• Journal of the Korean Mathematical Society
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• v.56 no.6
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• pp.1463-1474
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• 2019

Let C be a nonsingular projective curve of genus ${\geq}2$ over an algebraically closed field of characteristic 0. For a point P in C, the Weierstrass semigroup H(P) is defined as the set of non-negative integers n for which there exists a rational function f on C such that the order of the pole of f at P is equal to n, and f is regular away from P. A point P in C is referred to as a weak Galois-Weierstrass point if P is a Weierstrass point and there exists a Galois morphism ${\varphi}:C{\rightarrow}{\mathbb{p}}^1$ such that P is a total ramification point of ${\varphi}$. In this paper, we investigate the number of weak Galois-Weierstrass points of which the Weierstrass semigroups are generated by two positive integers.

• Communications of the Korean Mathematical Society
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• v.19 no.3
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• pp.507-517
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• 2004

Let (2,2,2,2) be ramification indices for the Riemann sphere. It is well known that the regular branched covering map corresponding to this, is the Weierstrass P function. Lattes [7] gives a rational function R(z)= ${\frac{z^4+{\frac{1}{2}}g2^{z}^2+{\frac{1}{16}}g{\frac{2}{2}}$ which is chaotic on ${\bar{C}}$ and is induced by the Weierstrass P function and the linear map L(z) = 2z on complex plane C. It is also known that there exist regular branched covering maps from $T^2$ onto ${\bar{C}}$ if and only if the ramification indices are (2,2,2,2), (2,4,4), (2,3,6) and (3,3,3), by the Riemann-Hurwitz formula. In this paper we will construct regular branched covering maps corresponding to the ramification indices (2,4,4), (2,3,6) and (3,3,3), as well as chaotic maps induced by these regular branched covering maps.

• Communications of the Korean Mathematical Society
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• v.21 no.4
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• pp.759-769
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• 2006

Let L : $C\;\rightarrow\;C$ be a hyperbolic automorphism. Then the hyperbolic toral automorphism $L_A\;T^2\;\rightarrow\;T^2$, induced by L, is a chaotic map ([2] pg.192). We characterize hyperbolic toral automorphisms by proving the converse of the above statement.

• Bulletin of the Korean Mathematical Society
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• v.47 no.2
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• pp.433-442
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• 2010

$\tilde{G}(P,\;Q)$, a new generalization of the set of gap numbers of a pair of points, was described in [1]. Here we study gap numbers of local subring $R\;=\;\cal{K}\;+\;C$ of algebraic function field over a finite field and we give a formula for the number of elements of $\tilde{G}(P,\;Q)$ depending on pure gaps and R.

• Communications of the Korean Mathematical Society
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• v.18 no.1
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• pp.105-115
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• 2003

It is well known that there exists a regular branched covering map from T$^2$ onto $\={C}$ iff the ramification indices are (2,2,2,2), (2,4,4), (2,3,6) and (3,3,3). In this paper we construct (count-ably many) chaotic homeomorphisms induced by hyperbolic toral automorphism and regular branched covering map corresponding to the ramification indices (2,2,2,2). And we also gave an example which shows that the above construction of a chaotic map is not true in general if the ramification indices is (2,4,4) and also show that there are no chaotic homeomorphisms induced by hyperbolic toral automorphism and regular branched covering map corresponding to the ramification indices (2,3,6) and (3,3,3).