scholarly journals A gap principle for dynamics

2010 ◽  
Vol 146 (4) ◽  
pp. 1056-1072 ◽  
Author(s):  
Robert L. Benedetto ◽  
Dragos Ghioca ◽  
Pär Kurlberg ◽  
Thomas J. Tucker
Keyword(s):  
Rational Functions ◽  
Arithmetic Progressions ◽  
Proper Subvariety ◽  
Gap Principle ◽  
Sparse Set

AbstractLet f1,…,fg∈ℂ(z) be rational functions, let Φ=(f1,…,fg) denote their coordinate-wise action on (ℙ1)g, let V ⊂(ℙ1)g be a proper subvariety, and let P be a point in (ℙ1)g(ℂ). We show that if 𝒮={n≥0:Φn(P)∈V (ℂ)} does not contain any infinite arithmetic progressions, then 𝒮 must be a very sparse set of integers. In particular, for any k and any sufficiently large N, the number of n≤N such that Φn(P)∈V (ℂ) is less than log kN, where log k denotes the kth iterate of the log function. This result can be interpreted as an analogue of the gap principle of Davenport–Roth and Mumford.

scholarly journals Powers in Orbits of Rational Functions: Cases of an Arithmetic Dynamical Mordell–Lang Conjecture

2019 ◽  
Vol 71 (4) ◽  
pp. 773-817
Author(s):  
Jordan Cahn ◽  
Rafe Jones ◽  
Jacob Spear
Keyword(s):  
Irreducible Component ◽  
Characteristic Zero ◽  
Rational Functions ◽  
Arithmetic Progressions ◽  
Additional Analysis ◽  
Finitely Generated ◽  
Index Set ◽  
Value Set

AbstractLet $K$ be a finitely generated field of characteristic zero. For fixed $m\geqslant 2$, we study the rational functions $\unicode[STIX]{x1D719}$ defined over $K$ that have a $K$-orbit containing infinitely many distinct $m$-th powers. For $m\geqslant 5$ we show that the only such functions are those of the form $cx^{j}(\unicode[STIX]{x1D713}(x))^{m}$ with $\unicode[STIX]{x1D713}\in K(x)$, and for $m\leqslant 4$ we show that the only additional cases are certain Lattès maps and four families of rational functions whose special properties appear not to have been studied before.With additional analysis, we show that the index set $\{n\geqslant 0:\unicode[STIX]{x1D719}^{n}(a)\in \unicode[STIX]{x1D706}(\mathbb{P}^{1}(K))\}$ is a union of finitely many arithmetic progressions, where $\unicode[STIX]{x1D719}^{n}$ denotes the $n$-th iterate of $\unicode[STIX]{x1D719}$ and $\unicode[STIX]{x1D706}\in K(x)$ is any map Möbius-conjugate over $K$ to $x^{m}$. When the index set is infinite, we give bounds on the number and moduli of the arithmetic progressions involved. These results are similar in flavor to the dynamical Mordell–Lang conjecture, and motivate a new conjecture on the intersection of an orbit with the value set of a morphism. A key ingredient in our proofs is a study of the curves $y^{m}=\unicode[STIX]{x1D719}^{n}(x)$. We describe all $\unicode[STIX]{x1D719}$ for which these curves have an irreducible component of genus at most 1, and show that such $\unicode[STIX]{x1D719}$ must have two distinct iterates that are equal in $K(x)^{\ast }/K(x)^{\ast m}$.

Application of the exp(-φ)-expansion method to the Pochhammer-Chree equation

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3347-3354 ◽  
Author(s):  
Nematollah Kadkhoda ◽  
Michal Feckan ◽  
Yasser Khalili
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exact Solutions ◽  
Present Article ◽  
Analytical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Rational Functions ◽  
Expansion Method ◽  
Exponential Functions

In the present article, a direct approach, namely exp(-?)-expansion method, is used for obtaining analytical solutions of the Pochhammer-Chree equations which have a many of models. These solutions are expressed in exponential functions expressed by hyperbolic, trigonometric and rational functions with some parameters. Recently, many methods were attempted to find exact solutions of nonlinear partial differential equations, but it seems that the exp(-?)-expansion method appears to be efficient for finding exact solutions of many nonlinear differential equations.

scholarly journals Global Well-Posedness of the Ocean Primitive Equations with Nonlinear Thermodynamics

2021 ◽  
Vol 23 (3) ◽  
Author(s):  
Peter Korn
Keyword(s):  
Equation Of State ◽  
Equations Of State ◽  
Boussinesq Equations ◽  
Climate Science ◽  
Rational Functions ◽  
Ocean Dynamics ◽  
Global Ocean ◽  
Primitive Equations ◽  
Well Posedness ◽  
Nonlinear Thermodynamics

AbstractWe consider the hydrostatic Boussinesq equations of global ocean dynamics, also known as the “primitive equations”, coupled to advection–diffusion equations for temperature and salt. The system of equations is closed by an equation of state that expresses density as a function of temperature, salinity and pressure. The equation of state TEOS-10, the official description of seawater and ice properties in marine science of the Intergovernmental Oceanographic Commission, is the most accurate equations of state with respect to ocean observation and rests on the firm theoretical foundation of the Gibbs formalism of thermodynamics. We study several specifications of the TEOS-10 equation of state that comply with the assumption underlying the primitive equations. These equations of state take the form of high-order polynomials or rational functions of temperature, salinity and pressure. The ocean primitive equations with a nonlinear equation of state describe richer dynamical phenomena than the system with a linear equation of state. We prove well-posedness for the ocean primitive equations with nonlinear thermodynamics in the Sobolev space $${{\mathcal {H}}^{1}}$$ H 1 . The proof rests upon the fundamental work of Cao and Titi (Ann. Math. 166:245–267, 2007) and also on the results of Kukavica and Ziane (Nonlinearity 20:2739–2753, 2007). Alternative and older nonlinear equations of state are also considered. Our results narrow the gap between the mathematical analysis of the ocean primitive equations and the equations underlying numerical ocean models used in ocean and climate science.

scholarly journals A dominated convergence theorem for Eisenstein series

2021 ◽  
Author(s):  
Johann Franke
Keyword(s):  
Fourier Series ◽  
Dirichlet Series ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Convergence Theorem ◽  
Rational Functions ◽  
New Approach ◽  
Series Representations ◽  
Dominated Convergence ◽  
Generalized Dirichlet

AbstractBased on the new approach to modular forms presented in [6] that uses rational functions, we prove a dominated convergence theorem for certain modular forms in the Eisenstein space. It states that certain rearrangements of the Fourier series will converge very fast near the cusp $$\tau = 0$$ τ = 0 . As an application, we consider L-functions associated to products of Eisenstein series and present natural generalized Dirichlet series representations that converge in an expanded half plane.

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