scholarly journals ON ERGODIC BEHAVIOR OF p-ADIC DYNAMICAL SYSTEMS

2001 ◽  
Vol 04 (04) ◽  
pp. 569-577 ◽  
Author(s):  
MATTHIAS GUNDLACH ◽  
ANDREI KHRENNIKOV ◽  
KARL-OLOF LINDAHL
Keyword(s):  
Dynamical Systems ◽  
Natural Number ◽  
Unit Circle ◽  
Complex Plane ◽  
Haar Measure ◽  
Analogous Result ◽  
Topologically Transitive ◽  
Ergodic Behavior ◽  
Adic Case

Monomial mappings, x ↦ xn, are topologically transitive and ergodic with respect to Haar measure on the unit circle in the complex plane. In this paper we obtain an analogous result for monomial dynamical systems over p-adic numbers. The process is, however, not straightforward. The result will depend on the natural number n. Moreover, in the p-adic case we will not have ergodicity on the unit circle, but on the circles around the point 1.

Analysis of Snapback Repellers Using Methods of Symbolic Computation

2019 ◽  
Vol 29 (04) ◽  
pp. 1950054 ◽  
Author(s):  
Bo Huang ◽  
Wei Niu
Keyword(s):  
Dynamical Systems ◽  
Fixed Points ◽  
Unit Circle ◽  
Symbolic Computation ◽  
Complex Plane ◽  
Discrete Dynamical Systems ◽  
Algorithmic Approach ◽  
Algebraic Criterion ◽  
Algebraic Problem

This paper presents an algebraic criterion for determining whether all the zeros of a given polynomial are outside the unit circle in the complex plane. This criterion is used to deduce critical algebraic conditions for the occurrence of chaos in multidimensional discrete dynamical systems based on a modified Marotto’s theorem developed by Li and Chen (called “Marotto–Li–Chen theorem”). Using these algebraic conditions we reduce the problem of analyzing chaos induced by snapback repeller to an algebraic problem, and propose an algorithmic approach to solve this algebraic problem by means of symbolic computation. The proposed approach is effective as shown by several examples and can be used to determine the possibility that all the fixed points are snapback repellers.

Beta dynamical systems over the formal fields

2014 ◽  
Vol 51 (4) ◽  
pp. 454-465
Author(s):  
Lu-Ming Shen ◽  
Huiping Jing
Keyword(s):  
Dynamical Systems ◽  
Finite Field ◽  
Haar Measure ◽  
Laurent Series ◽  
Formal Laurent Series ◽  
Almost All

Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q ((X^{ - 1} ))$$ \end{document} denote the formal field of all formal Laurent series x = Σ n=ν∞anX−n in an indeterminate X, with coefficients an lying in a given finite field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}. For any \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} with deg β > 1, it is known that for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$x \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} (with respect to the Haar measure), x is β-normal. In this paper, we show the inverse direction, i.e., for any x, for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document}, x is β-normal.

scholarly journals Stability analysis of flame fronts: Dynamical systems approach in the complex plane

1999 ◽  
Vol 59 (3) ◽  
pp. 2587-2593 ◽  
Author(s):  
Oleg Kupervasser ◽  
Zeev Olami ◽  
Itamar Procaccia
Keyword(s):  
Stability Analysis ◽  
Dynamical Systems ◽  
Complex Plane ◽  
Systems Approach

scholarly journals Hyponormal Toeplitz operators on H2(T) with polynomial symbols

1996 ◽  
Vol 144 ◽  
pp. 179-182 ◽  
Author(s):  
Dahai Yu
Keyword(s):  
Functional Equation ◽  
Hardy Space ◽  
Closed Form ◽  
Unit Circle ◽  
Toeplitz Operator ◽  
Explicit Expression ◽  
Complex Plane ◽  
Toeplitz Operators ◽  
Computing Process

Let T be the unit circle on the complex plane, H2(T) be the usual Hardy space on T, Tø be the Toeplitz operator with symbol Cowen showed that if f1 and f2 are functions in H such that is in Lø, then Tf is hyponormal if and only if for some constant c and some function g in H∞ with Using it, T. Nakazi and K. Takahashi showed that the symbol of hyponormal Toeplitz operator Tø satisfies and and they described the ø solving the functional equation above. Both of their conditions are hard to check, T. Nakazi and K. Takahashi remarked that even “the question about polynomials is still open” [2]. Kehe Zhu gave a computing process by way of Schur’s functions so that we can determine any given polynomial ø such that Tø is hyponormal [3]. Since no closed-form for the general Schur’s function is known, it is still valuable to find an explicit expression for the condition of a polynomial á such that Tø is hyponormal and depends only on the coefficients of ø, here we have one, it is elementary and relatively easy to check. We begin with the most general case and the following Lemma is essential.

scholarly journals On the Asymptotic Behavior of Functions Harmonic in a Disc

1966 ◽  
Vol 28 ◽  
pp. 187-191
Author(s):  
J. E. Mcmillan
Keyword(s):  
Asymptotic Behavior ◽  
Unit Circle ◽  
Complex Plane ◽  
Unit Disc ◽  
Open Unit ◽  
Continuous Curve ◽  
Open Unit Disc ◽  
Boundary Path

Let D be the open unit disc, and let C be the unit circle in the complex plane. Let f be a (real-valued) function that is harmonic in D. A simple continuous curve β: z(t) (0≦t<1) contained in D such that |z(t)|→1 as t→1 is a boundary path with end (the bar denotes closure).

scholarly journals On a Uniqueness Theorem

1969 ◽  
Vol 35 ◽  
pp. 151-157 ◽  
Author(s):  
V. I. Gavrilov
Keyword(s):  
Unit Circle ◽  
Uniqueness Theorem ◽  
Unit Disk ◽  
Complex Plane ◽  
Open Unit ◽  
Open Unit Disk ◽  
Extended Complex ◽  
Extended Complex Plane

1. Let D be the open unit disk and r be the unit circle in the complex plane, and denote by Q the extended complex plane or the Rie-mann sphere.

MONOGENEITY IN CYCLOTOMIC FIELDS

2010 ◽  
Vol 06 (07) ◽  
pp. 1589-1607 ◽  
Author(s):  
LEANNE ROBERTSON
Keyword(s):  
Unit Circle ◽  
Number Field ◽  
Complex Plane ◽  
Cyclotomic Field ◽  
Difficult Problem ◽  
Cyclotomic Fields ◽  
Ring Extension ◽  
Integral Basis ◽  
Ring Of Integers ◽  
Integral Bases

A number field is said to be monogenic if its ring of integers is a simple ring extension ℤ[α] of ℤ. It is a classical and usually difficult problem to determine whether a given number field is monogenic and, if it is, to find all numbers α that generate a power integral basis {1, α, α2, …, αk} for the ring. The nth cyclotomic field ℚ(ζn) is known to be monogenic for all n, and recently Ranieri proved that if n is coprime to 6, then up to integer translation all the integral generators for ℚ(ζn) lie on the unit circle or the line Re (z) = 1/2 in the complex plane. We prove that this geometric restriction extends to the cases n = 3k and n = 4k, where k is coprime to 6. We use this result to find all power integral bases for ℚ(ζn) for n = 15, 20, 21, 28. This leads us to a conjectural solution to the problem of finding all integral generators for cyclotomic fields.

scholarly journals Shadowing and -limit sets of circular Julia sets

2013 ◽  
Vol 35 (4) ◽  
pp. 1045-1055 ◽  
Author(s):  
ANDREW D. BARWELL ◽  
JONATHAN MEDDAUGH ◽  
BRIAN E. RAINES
Keyword(s):  
Unit Circle ◽  
Complex Plane ◽  
Julia Sets ◽  
Closed Set ◽  
Quadratic Polynomials ◽  
Limit Sets ◽  
Kneading Sequence

AbstractIn this paper we consider quadratic polynomials on the complex plane${f}_{c} (z)= {z}^{2} + c$and their associated Julia sets,${J}_{c} $. Specifically, we consider the case that the kneading sequence is periodic and not an$n$-tupling. In this case${J}_{c} $contains subsets that are homeomorphic to the unit circle, usually infinitely many disjoint such subsets. We prove that${f}_{c} : {J}_{c} \rightarrow {J}_{c} $has shadowing, and we classify all$\omega $-limit sets for these maps by showing that a closed set$R\subseteq {J}_{c} $is internally chain transitive if, and only if, there is some$z\in {J}_{c} $with$\omega (z)= R$.

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