12.4 Preperiodic points
9 min read•august 21, 2024
are key players in arithmetic geometry, linking dynamical systems and number theory. These points have finite orbits under iteration, offering insights into long-term behavior and uncovering patterns in algebraic structures and their geometric representations.
Understanding preperiodic points involves exploring their properties, relationships to , and importance in dynamical systems. This study bridges discrete mathematics and continuous , shedding light on the overall structure of these systems and their arithmetic properties.
Definition of preperiodic points
- Preperiodic points play a crucial role in arithmetic geometry by bridging dynamical systems and number theory
- These points exhibit finite orbits under iteration, providing insights into the long-term behavior of dynamical systems
- Understanding preperiodic points helps uncover patterns in algebraic structures and their geometric representations
Formal mathematical definition
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- A point x is preperiodic for a function f if there exist integers n > m ≥ 0 such that
- The smallest such n is called the period, while m represents the preperiod
- Preperiodic points include both periodic points (when m = 0) and points that eventually become periodic
- The set of all preperiodic points for a function f is denoted by PrePer(f)
Relation to periodic points
- Periodic points form a subset of preperiodic points with preperiod 0
- Every preperiodic point eventually enters a periodic cycle after a finite number of iterations
- The orbit of a preperiodic point consists of a finite "tail" followed by a repeating cycle
- Studying preperiodic points often involves analyzing the structure of periodic orbits
Importance in dynamical systems
- Preperiodic points serve as fixed reference points in the study of dynamical systems
- They provide valuable information about the global behavior of iterative processes
- Analysis of preperiodic points helps identify attractors, repellers, and other significant features of dynamical systems
- In arithmetic geometry, preperiodic points offer insights into the arithmetic properties of algebraic varieties
Properties of preperiodic points
- Preperiodic points exhibit unique characteristics that make them essential in studying dynamical systems
- These properties bridge the gap between discrete mathematics and continuous dynamics
- Understanding the behavior of preperiodic points sheds light on the overall structure of dynamical systems
Finiteness theorems
- guarantees finiteness of preperiodic points for certain classes of maps over number fields
- The theorem applies to maps with good reduction modulo all but finitely many primes
- Finiteness results often depend on the degree of the map and the field of definition
- These theorems provide a foundation for studying the distribution of preperiodic points
Orbit structure
- Preperiodic orbits consist of a preperiodic part (tail) and a periodic part (cycle)
- The length of the tail plus the length of the cycle equals the total period of the preperiodic point
- Orbit diagrams visually represent the structure of preperiodic orbits (trees or directed graphs)
- Studying orbit structures helps identify patterns and symmetries in dynamical systems
Algebraic characterization
- Preperiodic points can be characterized as solutions to certain polynomial equations
- For a polynomial map f, preperiodic points satisfy for some n > m ≥ 0
- The algebraic nature of preperiodic points allows for the application of powerful tools from algebraic geometry
- Resultants and elimination theory play crucial roles in computing and analyzing preperiodic points
Preperiodic points in rational maps
- Rational maps form a fundamental class of dynamical systems in arithmetic geometry
- Studying preperiodic points of rational maps provides insights into the behavior of more general algebraic dynamical systems
- The analysis of preperiodic points in rational maps often involves techniques from both algebra and analysis
Quadratic polynomials
- Quadratic polynomials of the form are extensively studied in complex dynamics
- The represents the set of c-values for which the critical point 0 is not in the escaping set
- Preperiodic points of quadratic polynomials exhibit intricate patterns (Misiurewicz points)
- The distribution of preperiodic points relates to the structure of and
Higher degree polynomials
- Polynomials of degree d > 2 exhibit more complex behavior in terms of preperiodic points
- The number of periodic points of a given period grows exponentially with the degree
- Multibrot sets generalize the Mandelbrot set for higher degree polynomials
- Studying preperiodic points of higher degree polynomials often involves sophisticated algebraic and geometric techniques
Möbius transformations
- Möbius transformations are rational functions of degree 1 that act on the Riemann sphere
- These transformations can be classified into elliptic, parabolic, and hyperbolic types based on their fixed points
- Preperiodic points of Möbius transformations relate to their classification and dynamical behavior
- Understanding preperiodic points of Möbius transformations provides insights into more general rational maps
Arithmetic aspects of preperiodic points
- Arithmetic dynamics combines techniques from number theory and dynamical systems
- The study of preperiodic points over number fields reveals deep connections between algebra and geometry
- Arithmetic properties of preperiodic points often reflect the underlying structure of the dynamical system
Heights and canonical heights
- Height functions measure the arithmetic complexity of points in projective space
- Canonical heights, introduced by Néron and Tate, are height functions adapted to specific dynamical systems
- For a rational map f, the canonical height ĥ satisfies the functional equation , where d is the degree of f
- Preperiodic points are characterized by having canonical height zero
Northcott property
- A h has the Northcott property if there are finitely many points of bounded height and bounded degree
- The Northcott property ensures finiteness of preperiodic points for many classes of maps over number fields
- This property is crucial in proving uniform boundedness results for preperiodic points
- The Northcott property fails over function fields, leading to different behavior of preperiodic points in that setting
Uniform boundedness conjecture
- The posits that the number of preperiodic points is uniformly bounded for maps of fixed degree over number fields of fixed degree
- This conjecture remains open and is considered one of the central problems in arithmetic dynamics
- Partial results have been obtained for specific families of maps (quadratic polynomials)
- The conjecture has implications for the distribution of preperiodic points across different number fields
Preperiodic points over number fields
- Number fields provide a rich setting for studying the arithmetic properties of preperiodic points
- The interplay between Galois theory and dynamics reveals deep structures in preperiodic point sets
- Studying preperiodic points over number fields often leads to connections with other areas of number theory
Galois action on preperiodic points
- The absolute Galois group of a number field acts on the set of preperiodic points
- This action preserves the dynamical structure, mapping preperiodic points to preperiodic points
- Analyzing Galois orbits of preperiodic points provides information about their field of definition
- The Galois action on preperiodic points relates to questions of rationality and algebraic independence
Density results
- Preperiodic points can be dense in certain subsets of the parameter space
- For complex quadratic polynomials, preperiodic points are dense in the boundary of the Mandelbrot set
- Over p-adic fields, the distribution of preperiodic points relates to the structure of p-adic Fatou and Julia sets
- Density results often involve techniques from ergodic theory and potential theory
Local-global principles
- Local-global principles investigate the relationship between local (p-adic) and global (rational) preperiodic points
- The Hasse principle for preperiodic points asks whether local preperiodicity implies global preperiodicity
- Counterexamples to the Hasse principle for preperiodic points have been constructed for certain families of maps
- Studying local-global principles for preperiodic points connects arithmetic dynamics to the broader field of diophantine geometry
Computational methods
- Computational techniques play a crucial role in studying preperiodic points in arithmetic geometry
- These methods allow for explicit calculations and provide empirical evidence for conjectures
- Computational approaches often reveal patterns and structures not easily discernible through purely theoretical means
Algorithms for finding preperiodic points
- Iteration-based algorithms involve directly computing orbits and checking for repetition
- Algebraic methods use resultants and Gröbner bases to solve systems of polynomial equations
- p-adic methods exploit the structure of p-adic dynamics to find preperiodic points efficiently
- Hybrid approaches combine multiple techniques to optimize performance for different types of maps
Complexity analysis
- The time complexity of preperiodic point algorithms often depends on the degree of the map and the desired period
- Space complexity considerations become important when dealing with high-precision computations
- Probabilistic algorithms can sometimes provide faster average-case performance for finding preperiodic points
- Complexity analysis helps in choosing appropriate algorithms for different problem instances
Software implementations
- Computer algebra systems (Sage, Magma) offer built-in functions for working with dynamical systems
- Specialized software packages focus on specific aspects of preperiodic point computation (period finding)
- Numerical methods, implemented in languages like Python or Julia, can approximate preperiodic points in complex dynamics
- Parallel computing techniques allow for efficient exploration of large parameter spaces in dynamical systems
Applications in arithmetic dynamics
- Preperiodic points serve as a bridge between abstract theory and concrete applications in arithmetic geometry
- The study of preperiodic points has led to new insights in various areas of mathematics
- Applications of preperiodic point theory extend beyond pure mathematics to fields such as cryptography and computer science
Dynamical Mordell-Lang conjecture
- The generalizes the classical Mordell-Lang conjecture to dynamical systems
- It predicts the structure of the intersection of an orbit with a subvariety
- Preperiodic points play a crucial role in formulating and studying this conjecture
- Results on the dynamical Mordell-Lang conjecture have implications for the distribution of preperiodic points
Arithmetic equidistribution
- Preperiodic points tend to equidistribute with respect to certain natural measures
- For complex dynamics, preperiodic points equidistribute with respect to the measure of maximal entropy
- In the p-adic setting, equidistribution results relate to the structure of p-adic Fatou and Julia sets
- Studying of preperiodic points connects dynamical systems to ergodic theory
Unlikely intersections
- The theory of studies atypical intersections between dynamical orbits and algebraic subvarieties
- Preperiodic points often arise as special cases of unlikely intersections
- The Zilber-Pink conjecture provides a framework for understanding unlikely intersections in general algebraic settings
- Studying unlikely intersections involving preperiodic points has led to new results in transcendence theory
Connections to other areas
- Preperiodic points in arithmetic geometry form connections with various mathematical disciplines
- These connections highlight the interdisciplinary nature of arithmetic dynamics
- Studying preperiodic points often requires techniques from multiple areas of mathematics
Complex dynamics vs arithmetic dynamics
- Complex dynamics studies iteration of holomorphic functions on complex manifolds
- Arithmetic dynamics focuses on iteration over number fields or function fields
- Preperiodic points serve as a bridge between these two perspectives
- Techniques from complex analysis (potential theory) often find analogues in arithmetic settings ()
Algebraic geometry connections
- Preperiodic points correspond to torsion points on certain algebraic varieties
- The theory of height functions in arithmetic geometry relates closely to the study of preperiodic points
- Moduli spaces of dynamical systems provide a geometric framework for studying families of preperiodic points
- Techniques from intersection theory and étale cohomology find applications in the study of preperiodic points
Number theory applications
- Preperiodic points relate to questions about rational points on curves and higher-dimensional varieties
- The study of preperiodic points has led to new insights in Diophantine approximation
- Galois representations associated with preperiodic points connect to broader themes in arithmetic geometry
- Preperiodic point theory has applications in cryptography, particularly in the construction of hash functions
Open problems and conjectures
- The field of arithmetic dynamics contains numerous open problems and conjectures related to preperiodic points
- These unresolved questions drive current research and highlight areas for future exploration
- Solving these problems often requires developing new techniques that bridge multiple areas of mathematics
Morton-Silverman conjecture
- Posits a uniform bound on the number of preperiodic points for maps of fixed degree over number fields of fixed degree
- Remains open in general, with partial results known for specific families of maps
- A positive resolution would have far-reaching consequences in arithmetic dynamics
- Approaches to this conjecture often involve techniques from height theory and algebraic geometry
Pink-Zilber conjecture
- Generalizes the Manin-Mumford and André-Oort conjectures to mixed Shimura varieties
- Has implications for the distribution of special points, including preperiodic points, in certain algebraic varieties
- Connects the study of preperiodic points to broader questions in arithmetic geometry and model theory
- Partial results towards this conjecture often use sophisticated techniques from algebraic and arithmetic geometry
Current research directions
- Exploring preperiodic points in higher-dimensional dynamical systems (Hénon maps)
- Investigating the structure of preperiodic points over function fields and in positive characteristic
- Developing more efficient algorithms for computing and analyzing preperiodic points
- Studying the relationship between preperiodic points and other special points in arithmetic geometry (CM points)
Key Terms to Review (26)
Algebraic dynamics: Algebraic dynamics is a field of mathematics that studies the behavior of algebraic objects under iterative processes, typically focusing on the dynamics of rational functions and endomorphisms of algebraic varieties. This area explores how points evolve through repeated applications of a function, connecting number theory, geometry, and dynamical systems. Key features include understanding preperiodic points, which are points that eventually repeat in their iteration, and the implications for conjectures like the Dynamical Mordell-Lang conjecture, which relates dynamical systems to arithmetic properties.
Arakelov Theory: Arakelov Theory is a framework that combines algebraic geometry and number theory, aiming to study arithmetic properties of algebraic varieties by incorporating a notion of height. It establishes a way to analyze the arithmetic aspects of varieties over global fields, such as rational numbers or finite fields, using tools from both geometry and number theory.
Arithmetic equidistribution: Arithmetic equidistribution is a concept that examines how a sequence of points, generated by a specific arithmetic process, becomes uniformly distributed in a given space over time. This concept often applies to studying the distribution of rational points or orbits of dynamical systems and can reveal insights into the behavior of preperiodic points, as it reflects how these points relate to one another within their respective structures.
Backward orbit: A backward orbit refers to the set of points that eventually lead to a preperiodic point under the iteration of a rational function. In other words, it describes how certain points in a dynamical system can trace back to specific preperiodic points when evaluated through a series of function iterations. Understanding backward orbits is crucial for analyzing the behavior of dynamical systems and their interactions with preperiodic points, as it highlights the pathways that connect various points in the context of rational maps.
Dynamical Mordell-Lang conjecture: The Dynamical Mordell-Lang conjecture proposes that for a dynamical system acting on an algebraic variety, the set of points with a certain property (like being preperiodic) is, under certain conditions, a finite union of translates of algebraic subvarieties. This conjecture connects the behavior of periodic and preperiodic points in dynamical systems with the underlying geometry of varieties, leading to deeper insights into their structure and behavior over iterations.
Dynamics: In the context of arithmetic geometry, dynamics refers to the study of the behavior of points under iterative processes defined by rational functions. This includes understanding how points behave when repeatedly applying a function, leading to concepts such as periodic and preperiodic points, which are essential in analyzing the long-term behavior of these functions.
Filled Julia Sets: Filled Julia sets are the set of points in the complex plane that remain bounded under iteration of a given complex polynomial. They represent the boundary between chaotic and stable behaviors in dynamical systems, especially when studying the behavior of sequences generated by these polynomials. Understanding filled Julia sets is crucial for analyzing preperiodic points, as they provide insight into the long-term behavior of points under iteration.
Forward Orbit: A forward orbit refers to the sequence of points generated by iterating a function on a given starting point in a dynamical system. Specifically, it is a crucial concept in understanding how points evolve over time under the action of a function, leading to insights into the behavior of preperiodic points, which are points that eventually fall into a periodic cycle but do not start in one.
Gisèle rémond: Gisèle Rémond is a mathematician known for her contributions to the study of preperiodic points in dynamical systems, particularly in arithmetic geometry. Her work often explores the interaction between algebraic structures and dynamical systems, focusing on how these preperiodic points behave under iterations of rational maps. This area of study is crucial for understanding the broader implications of dynamics on algebraic varieties and their arithmetic properties.
Height Function: The height function is a way to measure the complexity or size of rational points on algebraic varieties, particularly in arithmetic geometry. It provides a quantitative tool to analyze the distribution of rational points, connecting deeply with concepts such as the Mordell-Weil theorem and the properties of elliptic curves represented by Weierstrass equations. This function plays a pivotal role in understanding not just rational points, but also preperiodic points and their dynamics within the framework of conjectures like the Dynamical Manin-Mumford conjecture.
Joseph Silverman: Joseph Silverman is a prominent mathematician known for his contributions to the fields of arithmetic geometry and number theory. He has authored influential texts that bridge these areas, particularly focusing on the interplay between algebraic geometry and arithmetic. His work emphasizes the importance of understanding rational points on varieties, which connects directly to the study of preperiodic points in dynamical systems.
Julia Sets: Julia sets are fractals associated with complex dynamics, defined for a given complex function. They represent the boundary between points that escape to infinity under iteration of the function and those that remain bounded, forming intricate and beautiful patterns. The structure of Julia sets reveals a lot about the behavior of preperiodic points, which are points that eventually enter a cycle but do not settle into one immediately.
Lang's Conjecture: Lang's Conjecture is a hypothesis in arithmetic geometry that posits certain relationships between algebraic varieties and their rational points. Specifically, it suggests that for a given variety defined over a number field, the set of its rational points should be closely linked to the geometry of the variety and the arithmetic properties of the field. This conjecture bridges concepts from Diophantine geometry and the study of algebraic curves, particularly in contexts involving higher-dimensional varieties.
Mandelbrot set: The Mandelbrot set is a collection of complex numbers that produces a particular kind of fractal when visualized. This set is defined by the behavior of the iterative function $$f(z) = z^2 + c$$, where $$z$$ and $$c$$ are complex numbers. The fascinating aspect of the Mandelbrot set lies in its boundary, which exhibits intricate and self-similar structures, showcasing how simple rules can lead to complex behavior.
Mordell-Weil Theorem: The Mordell-Weil Theorem states that the group of rational points on an elliptic curve over a number field is finitely generated. This fundamental result connects the theory of elliptic curves with algebraic number theory, revealing the structure of rational solutions and their relationship to torsion points and complex multiplication.
Morton-Silverman Conjecture: The Morton-Silverman Conjecture is a conjecture in the field of arithmetic geometry that predicts a relationship between the number of rational points on an elliptic curve and the number of preperiodic points under a given morphism. This conjecture aims to connect the dynamics of rational maps with arithmetic properties, particularly focusing on how the structure of these points can provide insights into the curve's behavior over number fields.
Néron models: Néron models are a way to extend the concept of abelian varieties over a local field to a proper model over the ring of integers of that field. They provide a means to study reduction properties of abelian varieties and their behavior under specialization, especially when considering reductions modulo a prime. These models help analyze the reduction of points, including preperiodic points, and their structures in arithmetic geometry.
Northcott's Theorem: Northcott's Theorem states that for certain types of varieties, specifically in the context of arithmetic geometry, there are only finitely many rational points of bounded height. This theorem connects the concepts of height functions and rational points by establishing a crucial link between the finiteness of solutions to equations and the growth of the height associated with those solutions, leading to deeper insights about preperiodic points in dynamical systems.
Periodic Points: Periodic points are points in a dynamical system that return to their original position after a fixed number of iterations of the system. In the context of projective spaces, these points play a crucial role in understanding the behavior of iterations of rational functions and their interactions with the geometric structure of the space. They help identify stable configurations and provide insight into the overall dynamics at play.
Pink-Zilber Conjecture: The Pink-Zilber conjecture proposes a connection between the dynamics of algebraic maps and the geometry of the underlying varieties, specifically concerning the behavior of preperiodic points. It suggests that for certain algebraic structures, the set of preperiodic points is either finite or forms a structured configuration, thereby influencing the way these points are studied within arithmetic geometry.
Points over finite fields: Points over finite fields refer to the solutions or coordinates of algebraic equations defined over finite fields, which are fields with a finite number of elements. These points are crucial in various areas of mathematics, especially in understanding the behavior of algebraic varieties and rational points on these varieties when considered over finite fields. They play a significant role in number theory, cryptography, and the study of preperiodic points in dynamical systems.
Points over rational numbers: Points over rational numbers refer to the solutions of equations or geometric objects defined over the field of rational numbers, typically denoted as $$ ext{Q}$$. These points are significant in arithmetic geometry as they provide insight into the structure and properties of algebraic varieties and their rational points. Understanding points over rational numbers allows mathematicians to explore the relationships between algebraic geometry, number theory, and Diophantine equations.
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.
Self-maps of varieties: Self-maps of varieties refer to morphisms from a variety to itself. These maps are crucial for studying the dynamics and algebraic properties of varieties, especially in understanding how points behave under iteration of the map. In particular, analyzing self-maps can reveal insights into the structure of the variety and help identify fixed and preperiodic points, as well as their stability.
Theorems of Faltings: Theorems of Faltings, developed by Gerd Faltings, refer to a set of results in arithmetic geometry that have deep implications for the study of rational points on algebraic varieties, particularly in the context of Diophantine equations. These theorems fundamentally changed the understanding of how rational and integral solutions behave, providing a framework that connects number theory and algebraic geometry. They specifically establish conditions under which certain varieties can be shown to have only finitely many rational points, making them pivotal in modern arithmetic geometry.
Unlikely Intersections: Unlikely intersections refer to situations in arithmetic geometry where two or more geometric objects intersect in a way that is considered to be exceptional or rare, particularly in the context of their dimensions and degrees. These intersections can reveal profound connections between algebraic geometry and number theory, often involving unexpected relationships between preperiodic points and algebraic varieties. Understanding unlikely intersections helps to uncover deeper insights into the dynamics of rational maps and their behavior over time.