TY - JOUR

T1 - Sharkovskii’s Ordering and Estimates of the Number of Periodic Trajectories of Given Period of a Self-Map of an Interval

AU - Ivanov, O. A.

PY - 2019/7/1

Y1 - 2019/7/1

N2 - Abstract: In 1964, A. N. Sharkovskii published a paper in which he introduced an ordering relation on the set of positive integers. His ordering had the property that if a continuous self-map of an interval has a periodic point of some period p, then it also has periodic points of any period larger than p in this ordering. The least number in this ordering is 3. Thus, if a continuous self-map of an interval has a point of period 3, then it has points of any period. In 1975, this result was rediscovered by Lie and Yorke, who published it in their paper “Period three implies chaos.” Their work has led to the international recognition of Sharkovskii’s theorem. Since then, numerous papers on properties of self-maps of an interval have appeared. In 1994, even a conference named “Thirty Years after Sharkovskii’s Theorem: New Perspectives” was held. One of the research directions is estimating the number of periodic trajectories which a map satisfying the conditions of Sharkovskii’s theorem must have. In 1985, Bau-Sen Du published a paper in which he obtained an exact lower bound for the number of periodic trajectories of given period. In the present paper, a new, significantly shorter and more natural, proof of this result is given.

AB - Abstract: In 1964, A. N. Sharkovskii published a paper in which he introduced an ordering relation on the set of positive integers. His ordering had the property that if a continuous self-map of an interval has a periodic point of some period p, then it also has periodic points of any period larger than p in this ordering. The least number in this ordering is 3. Thus, if a continuous self-map of an interval has a point of period 3, then it has points of any period. In 1975, this result was rediscovered by Lie and Yorke, who published it in their paper “Period three implies chaos.” Their work has led to the international recognition of Sharkovskii’s theorem. Since then, numerous papers on properties of self-maps of an interval have appeared. In 1994, even a conference named “Thirty Years after Sharkovskii’s Theorem: New Perspectives” was held. One of the research directions is estimating the number of periodic trajectories which a map satisfying the conditions of Sharkovskii’s theorem must have. In 1985, Bau-Sen Du published a paper in which he obtained an exact lower bound for the number of periodic trajectories of given period. In the present paper, a new, significantly shorter and more natural, proof of this result is given.

KW - map of an interval

KW - path in a directed graph

KW - periodic trajectory

KW - Sharkovskii’s order

UR - http://www.scopus.com/inward/record.url?scp=85071968339&partnerID=8YFLogxK

U2 - 10.1134/S1063454119030099

DO - 10.1134/S1063454119030099

M3 - Article

AN - SCOPUS:85071968339

VL - 52

SP - 281

EP - 285

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 3

ER -