given by
This dynamical characterization of led Ledrappier to the following criterion for the validity of Mandelbrot’s conjecture when is an integer.
Denote by the alphabet . The unstable manifolds of through have slopes of the form
where , , , and
In this context, the push-forwards of the Bernoulli measure on (induced by the discrete measure assigning weight to each letter of the alphabet ) play the role of conditional measures along vertical fibers of the unique Sinai-Ruelle-Bowen (SRB) measure of the expanding endomorphism ,
where and . In plain terms, this means that
where is the unique -invariant probability measure which is absolutely continuous along unstable manifolds (see Tsujii’s paper).
As it was shown by Ledrappier in 1992, the fractal geometry of the conditional measures have important consequences for the fractal geometry of the graph :
Theorem 2 (Ledrappier)Suppose that for Lebesgue almost every the conditional measures have dimension , i.e.,
Then, the graph has Hausdorff dimension
Remark 8Very roughly speaking, the proof of Ledrappier theorem goes as follows. By Remark 4, it suffices to prove that . By Frostman lemma, we need to construct a Borel measure supported on such that
where . Finally, the main point is that the assumptions in Ledrappier theorem allow to prove that the measure given by the lift to of the Lebesgue measure on via the map satisfies
An interesting consequence of Ledrappier theorem and the equation 1 is the following criterion for Mandelbrot’s conjecture:
Corollary 3If is absolutely continuous with respect to the Lebesgue measure , then
Proof: By (1), the absolute continuity of implies that is absolutely continuous with respect to for Lebesgue almost every .
Since for almost every implies that for almost every , the desired corollary now follows from Ledrappier’s theorem.
3. Tsujii’s theorem
The relevance of Corollary 3 is explained by the fact that Tsujii found an explicittransversality condition implying the absolute continuity of .
More precisely, Tsujii firstly introduced the following definition:
Given , and , we say that two infinite words are -transverse at if either
Given , , and , we say that two finite words are -transverse at if , are -transverse at for all pairs of infinite words ; otherwise, we say that and are-tangent at ;
Next, Tsujii proves the following result:
Theorem 5 (Tsujii)If there exists integer such that , then
Remark 9Intuitively, Tsujii’s theorem says the following. The transversality condition implies that the majority of strong unstable manifolds are mutually transverse, so that they almost fill a small neighborhood of some point (see the figure below extracted from this paper of Tsujii). Since the SRB measure is absolutely continuous along strong unstable manifolds, the fact that the ‘s almost fill implies that becomes “comparable” to the restriction of the Lebesgue measure to .
Remark 10In this setting, Barańsky-Barány-Romanowska obtained their main result by showing that, for adequate choices of the parameters and , one has . Indeed, once we know that , since , they can apply Tsujii’s theorem and Ledrappier’s theorem (or rather Corollary 3) to derive the validity of Mandelbrot’s conjecture for certain parameters and .
For the sake of exposition, we will give just a flavor of the proof of Theorem 1 by sketching the derivation of the following result:
Proposition 6Let . If and is sufficiently large, then
In particular, by Corollary 3 and Tsujii’s theorem, if and is sufficiently large, then Mandelbrot’s conjecture is valid, i.e.,
Remark 11The proof of Theorem 1 in full generality (i.e., for integer and ) requires the introduction of a modified version of Tsujii’s transversality condition: roughly speaking, Shen defines a function (inspired from Peter-Paul inequality) and he proves
(a) a variant of Proposition 6: if integer and , then for some integer ;
(b) a variant of Tsujii’s theorem: if for some integer , then .
See Sections 2, 3, 4 and 5 of Shen’s paper for more details.
We start the (sketch of) proof of Proposition 6 by recalling that the slopes of unstable manifolds are given by
for , , so that
Remark 12Since , the series defining converges faster than the series defining .
By studying the first term of the expansion of and (while treating the remaining terms as a “small error term”), it is possible to show that if , then
(cf. Lemma 3.2 in Shen’s paper).
Using these estimates, we can find an upper bound for as follows. Take with , and let be such that distinct elements listed in such a way that
Chapter 5. Ergodic Properties and Invariant Measures.
1. Ergodicity, Attractors and Bowen-Ruelle-Sinai Measures.
A distortion result for unimodal maps with recurrence
Given a unimodal map , we say that an interval is symmetric if where is so that and if . Furthermore, for each symmetric interval let
for let be the minimal positive integer with and let
We call the Poincare map or transfer map to and the transfer time of to . The distortion result states that one can fined a sequence of symmetric neighbourhoods of the turning point such that the Poincare maps to these intervals have a distortion which is universally bounded:
Theorem 1.1. Let be a unimodal map with one non-flat critical point with negative Schwarzian derivative and without attracting periodic points. Then there exists and a sequence os symmetric intervals around the turning point which shrink to such that contains a scaled neighbourhood of and such that the following properties hold.
1. The transfer time on each component of is constant.
2. Let be a component of the domain of the transfer map to which does not intersect . Then there exists an interval such that is monotone, and . Here is the transfer time on , i.e., .
Corollary. There exists such that
1. for each component of not intersecting , the transfer map to sends diffeomorphically onto and the distortion of on is bounded from above by .
2. on each component of which is contained in , the map can be written as where the distortion of is universally bounded by .
As before, we say that is ergodic with respect to the Lebesgue measure if each completely invariant set (Here is called completely invariant if ) has either zero or full Lebesgue measure. An alternative way to define this notation of ergodicity goes as follows: is ergodic if for each two forward invariant sets and such that has Lebesgue measure zero, at most one of these sets has positive Lebesgue measure. (Here is called forward invariant if .)
Theorem 1.2 (Blokh and Lyubich). Let be a unimodal map with a non-flat critical point with negative Schwarzian derivative and without an attracting periodic points. Then is ergodic with respect to the Lebesgue measure.
Theorem 1.3. Let be a unimodal map with a non-flat critical point with negative Schwarzian derivative. Then has a unique attractor , for almost all and either consists of intervals or has Lebesgue measure zero. Furthermore, one has the following:
1. if has an attracting periodic orbit then is this periodic orbit;
2. if is infinitely often renormalizable then is the attracting Cantor set (in which case it is called a solenoidal attractor);
3. is only finitely often renormalizable then either
(a) coincides with the union of the transitive intervals, or,
(b) is a Cantor set and equal to .
If is not a minimal set then is as in case 3.a and each closed forward invariant set either contains intervals or has Lebesgue measure zero. Moreover, if does not contain intervals, then has Lebesgue measure zero.
Remark. Here a forward invariant set is said to be minimal if the closure of the forward orbit of a point in is always equal to . The attractors in case 3.b is called a non-renormalizable attracting Cantor set, or absorbing Cantor attractor or wild Cantor attractor. Such an attractor really exists which is proven in [BKNS], and one has the following strange phenomenon: there exist many orbits which are dense in some finite union of intervals and yet almost all points tend to a minimal Cantor set of Lebesgue measure zero (this Cantor set is ). The Fibonacci map is non-renormalizable and for which is a Cantor set. It was shown by Lyubich and Milnor that the quadratic map with this dynamics has no absorbing Cantor attractors. More generally, Jakobson and Swiatek proved that maps with negative Schwarzian derivative and which are close to the map do not have such Cantor attractors. Moreover, Lyubich has shown that these absorbing Cantor attractors can not exist if the critical point is quadratic. However, Bruin, Keller, Nowicki and Van Strien showed that the absorbing Cantor attractors exist for Fibonacci maps when the critical order is sufficiently large enough.
Theorem (Lyubich). If is unimodal, has a quadratic critical point, has negative Schwarzian derivative and has no periodic attractors, then each closed forward invariant set which has positive Lebesgue measure contains an interval.
The next result, which is due to Martens (1990), shows that if these absorbing Cantor attractors do not exist then one has a lot of ‘expansion’. Let not be in the pre orbit of and define to be the maximal interval on which is monotone. Let and be the components of and define be the minimum of the length of and .
Theorem 1.4 (Martens). Let be a unimodal map with negative Schwarian derivative whose critical point is non-flat. Then the following three properties are equivalent.
1. has no absorbing Cantor attractor;
2. for almost all ;
3. there exist neighbourhoods of with such that for almost every there exists a positive integer and an interval neighbourhood of such that is monotone, and .
Excerpt From: Falconer, Kenneth. “Fractals: A Very Short Introduction (Very Short Introductions).” iBooks.
A little history
Geometry, with its highly visual and practical nature, is one of the oldest branches of mathematics. Its development through the ages has paralleled its increasingly sophisticated applications. Construction, crafts, and astronomy practised by ancient civilizations led to the need to record and analyse the shapes, sizes, and positions of objects. Notions of angles, areas, and volumes developed with the need for surveying and building. Two shapes were especially important: the straight line and the circle, which occurred naturally in many settings but also underlay the design of many artefacts. As well as fulfilling practical needs, philosophers were motivated by aesthetic aspects of geometry and sought simplicity in geometric structures and their applications. This reached its peak with the Greek School, notably with Plato (c 428–348 BC) and Euclid (c 325–265 BC), for whom constructions using a straight edge and compass, corresponding to line and circle, were the essence of geometric perfection.
As time progressed, ways were found to express and solve geometrical problems using algebra. A major advance was the introduction by René Descartes (1596–1650) of the Cartesian coordinate system which enabled shapes to be expressed concisely in terms of equations. This was a necessary precursor to the calculus, developed independently by Isaac Newton (1642–1727) and Gottfried Leibniz (1646–1714) in the late 17th century. The calculus provided a mathematical procedure for finding tangent lines that touched smooth curves as well as a method for computing areas and volumes of an enormous variety of geometrical objects. Alongside this, more sophisticated geometric figures were being observed in nature and explained mathematically. For example, using Tycho Brahe’s observations, Johannes Kepler proposed that planets moved around ellipses, and this was substantiated as a mathematical consequence of Newton’s laws of motion and gravitation.
The tools and methods were now available for tremendous advances in mathematics and the sciences. All manner of geometrical shapes could be analysed. Using the laws of motion together with the calculus, one could calculate the trajectories of projectiles, the motion of celestial bodies, and, using differential equations which developed from the calculus, more complex motions such as fluid flows. Although the calculus underlay Graph of a Brownian process8I to think of all these applications, its foundations remained intuitive rather than rigorous until the 19th century when a number of leading mathematicians including Augustin Cauchy (1789–1857), Bernhard Riemann (1826–66), and Karl Weierstrass (1815–97) formalized the notions of continuity and limits. In particular, they developed a precise definition for a curve to be ‘differentiable’, that is for there to be a tangent line touching the curve at a point. Many mathematicians worked on the assumption that all curves worthy of attention were nice and smooth so had tangents at all their points, enabling application of the calculus and its many consequences. It was a surprise when, in 1872, Karl Weierstrass constructed a ‘curve’ that was so irregular that at no point at all was it possible to draw a tangent line. The Weierstrass graph might be regarded as the first formally defined fractal, and indeed it has been shown to have fractal dimension greater than 1.
In 1883, the German Georg Cantor (1845–1918) wrote a paper introducing the middle-third Cantor set, obtained by repeatedly removing the middle thirds of intervals (see Figure 44). The Cantor set is perhaps the most basic self-similar fractal, made up of 2 scale copies of itself, although of more immediate interest to Cantor were its topological and set theoretic properties, such as it being totally disconnected, rather than its geometry. (Several other mathematicians studied sets of a similar form around the same time, including the Oxford mathematician Henry Smith (1826–83) in an article in 1874.) In 1904, Helge von Koch introduced his curve, as a simpler construction than Weierstrass’s example of a curve without any tangents. Then, in 1915, the Polish mathematician Wacław Sierpiński (1882–1969) introduced his triangle and, in 1916, the Sierpiński carpet. His main interest in the carpet was that it was a ‘universal’ set, in that it contains continuously deformed copies of all sets of ‘topological dimension’ 1. Although such objects have in recent years become the best-known fractals, at the time properties such as self-similarity were almost irrelevant, their main use being to provide specific examples or counter-examples in topology and calculus.
It was in 1918 that Felix Hausdorff proposed a natural way of ‘measuring’ the middle-third Cantor set and related sets, utilizing a general approach due to Constantin Carathéodory (1873–1950). Hausdorff showed that the middle-third Cantor set had dimension of log2/log3 = 0.631, and also found the dimensions of other self-similar sets. This was the first occurrence of an explicit notion of fractional dimension. Now termed ‘Hausdorff dimension’, his definition of dimension is the one most commonly used by mathematicians today. (Hausdorff, who did foundational work in several other areas of mathematics and philosophy, was a German Jew who tragically committed suicide in 1942 to avoid being sent to a concentration camp.) Box-dimension, which in many ways is rather simpler than Hausdorff dimension, appeared in a 1928 paper by Georges Bouligand (1889–1979), though the idea underlying an equivalent definition had been mentioned rather earlier by Hermann Minkowski (1864–1909), a Polish mathematician known especially for his work on relativity.
For many years, few mathematicians were very interested in fractional dimensions, with highly irregular sets continuing to be regarded as pathological curiosities. One notable exception was Abram Besicovitch (1891–1970), a Russian mathematician who held a professorship in Cambridge for many years. He, along with a few pupils, investigated the dimension of a range of fractals as well as investigating some of their geometric properties.
Excerpt From: Falconer, Kenneth. “Fractals: A Very Short Introduction (Very Short Introductions).” iBooks.