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Least Number of periodic points of Self-maps of Lie Groups

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Acta Mathematica Sinica,English Series 2014年第9期30卷 1477-1494页

作者： Jerzy JEZIERSKI Department of Applied Informatics and Mathematics Warsaw University of Life Sciences

There are two algebraic lower bounds of the number of n-periodic points of a self-map f ： M → M of a compact smooth manifold of dimension at least 3： NFn（f） = min{#Fix（gn）;g - f; g is continuous} and NJDn（f） = min{#Fix（gn）; g - f; g is smooth}. In general, NJDn（f） may be much greater than NFn（f）. If M is a torus, then the invariants are equal. We show that for a self-map of a nonabelian compact Lie group, with free fundamental group, the equality holds 〈=〉 all eigenvalues of a quotient cohomology homomorphism induced by f have moduli ≤ 1.

关键词： Fixed point periodic point Nielsen fixed point theory Dold congruences least number of periodic points

ON THE NONEXISTENCE OF NONTRIVIAL SMALL CYCLES OF THE μ FUNCTION IN 3x+ 1 CONJECTURE

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Journal of Systems Science & Complexity 2012年第6期25卷 1215-1222页

作者： Dengguo FENG Xiubin FAN Liping DING Zhangyi WANG State Key Laboratory of Information Security Institute of SoftwareChinese Academy of Sciences Institute of Software Chinese Academy of Sciences

This paper studies the property of the recursive sequences in the 3x ＋ 1 conjecture. The authors introduce the concept of μ function, with which the 3x ＋ 1 conjecture can be transformed into two other conjectures： one is eventually periodic conjecture of the μ function and the other is periodic point conjecture. The authors prove that the 3x ＋ 1 conjecture is equivalent to the two conjectures above. In 2007, J. L. Simons proved the non-existence of nontrivial 2-cycle for the T function. In this paper, the authors prove that the μ function has nol-periodic points for 2 ≤ 1 ≤12. In 2005, J. L. Simons and B. M. M de Weger proved that there is no nontrivial/-cycle for the T function for 1 ≤68, and in this paper, the authors prove that there is no nontrivial l-cycle for the μ function for 2 ≤ 1≤ 102.

关键词： Diophantine equation eventual period periodic point 3x + 1 conjecture.

Kneading Sequences for Unimodal Expanding Maps of the Interval

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Acta Mathematica Sinica,English Series 1998年第4期14卷 457-462页

作者： Zeng Fanping, Institute of Mathematics, Guangxi University, Nanning 530004, China Institute of Mathematics Guangxi University Nanning China

Let P and AC be two primary sequences with min{P, AC} RLR∞,ρ（P） and p（AC） be the eigenvalues of P and AC, respectively. Let f ∈ C0（I, I） be a unimodal expanding map with expanding constant λ and m be a nonegative integer. It is proved that f has the kneading sequence K（f）（RC）*m * P if λ （ρ（P））1/2m, and K（f）>（RC）*m * AC * E for any shift maximal sequence E if λ>（ρ（AC））1/2m. The value of （ρ（P））1/2m or （ρ（AC））1/2m is the best possible in the sense that the related conclusion may not be true if it is replaced by any smaller one.

关键词： periodic point Kneading sequence Topological entropy

Topological Entropy of a Graph Map

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Acta Mathematica Sinica,English Series 2018年第2期34卷 194-208页

作者： Tai Xiang SUN Guangxi Key Laboratory Cultivation Base of Cross-border E-commerce Intelligent Information Processing Guangxi University of Finance and Economics

Let G be a graph and f： G → G be a continuous map. Denote by h（f）, P（f）, AP（f）, R（f） and w（x, f） the topological entropy of f, the set of periodic points of f, the set of almost periodic points of f, the set of recurrent points of f and the w-limit set of x under f, respectively. In this paper, we show that the following statements are equivalent：（1） h（f）〉 O. （2） There exists an x ∈ G such that w（x, f） ∩ P（f） ≠θ and w（x, f） is an infinite set. （3） There exists an x ∈ G such that w（x, f） contains two minimal sets. （4） There exist x, y ∈G such that w（x, f） - w（y, f） is an uncountable set andw（y,f）∩w（x,f）≠θ. （5） There exist anx C Gand a closed subset A w（x,f） with f（A） A such that w（x,f） - A is an uncountable set. （6） R（f） - nP（f） ≠θ. （7） f｜P（f） is not pointwise equicontinuous.

关键词： Topological entropy periodic point w-limit set recurrent point

On the density of Birkhoff sums for Anosov diffeomorphisms

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Science China Mathematics 2022年第2期65卷 319-330页

作者： Shaobo Gan Yi Shi Mingyang Xia School of Mathematical Sciences Peking UniversityBeijing 100871China

Let f : M → M be an Anosov diffeomorphism on a nilmanifold. We consider Birkhoff sums for a Holder continuous observation along periodic orbits. We show that if there are two Birkhoff sums distributed at both sides o... 详细信息

关键词： Anosov diffeomorphism Birkhoff sum periodic point nilmanifold

An Arc as the Inverse Limit of a Single Bonding Map of Type N on [0,1]

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Acta Mathematica Sinica,English Series 2004年第5期20卷 925-932页

作者： HongMO SongQuanSHI FangPingZENG JieHuaMAI DepartmentofAppliedMathematics ShanghaiUniversityofFinanceEconomicsShanghai200433P.R.China TheKeyLaboratoryofComplexSystemandIntelligentScience InstituteofAutomationChineseAcademyofSciencesBeifing100080P.R.China InstituteofMathematics GuangxiUniversityNanning530004P.R.China InstituteofMathematics ShantouUniversityShantou515062P.R.China

Let I=[0,1],c_1,c_2 ∈(0,1)with c_1a(resp.f(c_1)f(c_2),(resp.f has more than one fixed point,furthermore,if A≠φ and B\{b}≠φ,f(c_1)≤b or if bc_2 and f(c_2)point,a single period two o... 详细信息

Let I=[0,1],c_1,c_2 ∈(0,1)with c_1point,a period two orbit, but no points of period greater than two or f has more than one fixed point but no points of other periods,furthermore,if A≠φ and B≠φ,then f(c2)>a(resp.f(c_1)point,furthermore,if B≠φ and A\{a}≠φ,f(c_2)≥a or if a_1f(c_2),(resp.f has more than one fixed point,furthermore,if A≠φ and B\{b}≠φ,f(c_1)≤b or if bc_2 and f(c_2)point,a single period two orbit lying in I\(u,v)but no points of period greater than two,where u,v ∈[c_1,c_2] such that f(u)=c_2 and f(v)=c_1.

关键词： Inverse limit Arc Uninmodel map periodic point

Equicontinuity of Maps on a Dendrite with Finite Branch points

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Acta Mathematica Sinica,English Series 2017年第8期33卷 1125-1130页

作者： Tai Xiang SUN Guang Wang SU Hong Jian XI Xin KONG College of Information and Statistics Guangxi University of Finance and Economics College of Mathematics and Information Science Guangxi University

Let （T, d） be a dendrite with finite branch points and f be a continuous map from T to T. Denote by w（x, f） and P（f） the w-limit set of x under f and the set of periodic points of f, respectively. Write Ω（x, f） = {yl there exist a sequence of points xk ∈ T and a sequence of positive integers n1 〈 n2 〈 … such that lim k→∞ Xk = X and lim k→∞ f nk （xk） = y}. In this paper, we show that the following statements are equivalent：（1） f is equicontinuous. （2） w（x, f） = Ω（x, f） for any x ∈ T. （3） ∩ ∞ n=1 f n（T） = P（f）, and w（x, f） is a periodic orbit for every x ∈ T and map h： x → w（x, f）（x ∈ T） is continuous. （4） Ω（x, f） is a periodic orbit for any x ∈ T.

关键词： Dendrite map equicontinuity periodic point ε-limit set

Special α-limit points and unilateral γ-limit points for graph maps

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Science China Mathematics 2011年第9期54卷 2013-2018页

作者： SUN TaiXiang XI HongJian LIANG HaiLan College of Mathematics and Information Science Guangxi University Nanning 530004 China Department of Mathematics Guangxi College of Finance and Economics Nanning 530003 China

Let G be a graph and f ： G → G be a continuous map. Denote by P（f）, R（f）, SA（f） and UF（f） the sets of periodic points, recurrent points, special α-limit points and unilateral γ-limit points of f, respectiv... 详细信息

关键词： graph map recurrent point periodic point special α-limit point unilateral γ-limit point

A Product Formula for the Twisted Lefschetz Zeta Function

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Northeastern Mathematical Journal 1998年第3期14卷 267-272页

作者：黄保军

ＡＰｒｏｄｕｃｔＦｏｒｍｕｌａｆｏｒｔｈｅＴｗｉｓｔｅｄＬｅｆｓｃｈｅｔｚＺｅｔａＦｕｎｃｔｉｏｎＨｕａｎｇＢａｏｊｕｎ（黄保军）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨｕａｉｂｅｉＣｏａｌＭｉｎｉｎｇＴ... 详细信息

ＡＰｒｏｄｕｃｔＦｏｒｍｕｌａｆｏｒｔｈｅＴｗｉｓｔｅｄＬｅｆｓｃｈｅｔｚＺｅｔａＦｕｎｃｔｉｏｎＨｕａｎｇＢａｏｊｕｎ（黄保军）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨｕａｉｂｅｉＣｏａｌＭｉｎｉｎｇＴｅａｃｈｅｒｓＣｏｌｅｇｅ，Ｈｕａ...

关键词： and Phrases Twisted Lefschetz Zeta Function periodic point Representation of a Groupring