Characterization of line width variation
Alfred K. Wong, Antoinette F. Molless, et al.
SPIE Advanced Lithography 2000
We study finitely generated, abelian groups Γ of continuous automorphisms of a compact, metrizable group X and introduce the descending chain condition for such pairs (X, Γ). If Γ acts expansively on X then (X, Γ) satisfies the descending chain condition, and (X, Γ) satisfies the descending chain condition if and only if it is algebraically and topologically isomorphic to a closed, shift-invariant subgroup of GΓ, where G is a compact Lie group. Furthermore every such subgroup of GΓ is a (higher dimensional) Markov shift whose alphabet is a compact Lie group. By using the descending chain condition we prove, for example, that the set of Γ-periodic points is dense in X whenever Γ acts expansively on X. Furthermore, if X is a compact group and (X, Γ) satisfies the descending chain condition, then every ergodic element of Γ has a dense set of periodic points. Finally we give an algebraic description of pairs (X, Γ) satisfying the descending chain condition under the assumption that X is abelian. © 1989, Cambridge University Press. All rights reserved.
Alfred K. Wong, Antoinette F. Molless, et al.
SPIE Advanced Lithography 2000
Andrew Skumanich
SPIE Optics Quebec 1993
Moutaz Fakhry, Yuri Granik, et al.
SPIE Photomask Technology + EUV Lithography 2011
Martin Charles Golumbic, Renu C. Laskar
Discrete Applied Mathematics