12.2 Height functions in dynamics
8 min read•august 21, 2024
Height functions are essential tools in arithmetic geometry, quantifying the complexity of algebraic objects. They assign non-negative real numbers to points, reflecting arithmetic intricacy and enabling the study of on varieties.
In dynamics, height functions are crucial for analyzing long-term behavior of systems on algebraic varieties. They help study periodic points, orbit growth rates, and provide a vital link between number theory and dynamical systems in arithmetic geometry.
Definition of height functions
- Height functions quantify the arithmetic complexity of algebraic objects in arithmetic geometry
- Serve as fundamental tools for studying rational points on algebraic varieties and dynamical systems
Concept of height
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- Measures the arithmetic complexity or size of algebraic numbers or points on varieties
- Assigns a non-negative real number to each point, reflecting its arithmetic intricacy
- Increases as the coordinates of a point become more arithmetically complex
- Provides a way to order points based on their arithmetic sophistication
Importance in dynamics
- Enables the study of long-term behavior of dynamical systems on algebraic varieties
- Facilitates the analysis of periodic points and their distribution in algebraic dynamical systems
- Helps in understanding the growth rates of orbits under iteration of rational maps
- Provides a crucial link between number theory and dynamical systems in arithmetic geometry
Types of height functions
Weil height
- Fundamental height function defined on projective spaces over number fields
- Measures the arithmetic complexity of projective coordinates
- Computed as the logarithm of the product of local contributions from all places
- Satisfies important properties like and the
- Forms the basis for more sophisticated height functions in arithmetic geometry
Néron-Tate height
- Refinement of the for points on
- Quadratic in nature, meaning it satisfies for any point P
- Vanishes precisely on torsion points of the abelian variety
- Crucial in the study of rational points on elliptic curves and higher-dimensional abelian varieties
- Used in the proof of the Mordell-Weil theorem on the finite generation of rational points
Canonical height
- Generalizes the to dynamical systems
- Defined for rational maps on projective spaces or more general varieties
- Satisfies a functional equation related to the dynamics of the map
- Helps in studying the distribution of periodic points and
- Used to prove equidistribution results in
Properties of height functions
Functoriality
- Height functions behave well under between algebraic varieties
- For a morphism f: X → Y, the height of f(P) relates to the height of P
- Allows for the transfer of height-related results between different varieties
- Crucial in studying rational points on varieties through their embeddings in projective spaces
- Enables the use of height functions in the study of moduli spaces and parameter spaces
Northcott property
- States that for any bounded height and degree, there are only finitely many algebraic numbers
- Implies that points of bounded height on a variety form a discrete set
- Essential in proving finiteness results for rational points on varieties
- Allows for effective computations in many number-theoretic algorithms
- Provides a link between arithmetic complexity and geometric distribution of points
Logarithmic nature
- Height functions typically grow logarithmically with respect to arithmetic operations
- Reflects the multiplicative nature of complexity in algebraic number theory
- Leads to connections with and other logarithmic measures in number theory
- Allows for the use of techniques from analysis and in arithmetic geometry
- Facilitates the study of asymptotic behavior of points under iteration in dynamical systems
Height functions for polynomials
Mahler measure
- Measures the size of a polynomial using its coefficients and roots
- Defined as the product of the absolute values of roots outside the unit circle
- Relates to the height of the point in projective space formed by the polynomial's coefficients
- Used in transcendence theory and the study of algebraic numbers
- Connects polynomial heights to heights of algebraic numbers and points on varieties
Polynomial height vs degree
- Height of a polynomial increases with both its degree and the size of its coefficients
- Relationship between height and degree crucial in many finiteness theorems
- Used in the study of integer polynomials and their factorization properties
- Impacts the complexity of algorithms for polynomial manipulation and factorization
- Provides insights into the distribution of algebraic numbers of bounded degree and height
Applications in dynamics
Fixed points detection
- Height functions help identify fixed points of rational maps
- Fixed points often have bounded height under iteration of the map
- Allows for effective algorithms to find all fixed points up to a given height
- Crucial in studying the dynamics of rational maps on projective spaces
- Provides a computational approach to questions in arithmetic dynamics
Periodic orbits analysis
- Height functions aid in detecting and studying periodic orbits of dynamical systems
- Periodic points often exhibit specific growth patterns in their heights under iteration
- Enables the study of the distribution of periodic points of different periods
- Helps in proving results about the finiteness or infinitude of periodic orbits
- Connects the arithmetic properties of periodic points to their dynamical behavior
Equidistribution of points
- Height functions used to study the asymptotic distribution of points of bounded height
- Points of small height often equidistribute with respect to certain measures
- Leads to connections between arithmetic geometry and ergodic theory
- Provides insights into the structure of rational points on varieties
- Used in proving density results for rational points on algebraic varieties
Height functions on varieties
Ample line bundles
- Height functions associated with on projective varieties
- Generalizes the notion of height from projective spaces to more general varieties
- Crucial in the study of rational points on higher-dimensional varieties
- Allows for the formulation of conjectures like the Batyrev-Manin conjecture
- Connects arithmetic properties of varieties to their geometric structure
Arithmetic vs geometric height
- considers contributions from all places of a number field
- focuses on the contribution from archimedean places
- Arithmetic height crucial for number-theoretic applications
- Geometric height more directly related to the geometry of the variety
- Interplay between arithmetic and geometric heights important in arithmetic geometry
Computational aspects
Algorithms for height calculation
- Efficient algorithms developed for computing heights of points on varieties
- Involves techniques from computational number theory and algebraic geometry
- Crucial for practical applications of height functions in number theory
- Requires careful handling of precision issues in floating-point arithmetic
- Enables large-scale computational experiments in arithmetic geometry
Complexity considerations
- Computational complexity of height calculations depends on the type of variety and height function
- Trade-offs between accuracy and speed in height computations
- Impacts the feasibility of large-scale searches for points of bounded height
- Influences the design of algorithms for solving
- Connects computational aspects of arithmetic geometry to complexity theory
Height bounds
Lehmer's conjecture
- Proposes a lower bound on the height of algebraic numbers that are not roots of unity
- States that the height of an algebraic number α of degree d satisfies for some constant c > 0
- Remains one of the most important open problems in algebraic number theory
- Has implications for the distribution of algebraic numbers and points on varieties
- Connects to questions about the minimal polynomial of algebraic integers
Bogomolov conjecture
- Concerns the distribution of points of small height on subvarieties of abelian varieties
- States that points of height zero on a subvariety are contained in a finite union of translates of abelian subvarieties
- Proved for curves by Ullmo and for subvarieties of abelian varieties by Zhang
- Has applications to the study of and higher-dimensional varieties
- Connects arithmetic properties of varieties to their geometric structure
Dynamical systems and heights
Arithmetic dynamics
- Studies dynamical systems from an arithmetic perspective using height functions
- Investigates the behavior of rational maps on algebraic varieties over number fields
- Explores connections between classical dynamics and number theory
- Uses height functions to analyze periodic points, preperiodic points, and orbits
- Applies techniques from ergodic theory and complex dynamics to arithmetic questions
Preperiodic points
- Points with finite forward orbit under a dynamical system
- Height functions used to study the distribution and structure of preperiodic points
- Conjectures like the uniform conjecture relate heights to preperiodicity
- Important in understanding the global dynamics of rational maps
- Connects to questions about rational points on varieties and algebraic cycles
Height functions in number theory
Diophantine equations
- Height functions crucial in the study of solutions to Diophantine equations
- Used in effective methods for finding all solutions up to a given height
- Helps in proving finiteness results for solutions to certain types of equations
- Connects to the theory of Diophantine approximation and transcendence theory
- Provides a quantitative approach to questions in Diophantine geometry
Rational points on curves
- Height functions essential in studying rational points on algebraic curves
- Used in algorithms for finding rational points of bounded height on curves
- Crucial in the proof of Faltings' theorem (Mordell conjecture) on rational points
- Helps in understanding the distribution of rational points on curves of higher genus
- Connects to deep conjectures in arithmetic geometry like the Birch and Swinnerton-Dyer conjecture
Advanced topics
Adelic metrics
- Generalizes height functions to adelic setting, considering all places simultaneously
- Provides a unified framework for studying arithmetic properties of varieties
- Allows for the formulation of equidistribution results in terms of adelic measures
- Connects to the theory of arithmetic intersection theory on arithmetic varieties
- Used in the study of arithmetic dynamical systems over global fields
Local height functions
- Height functions defined with respect to a single place of a number field
- Sum of local heights at all places gives the global height function
- Important in studying the p-adic and archimedean behavior of points on varieties
- Used in the theory of p-adic dynamics and non-archimedean analysis
- Connects local analytic properties of varieties to their global arithmetic behavior
Connections to other areas
Ergodic theory
- Height functions used to define measures on algebraic varieties
- Equidistribution results for points of bounded height connect to ergodic theory
- Techniques from ergodic theory applied to study the distribution of rational points
- Connects arithmetic questions to dynamical systems on homogeneous spaces
- Provides a bridge between number theory and measure-theoretic dynamics
Algebraic geometry
- Height functions intrinsically connected to the geometry of algebraic varieties
- Used in the study of moduli spaces and parameter spaces in algebraic geometry
- Connects arithmetic properties of varieties to their complex geometric structure
- Important in the theory of arithmetic surfaces and higher-dimensional arithmetic varieties
- Provides a quantitative approach to questions in birational geometry and minimal model theory
Key Terms to Review (40)
Abelian varieties: Abelian varieties are higher-dimensional generalizations of elliptic curves, defined as complete algebraic varieties that have a group structure. These varieties play a critical role in various areas of mathematics, including number theory and algebraic geometry, and they exhibit deep connections to concepts like complex multiplication, zeta functions, and modular forms.
Adelic metrics: Adelic metrics are a way to measure the size of elements in an algebraic variety, capturing both local and global properties through a unified framework. They arise from the concept of adeles, which allow us to study arithmetic properties in a more comprehensive way by considering all completions of the number field simultaneously. This duality between local and global aspects is particularly important when analyzing height functions, which play a crucial role in dynamics, particularly in understanding the growth rates of sequences and their convergence behaviors.
Algorithms for height calculation: Algorithms for height calculation refer to systematic procedures used to determine the height of points on algebraic varieties, particularly in the context of arithmetic geometry. These algorithms play a crucial role in understanding the dynamics of rational maps and their interactions with the geometry of these varieties, contributing to the broader study of height functions in dynamical systems.
Ample Line Bundles: An ample line bundle is a type of line bundle on a projective variety that allows for the embedding of the variety into projective space. This concept is crucial in algebraic geometry as it helps in understanding the properties of varieties through their associated divisors and intersections. Ample line bundles ensure that there is a sufficient number of global sections, which can be used to construct morphisms to projective spaces, reflecting the geometric structure of the variety.
Arithmetic Dynamics: Arithmetic dynamics is the study of the interplay between arithmetic properties of numbers and dynamical systems, often focusing on how algebraic and geometric structures evolve under iteration. It connects number theory with dynamical systems, revealing how properties like orbits, fixed points, and stability can have deep implications for understanding complex numbers and their relationships. This field is particularly relevant when discussing complex multiplication and the use of height functions to analyze dynamical behaviors over time.
Arithmetic height: Arithmetic height is a function that assigns a numerical value to points on an algebraic variety or to rational points, capturing the complexity of these points in terms of their coordinates. It connects with concepts like dynamics and number theory, offering insights into the distribution of rational points and their behavior under iterative processes. By measuring the size of these points, arithmetic height plays a crucial role in understanding their properties in both algebraic geometry and dynamical systems.
Bogomolov Conjecture: The Bogomolov Conjecture is a statement in algebraic geometry that suggests the existence of a bounded degree for the set of rational points on certain types of varieties, particularly those with specific geometric properties. This conjecture is closely tied to the study of height functions and dynamical systems, asserting that for families of varieties with a particular structure, there are restrictions on how rational points can behave, impacting their distribution.
Boundedness: Boundedness refers to the property of a mathematical object being confined within a certain limit or boundary. In the context of height functions in dynamics, boundedness indicates that the values of these height functions do not exceed a specified upper limit, which is crucial for understanding the stability and behavior of dynamical systems.
Canonical height: Canonical height is a measure used in arithmetic geometry to quantify the complexity of rational points on algebraic varieties, especially in the context of abelian varieties. This concept helps to connect the geometry of the variety with the arithmetic properties of its points, facilitating the study of dynamical systems and the behavior of points under iteration. The canonical height allows for the computation of an efficient and standardized way to assess how 'large' or 'complicated' a point is within its geometric framework.
Complexity considerations: Complexity considerations refer to the analysis of the computational and structural challenges involved in mathematical concepts and frameworks, particularly how these challenges can affect the understanding and outcomes of various problems. In arithmetic geometry, these considerations help in evaluating the feasibility of certain operations, understanding the dynamics of functions, and assessing periodic behavior in systems.
David Mumford: David Mumford is a prominent mathematician known for his work in algebraic geometry, particularly in the areas of modular forms and algebraic curves. His contributions have significantly advanced the understanding of complex tori, modular curves, and other structures relevant to arithmetic geometry.
Diophantine equations: Diophantine equations are polynomial equations where the solutions are required to be integers or whole numbers. They are central to number theory and often relate to the search for rational points on algebraic varieties, connecting various mathematical concepts like algebraic geometry, arithmetic, and modular forms.
Dynamical Degrees: Dynamical degrees are numerical invariants that measure the growth rate of orbits in dynamical systems, particularly in the context of algebraic varieties. They reflect how the complexity of the dynamical behavior changes under iteration, linking geometric properties of varieties with their dynamical behavior. By analyzing these degrees, one can gain insights into the structure and stability of dynamical systems in algebraic geometry.
Endomorphisms: Endomorphisms are morphisms from a mathematical object to itself, preserving the structure of that object. In the context of algebraic geometry, these transformations can describe the dynamics of points on varieties, making them crucial for understanding properties like height functions. The study of endomorphisms often leads to insights into the behavior of cycles and how they evolve under repeated applications of the transformation.
Equidistribution of Points: Equidistribution of points refers to the phenomenon where a sequence of points becomes uniformly distributed over a certain space as the sequence progresses. This concept is particularly important in dynamics and height functions, where it allows for understanding how points behave under iterations of maps or transformations, often revealing patterns and symmetries in the data being studied.
Ergodic theory: Ergodic theory is a branch of mathematics that studies the long-term average behavior of dynamical systems. It connects statistical mechanics and measure theory, focusing on the idea that, under certain conditions, the time average of a function along the trajectories of the system equals the space average over the entire space. This concept is crucial for understanding how height functions behave over time in dynamic systems.
Ergodicity: Ergodicity is a property of dynamical systems where, over a long period, the time spent by a system in a particular state is proportional to the volume of the state in phase space. This concept connects to the idea that average behavior over time reflects the overall structure of the system. In essence, ergodic systems exhibit predictable long-term statistical behavior that aligns with their invariant measures, making them significant in studying both dynamics and number theory.
Expanding Maps: Expanding maps are dynamical systems where points in a space move away from each other over time, reflecting a form of growth or expansion. These maps are essential in the study of dynamical systems and often exhibit chaotic behavior, showcasing sensitive dependence on initial conditions. Understanding expanding maps helps analyze the behavior of iterative processes and their impact on height functions in various contexts.
Fixed Points Detection: Fixed points detection refers to the process of identifying points in a dynamical system where the system remains unchanged under a given transformation or map. In the context of height functions in dynamics, this concept is crucial as it helps to analyze the stability and behavior of orbits and maps by locating points that do not change even when the system evolves over time.
Functoriality: Functoriality is a fundamental concept in mathematics that refers to the property of a mathematical structure or operation preserving certain relationships when mappings between objects are applied. This notion ensures that if one has a morphism (or map) between two structures, one can induce a corresponding morphism between their images, thus maintaining the integrity of the relationships involved. It is essential in various areas such as algebraic geometry and number theory, where it connects different mathematical objects through coherent transformations.
Geometric Height: Geometric height is a numerical measure associated with points on an algebraic variety, which quantifies the complexity of the point in relation to a given height function. It reflects the growth rate of the coordinates of the point when expressed in a suitable projective embedding. This concept plays a critical role in understanding dynamical systems on varieties, especially in relation to how points move under iteration and how their heights evolve.
Green's Conjecture: Green's Conjecture is a statement in algebraic geometry that relates to the behavior of certain algebraic curves and their embeddings into projective space. It asserts that the syzygies of a smooth projective curve can be fully described by the properties of its degree and genus, particularly in relation to the canonical embedding. This conjecture links algebraic geometry with the study of height functions and dynamics through its implications on the growth of these functions as they pertain to curve families.
Jean-Pierre Serre: Jean-Pierre Serre is a prominent French mathematician known for his significant contributions to algebraic geometry, topology, and number theory. His work has deeply influenced various fields within mathematics, particularly in relation to the development of modern concepts and conjectures surrounding arithmetic geometry.
Kullback-Leibler Divergence: Kullback-Leibler divergence is a statistical measure that quantifies how one probability distribution diverges from a second, expected probability distribution. It's commonly used in various fields like information theory and machine learning to assess the difference between distributions, which can be important when evaluating models or understanding data dynamics. This concept helps in measuring how much information is lost when one distribution is used to approximate another.
Lehmer's Conjecture: Lehmer's Conjecture proposes that there are only finitely many non-zero integers with a specific property related to their heights, particularly concerning their integer sequences and growth rates. This conjecture plays a critical role in the understanding of height functions in dynamics, as it relates to how certain sequences behave under iteration and the associated growth properties that emerge from dynamical systems.
Local height functions: Local height functions are tools used in arithmetic geometry to measure the complexity of points on algebraic varieties. They provide a way to quantify how 'high' or 'far' a point is from a specified subvariety, essentially helping to analyze dynamics and the behavior of rational points in various settings. By relating these heights to the dynamics of rational functions, local height functions become essential in studying the distribution of points and their growth in arithmetic contexts.
Logarithmic nature: The logarithmic nature refers to a relationship or function that exhibits growth or decay at a rate proportional to the logarithm of a variable. This concept is crucial in understanding height functions in dynamics, as it helps describe how certain mathematical properties scale and behave over time, particularly in the context of dynamical systems where exponential growth can lead to complex behaviors.
Mahler Measure: The Mahler measure is a concept in mathematics that quantifies the size of a polynomial based on its roots, particularly focusing on the growth of these roots in relation to the unit circle. It is often defined as the product of the absolute values of its roots, scaled by a logarithmic transformation. This measure connects deeply with number theory and algebraic geometry, revealing insights about polynomial behavior and dynamics under iterations.
Metric spaces: A metric space is a set equipped with a function that defines a distance between any two elements in the set. This function, known as a metric, must satisfy certain properties: it is non-negative, symmetric, and satisfies the triangle inequality. Understanding metric spaces is essential for discussing concepts such as convergence, continuity, and compactness in various mathematical contexts, including height functions in dynamics.
Mori's Theorem: Mori's Theorem is a fundamental result in algebraic geometry that addresses the behavior of certain rational maps between varieties, particularly in the context of minimal models and the structure of higher-dimensional varieties. This theorem establishes a link between the geometry of varieties and the properties of their divisors, allowing for a better understanding of how these spaces can be manipulated and classified. The theorem plays a crucial role in the study of height functions and periodic points by providing insights into how these concepts interact within the framework of dynamics on algebraic varieties.
Morphisms: Morphisms are structure-preserving mappings between two mathematical objects, typically between algebraic varieties or geometric spaces. They help to understand how different mathematical structures relate to one another and are fundamental in various areas, including dynamics and height functions, as they enable the translation of properties and behaviors from one space to another while maintaining their essential characteristics.
Néron-Tate Height: The Néron-Tate height is a specific type of height function used in the study of rational points on algebraic varieties, particularly abelian varieties. It plays a crucial role in measuring the arithmetic complexity of these points, providing a way to analyze their distribution and relationships, especially in relation to the Mordell-Weil theorem and the heights of points in dynamics, polarizations, and arithmetic surfaces.
Northcott Property: The Northcott Property refers to a condition in number theory related to the boundedness of rational points on algebraic varieties. Specifically, it states that for a given projective variety defined over a number field, the set of rational points of bounded height is finite. This concept is particularly relevant in understanding the behavior of points on arithmetic threefolds, as well as in the dynamics of height functions and their periodic points.
P-adic metrics: p-adic metrics are a type of distance function used in number theory that measure how 'close' two numbers are based on the divisibility by a prime number p. This metric is fundamental in understanding the topology of p-adic numbers, allowing mathematicians to study properties like convergence, continuity, and compactness in a unique way compared to the standard real number metric. The p-adic metric is especially significant in dynamics as it helps to analyze height functions and their behavior under various transformations.
Periodic Orbits Analysis: Periodic orbits analysis refers to the study of trajectories in dynamical systems that repeat after a fixed interval of time. This concept is crucial for understanding the behavior of systems over time, particularly in relation to height functions, as it helps in determining how the dynamics interact with the geometric properties of the system. By examining periodic orbits, one can derive insights about stability, bifurcations, and the long-term behavior of points under iterative processes.
Preperiodic points: Preperiodic points are points in a dynamical system that eventually enter a periodic orbit after a finite number of iterations. These points are significant in understanding the behavior of maps, particularly in arithmetic dynamics, where they help connect various concepts like height functions and rational points on algebraic varieties. They illustrate how the structure of dynamical systems can lead to intricate relationships between algebraic geometry and dynamical behavior.
Rational Points: Rational points are solutions to equations that can be expressed as fractions of integers, typically where the coordinates are in the form of rational numbers. These points are crucial in the study of algebraic varieties, especially in understanding the solutions over different fields, including the rational numbers, which can reveal deeper properties of the geometric objects involved.
Rational Points on Curves: Rational points on curves refer to the points on an algebraic curve that have coordinates that are rational numbers. These points are significant in number theory and algebraic geometry as they can provide insight into the solutions of equations defining the curve, and understanding their distribution is key to solving Diophantine equations.
Subadditivity: Subadditivity is a property of functions that states the value of the function at the sum of two inputs is less than or equal to the sum of the function values at each input. This concept is crucial in various mathematical fields, including dynamics, as it helps in understanding how certain metrics behave when analyzing systems. In the context of height functions, subadditivity can provide insights into the growth rates and stability of dynamical systems.
Weil Height: Weil height is a measure that quantifies the complexity of rational points on algebraic varieties, often in the context of number theory and arithmetic geometry. It helps in understanding the distribution of these points by assigning a numerical value to them, which reflects their 'size' in relation to the underlying field. This concept plays a crucial role in dynamics by connecting heights to the growth rates of orbits under iteration of rational maps.