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In [[mathematics]], a '''quasicircle''' is a [[Jordan curve]] in the [[complex plane]] that is the image of a [[circle]] under a [[quasiconformal mapping]] of the plane onto itself. Originally introduced independently by {{harvtxt|Pfluger|1961}} and {{harvtxt|Tienari|1962}}, in the older literature (in German) they were referred to as '''quasiconformal curves''', a terminology which also applied to [[arc (geometry)|arc]]s.<ref>{{harvnb|Lehto|Virtanen|1973}}</ref><ref>{{harvnb|Lehto|1983|p=49}}</ref> In [[complex analysis]] and [[geometric function theory]], quasicircles play a fundamental role in the description of the  [[universal Teichmüller space]], through [[quasisymmetric map|quasisymmetric homeomorphism]]s of the circle. Quasicircles also play an important role in [[complex dynamical system]]s.


==Definitions==
A quasicircle is defined as the image of a circle under a [[quasiconformal mapping]] of the [[extended complex plane]]. It is called a ''K''-quasicircle if the quasiconformal mapping has dilatation ''K''. The definition of quasicircle generalizes the characterization of a [[Jordan curve]] as the image of a circle under a homeomorphism of the plane. In particular a quasicircle is a Jordan curve. The interior of a quasicircle is called a ''quasidisk''.<ref>{{harvnb|Lehto|1987|p=38}}</ref>


As shown in {{harvtxt|Lehto|Virtanen|1973}}, where the older term "quasiconformal curve" is used, if a Jordan curve is the image of a circle under a quasiconformal map in a neighbourhood of the curve, then it is also the image of a circle under a quasiconformal mapping of the extended plane and thus a quasicircle. The same is true for "quasiconformal arcs" which can be defined as quasiconformal images of a circular arc either in an open set or equivalently in the extended plane.<ref>{{harvnb|Lehto|Virtanen|1973|pp=97–98}}</ref>
Four or five years ago, a reader of some of my columns bought the domain name jamesaltucher.com and gave it to me as a birthday gift. It was a total surprise to me. [http://Search.About.com/?q=I+didn%27t I didn't] even know the reader. I hope one day we meet.<br>Two years ago a friend of mine, Tim Sykes, insisted I had to have a blog. He set it up for me. He even wrote the "About Me". I didn't want a blog. I had nothing to say. But about 6 or 7 months ago I decided I wanted to take this blog seriously. I kept putting off changing the "About Me" which was no longer really about me and maybe never was.<br>A few weeks ago I did a chapter in one of the books in Seth Godin's "The Domino Project". The book is out and called "No Idling". Mohit Pawar organized it (here's Mohit's blog) and sent me a bunch of questions recently. It's intended to be an interview on his blog but I hope Mohit forgives me because I want to use it as my new "About Me" also.<br>1. You are a trader, investor, writer, and entrepreneur? Which of these roles you enjoy the most and why?<br>When I first moved to New York City in 1994 I wanted to be everything to everyone. I had spent the six years prior to that writing a bunch of unpublished novels and unpublished short stories. I must've sent out 100s of stories to literary journals. I got form rejections from every publisher, journal, and agent I sent my novels and stories to.<br>Now, in 1994, everything was possible. The money was in NYC. Media was here. I lived in my 10�10 room and pulled suits out of a garbage bag every morning but it didn't matter...the internet was revving up and I knew how to build a website. One of the few in the city. My sister warned me though: nobody here is your friend. Everybody wants something<br>
 
And I wanted something. I wanted the fleeting feelings of success, for the first time ever, in order to feel better about myself. I wanted a girl next to me. I wanted to build and sell companies and finally prove to everyone I was the smartest. I wanted to do a TV show. I wanted to write books<br>
==Geometric characterizations==
But everything involved having a master. Clients. Employers. Investors. Publishers. The market (the deadliest master of all). Employees. I was a slave to everyone for so many years. And the more shackles I had on, the lonelier I got<br>
{{harvtxt|Ahlfors|1963}} gave a geometric characterization of quasicircles as those [[Jordan curve]]s for which the absolute value of the [[cross-ratio]] of any four points, taken in cyclic order, is bounded below by a positive constant.
(Me in the Fortress of Solitude<br>
 
Much of the time, even when I had those moments of success, I didn't know how to turn it into a better life. I felt ugly and then later, I felt stupid when I would let the success dribble away down the sink<br>
Ahlfors also proved that quasicircles can be characterized in terms of a reverse triangle inequality for three points: there should be a constant ''C'' such that if two points ''z''<sub>1</sub> and ''z''<sub>2</sub> are chosen on the curve and ''z''<sub>3</sub> lies on the shorter of the resulting arcs, then<ref name=autogenerated1>{{harvnb|Carleson|Gamelin|1993|p=102}}</ref>
I love writing because every now and then that ugliness turns into honesty. When I write, I'm only a slave to myself. When I do all of those other things you ask about, I'm a slave to everyone else<br>
 
Some links<br>
:<math> |z_1-z_3| + |z_2-z_3| \le C |z_1-z_2|.</math>
33 Unusual Tips to Being a Better Write<br>
 
"The Tooth<br>
This property is also called ''bounded turning''<ref>{{harvnb|Lehto|Virtanen|pp=100–102}}</ref> or the ''arc condition''.<ref>{{harvnb|Lehto|1983|p=45}}</ref>
(one of my favorite posts on my blog<br><br>
 
2. What inspires you to get up and start working/writing every day<br>
For Jordan curves in the extended plane passing through ∞, {{harvtxt|Ahlfors|1966}} gave a simpler necessary and sufficient condition to be a quasicircle.<ref>{{harvnb|Ahlfors|1966|p=81}}</ref><ref>{{harvnb|Lehto|1983|pp=48–49}}</ref>  There is a constant ''C'' > 0 such that if
The other day I had breakfast with a fascinating guy who had just sold a piece of his fund of funds. He told me what "fracking" was and how the US was going to be a major oil player again. We spoke for two hours about a wide range of topics, including what happens when we can finally implant a google chip in our brains<br>
''z''<sub>1</sub>, ''z''<sub>2</sub> are any points on the curve and  ''z''<sub>3</sub> lies on the segment between them, then
After that I had to go onto NPR because I firmly believe that in one important respect we are degenerating as a country - we are graduating a generation of indentured servants who will spend 50 years or more paying down their student debt rather than starting companies and curing cancer. So maybe I made a difference<br>
 
Then I had lunch with a guy I hadn't seen in ten years. In those ten years he had gone to jail and now I was [http://www.encyclopedia.com/searchresults.aspx?q=finally finally] taking the time to forgive him for something he never did to me. I felt bad I hadn't helped him when he was at his  [http://www.pcs-systems.co.uk/Images/celinebag.aspx http://www.pcs-systems.co.uk/Images/celinebag.aspx] low point. Then I came home and watched my kid play clarinet at her school. Then I read until I fell asleep. Today I did nothing but write. Both days inspired me<br>
:<math>\displaystyle{\left|z_3 -{z_1+z_2\over 2}\right|\le C |z_1-z_2|.}</math>
It also inspires me that I'm being asked these questions. Whenever anyone asks me to do anything I'm infinitely grateful. Why me? I feel lucky. I like it when someone cares what I think. I'll write and do things as long as anyone cares. I honestly probably wouldn't write if nobody cared. I don't have enough humility for that, I'm ashamed to admit<br><br>
 
3. Your new book "How to be the luckiest person alive" has just come out. What is it about<br>
These metric characterizations imply that an arc or closed curve is quasiconformal whenever it arises as the image of an interval or the circle under a [[Lipschitz continuity|bi-Lipschitz map]] ''f'', i.e. satisfying
When I was a kid I thought I needed certain things: a college education from a great school, a great home, a lot of money, someone who would love me with ease. I wanted people to think I was smart. I wanted people to think I was even special.  And as I grew older more and more goals got added to the list: a high chess rating, a published book, perfect weather, good friends,  respect in various fields, etc. I lied to myself that I needed these things to be happy. The world was going to work hard to give me these things, I thought. But it turned out the world owed me no favors<br>
 
And gradually, over time, I lost everything I had ever gained. Several times.  I've paced at night so many times wondering what the hell was I going to do next or trying not to care. The book is about regaining your sanity, regaining your happiness, finding luck in all the little pockets of life that people forget about. It's about turning away from the religion you've been hypnotized into believing into the religion you can find inside yourself every moment of the day<br><br>
:<math> C_1|s-t|\le |f(s)-f(t)| \le C_2 |s-t|</math>
[Note: in a few days I'm going to do a post on self-publishing and also how to get the ebook for free. The link above is to the paperback. Kindle should be ready soon also.<br>
 
Related link: Why I Write Books Even Though I've Lost Money On Every Book I've Ever Writte<br>
for positive constants ''C''<sub>''i''</sub>.<ref>{{harvnb|Lehto|Virtanen|pp=104–105}}</ref>
4. Is it possible to accelerate success? If yes, how<br><br><br>
 
Yes, and it's the only way I know actually to achieve success. Its by following the Daily Practice I outline in this post:<br>
==Quasicircles and quasisymmetric homeomorphisms==
It's the only way I know to exercise every muscle from the inside of you to the outside of you. I firmly believe that happiness starts with that practice<br>
If φ is a [[quasisymmetric map|quasisymmetric homeomorphism]] of the circle, then there are conformal maps ''f'' of [''z''| < 1 and ''g'' of  |''z''|>1 into disjoint regions such that the complement of the images of ''f'' and ''g'' is a Jordan curve. The maps ''f'' and ''g'' extend continuously to the circle |''z''| = 1 and the sewing equation
5. You say that discipline, persistence and psychology are important if one has to achieve success. How can one work on improving "psychology" part<br>
 
Success doesn't really mean anything. People want to be happy in a harsh and unforgiving world. It's very difficult. We're so lucky most of us live in countries without major wars. Our kids aren't getting killed by random gunfire. We all have cell phones. We all can communicate with each other on the Internet. We have Google to catalog every piece of information in history! We are so amazingly lucky already<br>
:<math> \varphi= g^{-1}\circ f </math>
How can it be I was so lucky to be born into such a body? In New York City of all places? Just by being born in such a way on this planet was an amazing success<br>
 
So what else is there? The fact is that most of us, including me, have a hard time being happy with such ready-made success. We quickly adapt and want so much more out of life. It's not wars or disease that kill us. It's the minor inconveniences that add up in life. It's the times we feel slighted or betrayed. Or even slightly betrayed. Or overcharged. Or we miss a train. Or it's raining today. Or the dishwasher doesn't work. Or the supermarket doesn't have the food we like. We forget how good the snow tasted when we were kids. Now we want gourmet food at every meal<br>
holds. The image of the circle is a quasicircle.
Taking a step back, doing the Daily Practice I outline in the question above. For me, the results of that bring me happiness. That's success. Today. And hopefully tomorrow<br>
 
6. You advocate not sending kids to college. What if kids grow up and then blame their parents about not letting them get a college education<br>
Conversely, using the [[Riemann mapping theorem]], the conformal maps ''f'' and ''g'' uniformizing the outside of a quasicircle give rise to a quasisymmetric homeomorphism through the above equation.
I went to one of my kid's music recitals yesterday. She was happy to see me. I hugged her afterwards. She played "the star wars theme" on the clarinet. I wish I could've played that for my parents. My other daughter has a dance recital in a few weeks. I tried to give her tips but she laughed at me. I was quite the breakdancer in my youth. The nerdiest breakdancer on the planet. I want to be present for them. To love them. To let them always know that in their own dark moments, they know I will listen to them. I love them. Even when they cry and don't always agree with me. Even when they laugh at me because sometimes I act like a clown<br>
 
Later, if they want to blame me for anything at all then I will still love them. That's my "what if"<br>
The quotient space of the group of quasisymmetric homeomorphisms by the subgroup of [[Möbius transformation]]s provides a model of [[universal Teichmüller space]]. The above correspondence shows that the space of quasicircles can also be taken as a model.<ref>{{harvnb|Lehto|1983}}</ref>
Two posts<br>
 
I want my daughters to be lesbian<br>
==Quasiconformal reflection==
Advice I want to give my daughter<br><br><br>
A quasiconformal reflection in a Jordan curve is an orientation-reversing quasiconformal map of period 2 which switches the inside and the outside of the curve fixing points on the curve. Since the map
7. Four of your favorite posts from The Altucher Confidential<br>
 
As soon as I publish a post I get scared to death. Is it good? Will people re-tweet? Will one part of the audience of this blog like it at the expense of another part of the audience. Will I get Facebook Likes? I have to stop clinging to these things but you also need to respect the audience. I don't know. It's a little bit confusing to me. I don't have the confidence of a real writer yet<br>
:<math>\displaystyle{R_0(z) = {1\over \overline{z}}} </math>
Here are four of my favorites<br>
 
How I screwed Yasser Arafat out of $2mm (and lost another $100mm in the process<br>
provides such a reflection for the unit circle, any quasicircle admits a quasiconformal reflection. {{harvtxt|Ahlfors|1963}} proved that this property characterizes quasicircles.
It's Your Fault<br>
 
I'm Guilty of Torturing Wome<br>
Ahlfors noted that this result can be applied to uniformly bounded [[holomorphic]] [[univalent function]]s ''f''(''z'') on the unit disk ''D''. Let Ω = ''f''(''D''). As Carathéodory had proved using his theory of [[prime end]]s, ''f'' extends continuously to the unit circle if and only if ∂Ω is locally connected, i.e. admits a covering by finitely many compact connected sets of arbitrarily small diameter. The extension to the circle is 1-1 if and only if ∂Ω  has no cut points, i.e. points which when removed from ∂Ω yield a disconnected set. [[Carathéodory's theorem (conformal mapping)|Carathéodory's theorem]] shows that a locally set without cut points is just a Jordan curve and that in precisely this case is the extension of ''f'' to the closed unit disk a homeomorphism.<ref>{{harvnb|Pommerenke|1975|pp=271–281}}</ref> If ''f'' extends to a quasiconformal mapping of the extended complex plane then ∂Ω is by definition a quasicircle. Conversely {{harvtxt|Ahlfors|1963}} observed that if ∂Ω is a quasicircle and ''R''<sub>1</sub> denotes the quasiconformal reflection in ∂Ω then the assignment
The Girl Whose Name Was a Curs<br>
 
Although these three are favorites I really don't post anything unless it's my favorite of that moment<br>
:<math> \displaystyle{f(z)=R_1f R_0(z)}</math>
8. 3 must-read books for aspiring entrepreneurs<br>
 
The key in an entrepreneur book: you want to learn business. You want to learn how to honestly communicate with your customers. You want to stand out<br>
for |''z''| > 1 defines a quasiconformal extension of ''f'' to the extended complex plane.
The Essays of Warren Buffett by Lawrence Cunningha<br>
 
"The Thank you Economy" by Gary Vaynerchu<br>
==Complex dynamical systems==
"Purple cow" by Seth Godi<br>
[[File:Flocke.PNG|thumb|[[Koch snowflake]]]]
9. I love your writing, so do so many others out there. Who are your favorite writers<br>
Quasicircles were known to arise as the [[Julia set]]s of rational maps ''R''(''z''). {{harvtxt|Sullivan|1985}} proved that if the [[Fatou set]] of ''R'' has two components and the action of ''R'' on the Julia set is "hyperbolic", i.e. there are constants ''c'' > 0 and ''A'' > 1 such that
"Jesus's Son" by Denis Johnson is the best collection of short stories ever written. I'm afraid I really don't like his novels though<br>
 
"Tangents" by M. Prado. A beautiful series of graphic stories about relationships<br>
:<math> |\partial_z R^n(z)| \ge c A^n </math>
Other writers: Miranda July, Ariel Leve, Mary Gaitskill, Charles Bukowski, Celine, Sam Lipsyte, William Vollmann, Raymond Carver. Arthur Nersesian. Stephen Dubner<br><br>
 
(Bukowski<br><br><br><br><br><br><br><br><br>
on the Julia set, then the Julia set is a quasicircle.<ref name=autogenerated1 />
Many writers are only really good storytellers. Most writers come out of a cardboard factory MFA system and lack a real voice. A real voice is where every word exposes ten levels of hypocrisy in the world and brings us all the way back to see reality. The writers above have their own voices, their own pains, and their unique ways of expressing those pains. Some of them are funny. Some a little more dark. I wish I could write 1/10 as good as any of them<br><br>
 
10. You are a prolific writer. Do you have any hacks that help you write a lot in little time<br>
There are many examples:<ref>{{harvnb|Carleson|Gamelin|1993|pp=123–126}}</ref><ref>{{harvnb|Rohde|1991}}</ref>
Coffee, plus everything else coffee does for you first thing in the morning<br>
 
Only write about things you either love or hate. But if you hate something, try to find a tiny gem buried in the bag of dirt so you can reach in when nobody is looking and put that gem in your pocket. Stealing a diamond in all the shit around us and then giving it away for free via writing is a nice little hack, Being fearless precisely when you are most scared is the best hack<br><br>
* quadratic polynomials ''R''(''z'') = ''z''<sup>2</sup> + ''c'' with an attracting fixed point
11. I totally get and love your idea about bleeding as a writer, appreciate if you share more with the readers of this blog<br>
* the [[Douady rabbit]] (''c'' = –0.122561 + 0.744862i, where ''c''<sup>3</sup> + 2 ''c''<sup>2</sup> + ''c'' + 1 = 0)
Most people worry about what other people think of them. Most people worry about their health. Most people are at a crossroads and don't know how to take the next step and which road to take it on. Everyone is in a perpetual state of 'where do I put my foot next'. Nobody, including me, can avoid that<br>
* quadratic polynomials ''z''<sup>2</sup> + λ''z'' with |λ| < 1
You and I both need to wash our faces in the morning, brush our teeth, shower, shit, eat, fight the weather, fight the colds that want to attack us if we're not ready. Fight loneliness or learn how to love and appreciate the people who want to love you back. And learn how to forgive and love the people who are even more stupid and cruel than we are. We're afraid to tell each other these things because they are all both disgusting and true<br>
* the [[Koch snowflake]]
You and I both have the same color blood. If I cut my wrist open you can see the color of my blood. You look at it and see that it's the same color as yours. We have something in common. It doesn't have to be shameful. It's just red. Now we're friends. No matter whom you are or where you are from. I didn't have to lie to you to get you to be my friend<br>
 
Related Links<br>
==Quasi-Fuchsian groups==
How to be a Psychic in Ten Easy Lesson<br>
[[Quasi-Fuchsian group]]s are obtained as quasiconformal deformations of [[Fuchsian group]]s. By definition their [[limit set]]s are quasicircles.<ref>{{harvnb|Bers|1961}}</ref><ref>{{harvnb|Bowen|1979}}</ref><ref>{{harvnb|Mumford|Series|Wright|2002}}</ref><ref>{{harvnb|Imayoshi|Taniguchi|1992|p=147}}</ref><ref>{{harvnb|Marden|2007|pp=79–80,134}}</ref>
My New Year's Resolution in 199<br><br><br>
 
12. What is your advice for young entrepreneurs<br>
Let Γ be a Fuchsian group of the first kind: a discrete subgroup of the Möbius group preserving the unit circle. acting properly discontinuously on the unit disk ''D'' and with limit set the unit circle.
Only build something you really want to use yourself. There's got to be one thing you are completely desperate for and no matter where you look you can't find it. Nobody has invented it yet. So there you go - you invent it. If there's other people like you, you have a business. Else. You fail. Then do it again. Until it works. One day it will<br>
 
Follow these 100 Rules<br>
Let μ(''z'') be a measurable function on ''D''  with
The 100 Rules for Being a Good Entrepreneur<br>
 
And, in particular this<br>
:<math>\|\mu\|_\infty < 1</math>
The Easiest Way to Succeed as an Entrepreneu<br>
 
In my just released book I have more chapters on my experiences as an entrepreneur<br>
such that μ is Γ-invariant, i.e.
13. I advocate the concept of working at a job while building your business. You have of course lived it. Now as you look back, what is your take on this? Is it possible to make it work while sailing on two boats<br><br>
 
Your boss wants everything out of you. He wants you to work 80 hours a week. He wants to look good taking credit for your work. He wants your infinite loyalty. So you need something back<br>
:<math>\mu(g(z)){\partial_{\overline{z}}g(z)\over \partial_z g(z)}=\mu(z)</math>
Exploit your employer. It's the best way to get good experience, clients, contacts. It's a legal way to steal. It's a fast way to be an entrepreneur because you see what large companies with infinite money are willing to pay for. If you can provide that, you make millions. It's how many great businesses have started and will always start. It's how every exit I've had started<br>
 
14. Who is a "person with true moral fiber"? In current times are there any role models who are people with true moral fiber<br><br><br>
for every ''g'' in Γ.  (μ is thus a "Beltrami differential" on the [[Riemann surface]] ''D'' /  Γ.)
I don't really know the answer. I think I know a few people like that. I hope I'm someone like that. And I pray to god the people I'm invested in are like that and my family is like that<br>
 
I find most people to be largely mean and stupid, a vile combination. It's not that I'm pessimistic or cynical. I'm very much an optimist. It's just reality. Open the newspaper or turn on the TV and watch these people<br>
Extend μ to a function on '''C''' by setting  μ(''z'') = 0 off ''D''.
Moral fiber atrophies more quickly than any muscle on the body. An exercise I do every morning is to promise myself that "I'm going to save a life today" and then leave it in the hands of the Universe to direct me how I can best do that. Through that little exercise plus the Daily Practice described above I hope to keep regenerating that fiber<br><br>
 
15.  Your message to the readers of this blog<br>
The [[Beltrami equation]]
Skip dinner. But follow me on Twitter.<br><br><br><br>
 
Read more posts on The Altucher Confidential �
:<math> \partial_{\overline{z}} f (z) =\mu(z)\partial_zf(z)</math>
More from The Altucher Confidentia<br>
 
Life is Like a Game. Here�s How You Master ANY Gam<br><br>
admits a solution unique up to composition with a Möbius transformation.
Step By Step Guide to Make $10 Million And Then Totally Blow <br><br>
 
Can You Do One Page a Day?
It is a quasiconformal homeomorphism of the extended complex plane.
 
If ''g'' is an element of Γ, then ''f''(''g''(''z'')) gives another solution of the Beltrami equation, so that
 
:<math>\alpha(g)=f\circ g \circ f^{-1}</math>
 
is a Möbius transformation.
 
The group α(Γ) is a quasi-Fuchsian group with limit set the quasicircle given by the image of the unit circle under ''f''.
 
==Hausdorff dimension==
[[File:Douady rabbit.png|thumb|The [[Douady rabbit]] is composed of quasicircles with Hausdorff dimension approximately 1.3934<ref>{{harvnb|Carleson|Gamelin|1993|p=122}}</ref> ]]
It is known that there are quasicircles for which no segment has finite length.<ref>{{harvnb|Lehto|Virtanen|1973|p=104}}</ref> The [[Hausdorff dimension]] of quasicircles was first investigated by {{harvtxt|Gehring|Väisälä|1973}}, who proved that it can take all values in the interval [1,2).<ref>{{harvnb|Lehto|1982|p=38}}</ref> {{harvtxt|Astala|1993}}, using the new technique of "holomorphic motions" was able to estimate the change in the Hausdorff dimension of any planar set under a quasiconformal map with dilatation ''K''. For quasicircles ''C'', there was a crude estimate for the Hausdorff dimension<ref>{{harvnb|Astala|Iwaniec|Martin|2009}}</ref>
 
:<math> d_H(C) \le 1 + k</math>
 
where
 
:<math>k={K-1\over K+1}.</math>
 
On the other hand, the Hausdorff dimension for the [[Julia set]]s ''J''<sub>c</sub> of the iterates of the [[rational map]]s
 
:<math>R(z) =z^2 +c</math>
 
had been estimated as result of the work of [[Rufus Bowen]] and [[David Ruelle]], who showed that
 
:<math>1 <  d_H(J_c) < 1 + {|c|^2 \over4\log 2} + o(|c|^2).</math>
 
Since these are quasicircles corresponding to a dilatation
 
:<math> K=\sqrt{1+t\over 1-t}</math>
 
where
 
:<math> t= |1-\sqrt{1-4c}|,</math>
 
this led {{harvtxt|Becker|Pommerenke|1987}} to show that for ''k'' small
 
:<math>1+ 0.36 k^2\le d_H(C) \le 1 + 37 k^2.</math>
 
Having improved the lower bound following calculations for the [[Koch snowflake]] with Steffen Rohde and [[Oded Schramm]],
{{harvtxt|Astala|1994}} conjectured that
 
:<math> d_H(C) \le 1 + k^2.</math>
 
This conjecture was proved  by {{harvtxt|Smirnov|2010}}; a complete account of his proof, prior to publication, was already given in {{harvtxt|Astala|Iwaniec|Martin|2009}}.
 
For a quasi-Fuchsian group {{harvtxt|Bowen|1978}} and {{harvtxt|Sullivan|1982}} showed that the Hausdorff dimension ''d'' of the limit set is always greater than 1. When ''d'' < 2, the quantity
 
:<math>\lambda=d(2-d)\,\in (0,1)</math>
 
is the lowest eigenvalue of the Laplacian of the corresponding [[hyperbolic 3-manifold]].<ref>{{harvnb|Astala|Zinsmeister|1994}}</ref><ref>{{harvnb|Marden|2007|p=284}}</ref>
 
==Notes==
{{reflist|2}}
 
==References==
*{{citation|last=Ahlfors|first=Lars V.|authorlink=Lars Ahlfors|title=Lectures on quasiconformal mappings|publisher=Van Nostrand|year=1966}}
*{{citation | title=Quasiconformal reflections | last= Ahlfors|first=L. | authorlink=Lars Ahlfors | journal=[[Acta Mathematica]] | volume=109 | year=1963 | pages=291–301 | zbl=0121.06403 }}
*{{citation|last=Astala|first= K.|title=Distortion of area and dimension under quasiconformal mappings in the plane|journal=
Proc. Nat. Acad. Sci. U.S.A. |volume=90 |year=1993|pages=11958–11959}}
*{{citation|last=Astala|first= K.|last2=Zinsmeister|first2= M.|title=Holomorphic families of quasi-Fuchsian groups|journal=Ergodic Theory Dynam. Systems|volume= 14|year=1994| pages=207–212}}
*{{citation|last=Astala|first=K.|title=Area distortion of quasiconformal mappings|journal=Acta Math.|volume= 173 |year=1994|pages= 37–60}}
*{{citation|title=Elliptic partial differential equations and quasiconformal mappings in the plane|volume= 48|series= Princeton mathematical series|
first=Kari|last= Astala|first2= Tadeusz |last2=Iwaniec|first3= Gaven|last3= Martin|publisher=Princeton University Press|year= 2009|
isbn=0-691-13777-3|pages=332=342}}, Section 13.2, Dimension of quasicircles.
*{{citation|last=Becker|first= J.|last2= Pommerenke|first2= C.|title=On the Hausdorff dimension of quasicircles|journal=Ann. Acad. Sci. Fenn. Ser. A I Math. |volume=12 |year=1987|pages= 329–333}}
*{{citation|last=Bowen|first= R.|title=Hausdorff dimension of quasicircles|journal=Inst. Hautes Études Sci. Publ. Math.|volume= 50|year=1979|pages= 11–25}}
*{{citation|last=Carleson|first=L.|last2= Gamelin|first2= T. D. W.|title=Complex dynamics|series=
Universitext: Tracts in Mathematics|publisher= Springer-Verlag|year=1993|isbn=0-387-97942-5}}
*{{citation|last=Gehring|first= F. W.|last2= Väisälä|first2= J.|title=Hausdorff dimension and quasiconformal mappings|journal=Journal of the London Mathematical Society|volume= 6|year= 1973|pages= 504–512}}
*{{citation|last=Gehring|first= F. W.|title=Characteristic properties of quasidisks|series=Séminaire de Mathématiques Supérieures |volume=84 |publisher=Presses de l'Université de Montréal|year= 1982|isbn= 2-7606-0601-5}}
*{{citation|first=Y. |last=Imayoshi|first2=M.|last2=Taniguchi|title=An Introduction to Teichmüller spaces|publisher=Springer-Verlag|year=1992|isbn=0-387-70088-9}}'' +
*{{citation|first=O.|last=Lehto|title=Univalent functions and  Teichmüller spaces|publisher=Springer-Verlag|year=1987|isbn=0-387-96310-3|pages=50–59, 111–118, 196–205}}
*{{citation|last=Lehto|first=O.|last2= Virtanen|first2= K. I.|title=Quasiconformal mappings in the plane|edition=Second|series= Die Grundlehren der mathematischen Wissenschaften|volume= 126|publisher=Springer-Verlag|year= 1973}}
*{{citation|last=Marden|first= A.|title=Outer circles. An introduction to hyperbolic 3-manifolds|publisher= Cambridge University Press|year= 2007|isbn= 978-0-521-83974-7}}
*{{citation|last=Mumford|first=D.|last2= Series|first2= C.|last3= Wright|first3= David|title=Indra's pearls. The vision of Felix Klein|publisher= Cambridge University Press|year=2002|isbn= 0-521-35253-3}}
*{{citation|last=Pfluger|first= A.|title=Ueber die Konstruktion Riemannscher Flächen durch Verheftung|journal=J. Indian Math. Soc.|volume= 24 |year=1961 |pages=401–412}}
*{{citation|last=Rohde|first= S.|title=On conformal welding and quasicircles|journal=Michigan Math. J. |volume=38 |year=1991|pages=111–116}}
*{{citation|last=Sullivan|first= D.|title=Discrete conformal groups and measurable dynamics|journal=Bull. Amer. Math. Soc.|volume= 6 |year=1982|pages=57–73}}
*{{citation|last=Sullivan|first=D.|title=Quasiconformal homeomorphisms and dynamics, I, Solution of the Fatou-Julia problem on wandering domains|journal= Annals of Mathematics|year= 1985|volume=122|pages= 401–418}}
*{{citation|last=Tienari|first= M.|title=Fortsetzung einer quasikonformen Abbildung über einen Jordanbogen|journal=
Ann. Acad. Sci. Fenn. Ser. A|volume= 321|year= 1962}}
*{{citation |first=S.|last= Smirnov | authorlink=Stanislav Smirnov | title=Dimension of quasicircles | journal=[[Acta Mathematica]] | volume=205 | year=2010 | pages=189–197 | mr=2011j:30027 | doi=10.1007/s11511-010-0053-8 | url=http://www.springerlink.com/content/5387215545350876/ }}
 
[[Category:Complex analysis]]
[[Category:Dynamical systems]]
[[Category:Fractals]]

Latest revision as of 17:56, 27 October 2014


Four or five years ago, a reader of some of my columns bought the domain name jamesaltucher.com and gave it to me as a birthday gift. It was a total surprise to me. I didn't even know the reader. I hope one day we meet.
Two years ago a friend of mine, Tim Sykes, insisted I had to have a blog. He set it up for me. He even wrote the "About Me". I didn't want a blog. I had nothing to say. But about 6 or 7 months ago I decided I wanted to take this blog seriously. I kept putting off changing the "About Me" which was no longer really about me and maybe never was.
A few weeks ago I did a chapter in one of the books in Seth Godin's "The Domino Project". The book is out and called "No Idling". Mohit Pawar organized it (here's Mohit's blog) and sent me a bunch of questions recently. It's intended to be an interview on his blog but I hope Mohit forgives me because I want to use it as my new "About Me" also.
1. You are a trader, investor, writer, and entrepreneur? Which of these roles you enjoy the most and why?
When I first moved to New York City in 1994 I wanted to be everything to everyone. I had spent the six years prior to that writing a bunch of unpublished novels and unpublished short stories. I must've sent out 100s of stories to literary journals. I got form rejections from every publisher, journal, and agent I sent my novels and stories to.
Now, in 1994, everything was possible. The money was in NYC. Media was here. I lived in my 10�10 room and pulled suits out of a garbage bag every morning but it didn't matter...the internet was revving up and I knew how to build a website. One of the few in the city. My sister warned me though: nobody here is your friend. Everybody wants something
And I wanted something. I wanted the fleeting feelings of success, for the first time ever, in order to feel better about myself. I wanted a girl next to me. I wanted to build and sell companies and finally prove to everyone I was the smartest. I wanted to do a TV show. I wanted to write books
But everything involved having a master. Clients. Employers. Investors. Publishers. The market (the deadliest master of all). Employees. I was a slave to everyone for so many years. And the more shackles I had on, the lonelier I got
(Me in the Fortress of Solitude
Much of the time, even when I had those moments of success, I didn't know how to turn it into a better life. I felt ugly and then later, I felt stupid when I would let the success dribble away down the sink
I love writing because every now and then that ugliness turns into honesty. When I write, I'm only a slave to myself. When I do all of those other things you ask about, I'm a slave to everyone else
Some links
33 Unusual Tips to Being a Better Write
"The Tooth
(one of my favorite posts on my blog

2. What inspires you to get up and start working/writing every day
The other day I had breakfast with a fascinating guy who had just sold a piece of his fund of funds. He told me what "fracking" was and how the US was going to be a major oil player again. We spoke for two hours about a wide range of topics, including what happens when we can finally implant a google chip in our brains
After that I had to go onto NPR because I firmly believe that in one important respect we are degenerating as a country - we are graduating a generation of indentured servants who will spend 50 years or more paying down their student debt rather than starting companies and curing cancer. So maybe I made a difference
Then I had lunch with a guy I hadn't seen in ten years. In those ten years he had gone to jail and now I was finally taking the time to forgive him for something he never did to me. I felt bad I hadn't helped him when he was at his http://www.pcs-systems.co.uk/Images/celinebag.aspx low point. Then I came home and watched my kid play clarinet at her school. Then I read until I fell asleep. Today I did nothing but write. Both days inspired me
It also inspires me that I'm being asked these questions. Whenever anyone asks me to do anything I'm infinitely grateful. Why me? I feel lucky. I like it when someone cares what I think. I'll write and do things as long as anyone cares. I honestly probably wouldn't write if nobody cared. I don't have enough humility for that, I'm ashamed to admit

3. Your new book "How to be the luckiest person alive" has just come out. What is it about
When I was a kid I thought I needed certain things: a college education from a great school, a great home, a lot of money, someone who would love me with ease. I wanted people to think I was smart. I wanted people to think I was even special. And as I grew older more and more goals got added to the list: a high chess rating, a published book, perfect weather, good friends, respect in various fields, etc. I lied to myself that I needed these things to be happy. The world was going to work hard to give me these things, I thought. But it turned out the world owed me no favors
And gradually, over time, I lost everything I had ever gained. Several times. I've paced at night so many times wondering what the hell was I going to do next or trying not to care. The book is about regaining your sanity, regaining your happiness, finding luck in all the little pockets of life that people forget about. It's about turning away from the religion you've been hypnotized into believing into the religion you can find inside yourself every moment of the day

[Note: in a few days I'm going to do a post on self-publishing and also how to get the ebook for free. The link above is to the paperback. Kindle should be ready soon also.
Related link: Why I Write Books Even Though I've Lost Money On Every Book I've Ever Writte
4. Is it possible to accelerate success? If yes, how


Yes, and it's the only way I know actually to achieve success. Its by following the Daily Practice I outline in this post:
It's the only way I know to exercise every muscle from the inside of you to the outside of you. I firmly believe that happiness starts with that practice
5. You say that discipline, persistence and psychology are important if one has to achieve success. How can one work on improving "psychology" part
Success doesn't really mean anything. People want to be happy in a harsh and unforgiving world. It's very difficult. We're so lucky most of us live in countries without major wars. Our kids aren't getting killed by random gunfire. We all have cell phones. We all can communicate with each other on the Internet. We have Google to catalog every piece of information in history! We are so amazingly lucky already
How can it be I was so lucky to be born into such a body? In New York City of all places? Just by being born in such a way on this planet was an amazing success
So what else is there? The fact is that most of us, including me, have a hard time being happy with such ready-made success. We quickly adapt and want so much more out of life. It's not wars or disease that kill us. It's the minor inconveniences that add up in life. It's the times we feel slighted or betrayed. Or even slightly betrayed. Or overcharged. Or we miss a train. Or it's raining today. Or the dishwasher doesn't work. Or the supermarket doesn't have the food we like. We forget how good the snow tasted when we were kids. Now we want gourmet food at every meal
Taking a step back, doing the Daily Practice I outline in the question above. For me, the results of that bring me happiness. That's success. Today. And hopefully tomorrow
6. You advocate not sending kids to college. What if kids grow up and then blame their parents about not letting them get a college education
I went to one of my kid's music recitals yesterday. She was happy to see me. I hugged her afterwards. She played "the star wars theme" on the clarinet. I wish I could've played that for my parents. My other daughter has a dance recital in a few weeks. I tried to give her tips but she laughed at me. I was quite the breakdancer in my youth. The nerdiest breakdancer on the planet. I want to be present for them. To love them. To let them always know that in their own dark moments, they know I will listen to them. I love them. Even when they cry and don't always agree with me. Even when they laugh at me because sometimes I act like a clown
Later, if they want to blame me for anything at all then I will still love them. That's my "what if"
Two posts
I want my daughters to be lesbian
Advice I want to give my daughter


7. Four of your favorite posts from The Altucher Confidential
As soon as I publish a post I get scared to death. Is it good? Will people re-tweet? Will one part of the audience of this blog like it at the expense of another part of the audience. Will I get Facebook Likes? I have to stop clinging to these things but you also need to respect the audience. I don't know. It's a little bit confusing to me. I don't have the confidence of a real writer yet
Here are four of my favorites
How I screwed Yasser Arafat out of $2mm (and lost another $100mm in the process
It's Your Fault
I'm Guilty of Torturing Wome
The Girl Whose Name Was a Curs
Although these three are favorites I really don't post anything unless it's my favorite of that moment
8. 3 must-read books for aspiring entrepreneurs
The key in an entrepreneur book: you want to learn business. You want to learn how to honestly communicate with your customers. You want to stand out
The Essays of Warren Buffett by Lawrence Cunningha
"The Thank you Economy" by Gary Vaynerchu
"Purple cow" by Seth Godi
9. I love your writing, so do so many others out there. Who are your favorite writers
"Jesus's Son" by Denis Johnson is the best collection of short stories ever written. I'm afraid I really don't like his novels though
"Tangents" by M. Prado. A beautiful series of graphic stories about relationships
Other writers: Miranda July, Ariel Leve, Mary Gaitskill, Charles Bukowski, Celine, Sam Lipsyte, William Vollmann, Raymond Carver. Arthur Nersesian. Stephen Dubner

(Bukowski








Many writers are only really good storytellers. Most writers come out of a cardboard factory MFA system and lack a real voice. A real voice is where every word exposes ten levels of hypocrisy in the world and brings us all the way back to see reality. The writers above have their own voices, their own pains, and their unique ways of expressing those pains. Some of them are funny. Some a little more dark. I wish I could write 1/10 as good as any of them

10. You are a prolific writer. Do you have any hacks that help you write a lot in little time
Coffee, plus everything else coffee does for you first thing in the morning
Only write about things you either love or hate. But if you hate something, try to find a tiny gem buried in the bag of dirt so you can reach in when nobody is looking and put that gem in your pocket. Stealing a diamond in all the shit around us and then giving it away for free via writing is a nice little hack, Being fearless precisely when you are most scared is the best hack

11. I totally get and love your idea about bleeding as a writer, appreciate if you share more with the readers of this blog
Most people worry about what other people think of them. Most people worry about their health. Most people are at a crossroads and don't know how to take the next step and which road to take it on. Everyone is in a perpetual state of 'where do I put my foot next'. Nobody, including me, can avoid that
You and I both need to wash our faces in the morning, brush our teeth, shower, shit, eat, fight the weather, fight the colds that want to attack us if we're not ready. Fight loneliness or learn how to love and appreciate the people who want to love you back. And learn how to forgive and love the people who are even more stupid and cruel than we are. We're afraid to tell each other these things because they are all both disgusting and true
You and I both have the same color blood. If I cut my wrist open you can see the color of my blood. You look at it and see that it's the same color as yours. We have something in common. It doesn't have to be shameful. It's just red. Now we're friends. No matter whom you are or where you are from. I didn't have to lie to you to get you to be my friend
Related Links
How to be a Psychic in Ten Easy Lesson
My New Year's Resolution in 199


12. What is your advice for young entrepreneurs
Only build something you really want to use yourself. There's got to be one thing you are completely desperate for and no matter where you look you can't find it. Nobody has invented it yet. So there you go - you invent it. If there's other people like you, you have a business. Else. You fail. Then do it again. Until it works. One day it will
Follow these 100 Rules
The 100 Rules for Being a Good Entrepreneur
And, in particular this
The Easiest Way to Succeed as an Entrepreneu
In my just released book I have more chapters on my experiences as an entrepreneur
13. I advocate the concept of working at a job while building your business. You have of course lived it. Now as you look back, what is your take on this? Is it possible to make it work while sailing on two boats

Your boss wants everything out of you. He wants you to work 80 hours a week. He wants to look good taking credit for your work. He wants your infinite loyalty. So you need something back
Exploit your employer. It's the best way to get good experience, clients, contacts. It's a legal way to steal. It's a fast way to be an entrepreneur because you see what large companies with infinite money are willing to pay for. If you can provide that, you make millions. It's how many great businesses have started and will always start. It's how every exit I've had started
14. Who is a "person with true moral fiber"? In current times are there any role models who are people with true moral fiber


I don't really know the answer. I think I know a few people like that. I hope I'm someone like that. And I pray to god the people I'm invested in are like that and my family is like that
I find most people to be largely mean and stupid, a vile combination. It's not that I'm pessimistic or cynical. I'm very much an optimist. It's just reality. Open the newspaper or turn on the TV and watch these people
Moral fiber atrophies more quickly than any muscle on the body. An exercise I do every morning is to promise myself that "I'm going to save a life today" and then leave it in the hands of the Universe to direct me how I can best do that. Through that little exercise plus the Daily Practice described above I hope to keep regenerating that fiber

15. Your message to the readers of this blog
Skip dinner. But follow me on Twitter.



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