Mandelbrot Set and Fermat’s Last
Theorem
P. B. Vinod Kumar
Abstract In this article a brief description of the work done with Prof. K. Babu
Joseph and some related problems are described. The notion of irrational points in
the Mandelbrot set ( M 2 ) is explained. Fermat’s last Theorem (FLT) om Topological
Fields is discussed. We ask the question: Given n Z + Do there exist (x, y, z))
M 2 × M 2 × M 2 such that x
n
+ y
n
= z
n ? An attempt is made to solve this problem.
Keywords Mandelbrot set · Julia set · Fermat’s theorem
1 Mandlebrot Set
In this section some results related to the Mandelbrot set are given. Carleson
and Gamelin [3] gives a wonderful description of Julia set and the Mandelbrot
set. It is interesting to study the continuity of the map : C → R, defined as
φ(c) = dim H (J c ). Curt Mcmullen has studied some properties of the map in [1].
But we show that if f c has a Siegel disc at c then φ is discontinuous at c.
M = {cC|J c is connected} is named as the Mandelbrot set. There are a lot of mysteries behind Mandelbrot set. We will define irrational points on M using the Hausdorff
dimension. Mcmullen has computed Hausdorff dimension of J c for different values
of c in [1].
Curt Mcmullen in [1] has proved the following result.
Theorem 1.1 [1] (i) For |c| small (near 0) the map φ : c → dim H (J c ) is analytic
and dim H (J c ) → 1 +
|c|
2
4log2
, as |c| → 0. dim H (J c ) →
2log2
log|c|
, as |c| → ∞.
A natural question arises at this point is about the map φ : C → R, defined as
φ(c) = dim H (J c ). We will prove the following result.
P. B. Vinod Kumar (B)
Department of Mathematics, Rajagiri School of Engineering and Technology,
Kochi, Kerala, India
e-mail: vinod_kumar@rajagiritech.edu.in
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
K. S. Sreelatha and V. Jacob (eds.), Modern Perspectives in Theoretical Physics,
https://doi.org/10.1007/978-981-15-9313-0_8
109
Theorem
P. B. Vinod Kumar
Abstract In this article a brief description of the work done with Prof. K. Babu
Joseph and some related problems are described. The notion of irrational points in
the Mandelbrot set ( M 2 ) is explained. Fermat’s last Theorem (FLT) om Topological
Fields is discussed. We ask the question: Given n Z + Do there exist (x, y, z))
M 2 × M 2 × M 2 such that x
n
+ y
n
= z
n ? An attempt is made to solve this problem.
Keywords Mandelbrot set · Julia set · Fermat’s theorem
1 Mandlebrot Set
In this section some results related to the Mandelbrot set are given. Carleson
and Gamelin [3] gives a wonderful description of Julia set and the Mandelbrot
set. It is interesting to study the continuity of the map : C → R, defined as
φ(c) = dim H (J c ). Curt Mcmullen has studied some properties of the map in [1].
But we show that if f c has a Siegel disc at c then φ is discontinuous at c.
M = {cC|J c is connected} is named as the Mandelbrot set. There are a lot of mysteries behind Mandelbrot set. We will define irrational points on M using the Hausdorff
dimension. Mcmullen has computed Hausdorff dimension of J c for different values
of c in [1].
Curt Mcmullen in [1] has proved the following result.
Theorem 1.1 [1] (i) For |c| small (near 0) the map φ : c → dim H (J c ) is analytic
and dim H (J c ) → 1 +
|c|
2
4log2
, as |c| → 0. dim H (J c ) →
2log2
log|c|
, as |c| → ∞.
A natural question arises at this point is about the map φ : C → R, defined as
φ(c) = dim H (J c ). We will prove the following result.
P. B. Vinod Kumar (B)
Department of Mathematics, Rajagiri School of Engineering and Technology,
Kochi, Kerala, India
e-mail: vinod_kumar@rajagiritech.edu.in
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
K. S. Sreelatha and V. Jacob (eds.), Modern Perspectives in Theoretical Physics,
https://doi.org/10.1007/978-981-15-9313-0_8
109