Ebook: Fractals and Chaos: The Mandelbrot Set and Beyond

"It is only twenty-three years since Benoit Mandelbrot published his famous picture of what is now called the Mandelbrot Set. The graphics were state of the art, though now they may seem primitive. But how that picture has changed our views of the mathematical and physical universe! Fractals, a term coined by Mandelbrot, are now so ubiquitous in the scientific conscience that it is difficult to remember the psychological shock of their arrival. What we see in this book is a glimpse of how Mandelbrot helped change our way of looking at the world. It is not just a book about a particular class of problems, but contains a view on how to approach the mathematical and physical universe. This view is certain not to fade, but to be part of the working philosophy of the next mathematical revolution, wherever it may take us. So read the book, look at the beautiful pictures that continue to fascinate and amaze, and enjoy! "

From the foreword by Peter W Jones, Yale University

This heavily illustrated book combines hard-to-find early papers by the author with additional chapters that describe the historical background and context. Key topics are quadratic dynamics and its Julia and Mandelbrot sets, nonquadratic dynamics, Kleinian limit sets, and the Minkowski measure.

Benoit B Mandelbrot is Sterling Professor of Mathematical Sciences at Yale University and IBM Fellow Emeritus (Physics) at the IBM T J Watson Research Center. He was awarded the Wolf Prize for Physics in 1993 and the Japan Prize for Science and Technology in 2003.

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"It is only twenty-three years since Benoit Mandelbrot published his famous picture of what is now called the Mandelbrot Set. The graphics were state of the art, though now they may seem primitive. But how that picture has changed our views of the mathematical and physical universe! Fractals, a term coined by Mandelbrot, are now so ubiquitous in the scientific conscience that it is difficult to remember the psychological shock of their arrival. What we see in this book is a glimpse of how Mandelbrot helped change our way of looking at the world. It is not just a book about a particular class of problems, but contains a view on how to approach the mathematical and physical universe. This view is certain not to fade, but to be part of the working philosophy of the next mathematical revolution, wherever it may take us. So read the book, look at the beautiful pictures that continue to fascinate and amaze, and enjoy! " 

From the foreword by Peter W Jones, Yale University

This heavily illustrated book combines hard-to-find early papers by the author with additional chapters that describe the historical background and context. Key topics are quadratic dynamics and its Julia and Mandelbrot sets, nonquadratic dynamics, Kleinian limit sets, and the Minkowski measure.

Benoit B Mandelbrot is Sterling Professor of Mathematical Sciences at Yale University and IBM Fellow Emeritus (Physics) at the IBM T J Watson Research Center. He was awarded the Wolf Prize for Physics in 1993 and the Japan Prize for Science and Technology in 2003.

---

 

"It is only twenty-three years since Benoit Mandelbrot published his famous picture of what is now called the Mandelbrot Set. The graphics were state of the art, though now they may seem primitive. But how that picture has changed our views of the mathematical and physical universe! Fractals, a term coined by Mandelbrot, are now so ubiquitous in the scientific conscience that it is difficult to remember the psychological shock of their arrival. What we see in this book is a glimpse of how Mandelbrot helped change our way of looking at the world. It is not just a book about a particular class of problems, but contains a view on how to approach the mathematical and physical universe. This view is certain not to fade, but to be part of the working philosophy of the next mathematical revolution, wherever it may take us. So read the book, look at the beautiful pictures that continue to fascinate and amaze, and enjoy! " 

From the foreword by Peter W Jones, Yale University

This heavily illustrated book combines hard-to-find early papers by the author with additional chapters that describe the historical background and context. Key topics are quadratic dynamics and its Julia and Mandelbrot sets, nonquadratic dynamics, Kleinian limit sets, and the Minkowski measure.

Benoit B Mandelbrot is Sterling Professor of Mathematical Sciences at Yale University and IBM Fellow Emeritus (Physics) at the IBM T J Watson Research Center. He was awarded the Wolf Prize for Physics in 1993 and the Japan Prize for Science and Technology in 2003.

Content:
Front Matter....Pages i-8
Introduction to papers on quadratic dynamics: a progression from seeing to discovering....Pages 9-26
Acknowledgments related to quadratic dynamics....Pages 27-36
Fractal aspects of the iteration of z??z(1-z) for complex ? and z....Pages 37-51
Cantor and Fatou dusts; self-squared dragons....Pages 52-72
The complex quadratic map and its ?-set....Pages 73-95
Bifurcation points and the “n-squared” approximation and conjecture, illustrated by M.L Frame and K Mitchell....Pages 96-99
The “normalized radical” of the ?-set....Pages 100-109
The boundary of the ?-set is of dimension 2....Pages 110-113
Certain Julia sets include smooth components....Pages 114-116
Domain-filling sequences of Julia sets, and intuitive rationale for the Siegel discs....Pages 117-124
Continuous interpolation of the quadratic map and intrinsic tiling of the interiors of Julia sets....Pages 125-136
Introduction to papers on chaos in nonquadratic dynamics: rational functions devised from doubling formulas....Pages 137-145
The map z ? ? (z+1/z) and roughening of chaos, from linear to planar (computer-assisted homage to K. Hokusai)....Pages 146-156
Two nonquadratic rational maps devised from Weierstrass doubling formulas....Pages 157-170
Introduction to papers on Kleinian groups, their fractal limit sets, and IFS: history, recollections, and acknowledgments....Pages 171-177
Self-inverse fractals, Apollonian nets, and soap....Pages 178-192
Symmetry with respect to several circles: dilation/reduction, fractals, and roughness....Pages 193-204
Self-inverse fractals osculated by sigma-discs: the limit sets of (“Kleinian”) inversion groups....Pages 205-220
Introduction to measures that vanish exponentially almost everywhere: DLA and Minkowski....Pages 221-230
Invariant multifractal measures in chaotic Hamiltonian systems and related structures (Gutzwiller & M 1988)....Pages 231-238
The Minkowski measure and multifractal anomalies in invariant measures of parabolic dynamic systems....Pages 239-250
Harmonic measure on DLA and extended self-similarity (M & Evertsz 1991)....Pages 251-258
The inexhaustible function z squared plus c....Pages 259-267
The Fatou and Julia stories....Pages 268-275
Mathematical analysis while in the wilderness....Pages 276-280
Back Matter....Pages 281-308

---

 

"It is only twenty-three years since Benoit Mandelbrot published his famous picture of what is now called the Mandelbrot Set. The graphics were state of the art, though now they may seem primitive. But how that picture has changed our views of the mathematical and physical universe! Fractals, a term coined by Mandelbrot, are now so ubiquitous in the scientific conscience that it is difficult to remember the psychological shock of their arrival. What we see in this book is a glimpse of how Mandelbrot helped change our way of looking at the world. It is not just a book about a particular class of problems, but contains a view on how to approach the mathematical and physical universe. This view is certain not to fade, but to be part of the working philosophy of the next mathematical revolution, wherever it may take us. So read the book, look at the beautiful pictures that continue to fascinate and amaze, and enjoy! " 

From the foreword by Peter W Jones, Yale University

This heavily illustrated book combines hard-to-find early papers by the author with additional chapters that describe the historical background and context. Key topics are quadratic dynamics and its Julia and Mandelbrot sets, nonquadratic dynamics, Kleinian limit sets, and the Minkowski measure.

Benoit B Mandelbrot is Sterling Professor of Mathematical Sciences at Yale University and IBM Fellow Emeritus (Physics) at the IBM T J Watson Research Center. He was awarded the Wolf Prize for Physics in 1993 and the Japan Prize for Science and Technology in 2003.

Content:
Front Matter....Pages i-8
Introduction to papers on quadratic dynamics: a progression from seeing to discovering....Pages 9-26
Acknowledgments related to quadratic dynamics....Pages 27-36
Fractal aspects of the iteration of z??z(1-z) for complex ? and z....Pages 37-51
Cantor and Fatou dusts; self-squared dragons....Pages 52-72
The complex quadratic map and its ?-set....Pages 73-95
Bifurcation points and the “n-squared” approximation and conjecture, illustrated by M.L Frame and K Mitchell....Pages 96-99
The “normalized radical” of the ?-set....Pages 100-109
The boundary of the ?-set is of dimension 2....Pages 110-113
Certain Julia sets include smooth components....Pages 114-116
Domain-filling sequences of Julia sets, and intuitive rationale for the Siegel discs....Pages 117-124
Continuous interpolation of the quadratic map and intrinsic tiling of the interiors of Julia sets....Pages 125-136
Introduction to papers on chaos in nonquadratic dynamics: rational functions devised from doubling formulas....Pages 137-145
The map z ? ? (z+1/z) and roughening of chaos, from linear to planar (computer-assisted homage to K. Hokusai)....Pages 146-156
Two nonquadratic rational maps devised from Weierstrass doubling formulas....Pages 157-170
Introduction to papers on Kleinian groups, their fractal limit sets, and IFS: history, recollections, and acknowledgments....Pages 171-177
Self-inverse fractals, Apollonian nets, and soap....Pages 178-192
Symmetry with respect to several circles: dilation/reduction, fractals, and roughness....Pages 193-204
Self-inverse fractals osculated by sigma-discs: the limit sets of (“Kleinian”) inversion groups....Pages 205-220
Introduction to measures that vanish exponentially almost everywhere: DLA and Minkowski....Pages 221-230
Invariant multifractal measures in chaotic Hamiltonian systems and related structures (Gutzwiller & M 1988)....Pages 231-238
The Minkowski measure and multifractal anomalies in invariant measures of parabolic dynamic systems....Pages 239-250
Harmonic measure on DLA and extended self-similarity (M & Evertsz 1991)....Pages 251-258
The inexhaustible function z squared plus c....Pages 259-267
The Fatou and Julia stories....Pages 268-275
Mathematical analysis while in the wilderness....Pages 276-280
Back Matter....Pages 281-308
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