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Some classes of substitutions with cycle lengths from a given set

Some classes of substitutions with cycle lengths from a given set -- We consider the classes Tn of substitutions of degree n whose cycle lengths belong to a set A C N, where the set is completely determined by a given regularly varying function g(t) and a finite union of intervals from [0, 1]. We find the asymptotics of the number of elements of Tn as --> oo. The limit theorems on the total number of cycles and the number of cycles of a fixed length in random substitutions uniformly distributed on Tn are proved. This paper continues the investigations started in [1]. 1. INTRODUCTION Let a function g(t) for t > 1 and a finite union of intervals from [0, 1] be given. A number m 6 is included in the set A if and only if {g(m)} e , where {a} is the fractional part of a number a. Let Tn denote the totality of substitutions of degree whose cycle lengths belong to A, let |Tn| denote the number of such substitutions, and let be the Lebesgue measure of . We assume that for some non-integer and a function l(t) slowly varying on infinity 5(0 = /(0, and for = 1, . . . , [] + http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete Mathematics and Applications de Gruyter

Some classes of substitutions with cycle lengths from a given set

Discrete Mathematics and Applications , Volume 3 (2) – Jan 1, 1993

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Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0924-9265
eISSN
1569-3929
DOI
10.1515/dma.1993.3.2.213
Publisher site
See Article on Publisher Site

Abstract

-- We consider the classes Tn of substitutions of degree n whose cycle lengths belong to a set A C N, where the set is completely determined by a given regularly varying function g(t) and a finite union of intervals from [0, 1]. We find the asymptotics of the number of elements of Tn as --> oo. The limit theorems on the total number of cycles and the number of cycles of a fixed length in random substitutions uniformly distributed on Tn are proved. This paper continues the investigations started in [1]. 1. INTRODUCTION Let a function g(t) for t > 1 and a finite union of intervals from [0, 1] be given. A number m 6 is included in the set A if and only if {g(m)} e , where {a} is the fractional part of a number a. Let Tn denote the totality of substitutions of degree whose cycle lengths belong to A, let |Tn| denote the number of such substitutions, and let be the Lebesgue measure of . We assume that for some non-integer and a function l(t) slowly varying on infinity 5(0 = /(0, and for = 1, . . . , [] +

Journal

Discrete Mathematics and Applicationsde Gruyter

Published: Jan 1, 1993

There are no references for this article.