110
P. B. Vinod Kumar
Theorem 1.2 [4] Let c ∈ M be a parameter value for which f c has a Siegel disc.
Then the map φ is discontinuous at c.
Proof Let z 0 be a Siegel periodic point of f c and denote the Siegel disc around
z 0 , p its period, and θ the rotation angle. Let U (c) be an open set containing c.
By the Implicit function theorem, there exists a holomorphic mapping ζ : U (c) →
C such that ζ(c) = z 0 and ζ(c) is the fixed number ( f c )
p . The mapping χ : c →
(( f c )
p
)
(ζ (c)) is holomorphic, hence it is either constant or open. If it is constant,
all quadratic polynomials have a Siegel disc (see [2] in which it has proved that a
quadratic polynomial has almost one cycle of Siegel discs). This is not possible:
for instance, f 1/4 has a parabolic fixed point and thus no other non-repelling cycles.
Therefore, χ is open and in particular there is a sequence of parameters c n → c such
that ζ(c n ) has multiplier e
2π pn
qn . Since ζ(c n ) is parabolic, it lies in the Julia set of J c n .
Hence, dist H (J c, J ) ≥
dist(c,ρ)
2
for large n. Hence the result.
Converse of above result is not true.
For example, c = 0.25 is parabolic i.e., f c has a parabolic cycle (z = 0.25). So it
is geometrically finite but not expanding. As c decreases from
1
4
along the real axis
the map f c undergoes a sequence of period doubling bifurcation at parabolic points
c n converging to the Feigenbaum point c F ≈ −1.40115. The map f c is expanding
for all c(c F ,
1
4
]. At c =
1
4
, φ is discontinuous but f c has no Siegel disc there.
In [1], Mcmullen proved the following result.
Theorem 1.3 [1] The function φ is continuous in (c F ,
1
4
].
2 Fermat’s Last Theorem
2.1 Topological Fields
Let (F, ∗, .) be a Field and T be a topology on F. Then (F, ∗, ., T ) is called a
topological Field. There are topological fields which look like as same if we view
from different angles, for example, R
n .
A topological space F is homogenous if given any x, y ∈ F, there exist f : F →
F, a homeomorphism on to F such that f (x) = y.
In (F, ∗, ., T ) neighbourhood of 0 is called additive nucleius.
We prove the following results in this section.
Theorem 2.1 [5] (i) Every topological field is homogenous. (ii) Every topological
field is Hausdorff.
Proof (i) Let x, y ∈ F. Let L a (b) = a.b. Then L
−1
yx (x) = y. So givenx, y ∈ F, there
exists L
−1
yx : F → F, which is a homeomorphism.
P. B. Vinod Kumar
Theorem 1.2 [4] Let c ∈ M be a parameter value for which f c has a Siegel disc.
Then the map φ is discontinuous at c.
Proof Let z 0 be a Siegel periodic point of f c and denote the Siegel disc around
z 0 , p its period, and θ the rotation angle. Let U (c) be an open set containing c.
By the Implicit function theorem, there exists a holomorphic mapping ζ : U (c) →
C such that ζ(c) = z 0 and ζ(c) is the fixed number ( f c )
p . The mapping χ : c →
(( f c )
p
)
(ζ (c)) is holomorphic, hence it is either constant or open. If it is constant,
all quadratic polynomials have a Siegel disc (see [2] in which it has proved that a
quadratic polynomial has almost one cycle of Siegel discs). This is not possible:
for instance, f 1/4 has a parabolic fixed point and thus no other non-repelling cycles.
Therefore, χ is open and in particular there is a sequence of parameters c n → c such
that ζ(c n ) has multiplier e
2π pn
qn . Since ζ(c n ) is parabolic, it lies in the Julia set of J c n .
Hence, dist H (J c, J ) ≥
dist(c,ρ)
2
for large n. Hence the result.
Converse of above result is not true.
For example, c = 0.25 is parabolic i.e., f c has a parabolic cycle (z = 0.25). So it
is geometrically finite but not expanding. As c decreases from
1
4
along the real axis
the map f c undergoes a sequence of period doubling bifurcation at parabolic points
c n converging to the Feigenbaum point c F ≈ −1.40115. The map f c is expanding
for all c(c F ,
1
4
]. At c =
1
4
, φ is discontinuous but f c has no Siegel disc there.
In [1], Mcmullen proved the following result.
Theorem 1.3 [1] The function φ is continuous in (c F ,
1
4
].
2 Fermat’s Last Theorem
2.1 Topological Fields
Let (F, ∗, .) be a Field and T be a topology on F. Then (F, ∗, ., T ) is called a
topological Field. There are topological fields which look like as same if we view
from different angles, for example, R
n .
A topological space F is homogenous if given any x, y ∈ F, there exist f : F →
F, a homeomorphism on to F such that f (x) = y.
In (F, ∗, ., T ) neighbourhood of 0 is called additive nucleius.
We prove the following results in this section.
Theorem 2.1 [5] (i) Every topological field is homogenous. (ii) Every topological
field is Hausdorff.
Proof (i) Let x, y ∈ F. Let L a (b) = a.b. Then L
−1
yx (x) = y. So givenx, y ∈ F, there
exists L
−1
yx : F → F, which is a homeomorphism.