12.1 Dynamical systems on projective spaces

Dynamical systems on projective spaces blend algebraic geometry with iteration theory. This fusion allows us to study how rational maps behave when applied repeatedly, revealing intricate patterns and structures in geometric objects.

By examining fixed points, periodic orbits, and long-term behavior, we gain insights into both the geometry of projective spaces and the arithmetic properties of the maps. This approach connects abstract algebra, number theory, and complex analysis in fascinating ways.

Projective spaces overview

• Projective spaces form a fundamental concept in algebraic geometry, providing a framework for studying geometric objects and their transformations
• In arithmetic geometry, projective spaces play a crucial role in understanding the behavior of polynomial equations and rational maps over various number fields

Definition of projective spaces

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• Projective space $\mathbb{P}^n$ defined as the set of lines passing through the origin in $\mathbb{R}^{n+1}$ or $\mathbb{C}^{n+1}$
• Points in projective space represented by equivalence classes of non-zero vectors
• Formalizes the notion of "points at infinity" in geometry
• Allows for a more unified treatment of geometric objects and their intersections

Homogeneous coordinates

• System for representing points in projective space using $(n+1)$ coordinates
• Points in $\mathbb{P}^n$ represented as $[x_0 : x_1 : \ldots : x_n]$, where not all $x_i$ are zero
• Scaling invariance: $[x_0 : x_1 : \ldots : x_n] = [λx_0 : λx_1 : \ldots : λx_n]$ for any non-zero scalar $λ$
• Enables algebraic manipulation of projective objects without choosing specific representatives
• Simplifies computations involving projective transformations and intersections

Projective transformations

• Linear transformations of projective space preserving collinearity and cross-ratio
• Represented by invertible $(n+1) \times (n+1)$ matrices acting on homogeneous coordinates
• Include translations, rotations, scaling, and perspective transformations
• Form the projective linear group $PGL(n+1, \mathbb{K})$ over a field $\mathbb{K}$
• Essential for studying symmetries and automorphisms of projective varieties

Dynamical systems fundamentals

• Dynamical systems theory provides a framework for studying the long-term behavior of evolving systems
• In arithmetic geometry, dynamical systems offer insights into the iteration of rational maps and their arithmetic properties

Discrete vs continuous systems

• Discrete systems evolve in distinct time steps (difference equations)
  • Described by maps or functions iteratively applied
  • Suitable for studying rational maps in projective spaces
• Continuous systems evolve smoothly over time (differential equations)
  • Described by vector fields or flows
  • Relevant for studying continuous symmetries of algebraic varieties
• Both types can exhibit complex behaviors (chaos, fractals)
• Discretization of continuous systems often used in numerical analysis

Fixed points and periodic orbits

• Fixed points remain unchanged under the system's evolution
  • Satisfy the equation $f(x) = x$ for discrete systems
  • Critical for understanding the long-term behavior of orbits
• Periodic orbits repeat after a finite number of iterations
  • Period-n orbit satisfies $f^n(x) = x$ but $f^k(x) \neq x$ for $0 < k < n$
  • Can form the backbone of more complex dynamics (heteroclinic cycles)
• Classification of fixed points and periodic orbits crucial for global dynamics
• Arithmetic properties of fixed points and periodic orbits important in number theory

Stability and attractors

• Stability determines the behavior of nearby orbits
  • Asymptotically stable fixed points attract nearby orbits
  • Unstable fixed points repel nearby orbits
• Lyapunov exponents quantify the rate of separation of nearby trajectories
  • Positive Lyapunov exponents indicate chaos
  • Negative Lyapunov exponents indicate stability
• Attractors represent the long-term behavior of a set of initial conditions
  • Can be fixed points, periodic orbits, or more complex sets (strange attractors)
  • Basin of attraction consists of all initial conditions converging to the attractor
• Stability analysis essential for understanding the global structure of the phase space

Projective dynamical systems

• Dynamical systems on projective spaces combine algebraic geometry with dynamical systems theory
• Provide a rich setting for studying iteration of rational maps and their geometric properties

Rational maps on projective spaces

• Maps between projective spaces defined by ratios of homogeneous polynomials
• General form: $f([x_0 : \ldots : x_n]) = [f_0(x_0, \ldots, x_n) : \ldots : f_n(x_0, \ldots, x_n)]$
• Well-defined except at points where all $f_i$ simultaneously vanish (indeterminacy locus)
• Include projective linear transformations as a special case
• Iteration of rational maps leads to complex dynamics and arithmetic questions

Homogeneous polynomial maps

• Special case of rational maps where all components are homogeneous polynomials of the same degree
• Preserve the degree of projective varieties under iteration
• Simplify certain computations and theoretical considerations
• Examples include:
  • Quadratic maps in $\mathbb{P}^2$: $[x : y : z] \mapsto [x^2 : xy : z^2]$
  • Fermat maps: $[x : y : z] \mapsto [x^d : y^d : z^d]$

Degree of projective maps

• Measures the "algebraic complexity" of a rational map
• Defined as the common degree of the homogeneous polynomials defining the map
• Preserved under composition: $\deg(f \circ g) = \deg(f) \cdot \deg(g)$
• Relates to the number of preimages of generic points
• Dynamical degree measures the asymptotic growth rate of degrees under iteration
  • Important invariant in complex dynamics and arithmetic geometry

Iteration in projective spaces

• Iteration of maps in projective spaces reveals rich dynamical behaviors
• Combines techniques from complex analysis, algebraic geometry, and number theory

Fatou and Julia sets

• Fatou set consists of points with stable behavior under iteration
  • Open set where the family of iterates is normal (equicontinuous)
  • Can contain attracting basins, parabolic basins, and Siegel disks
• Julia set is the complement of the Fatou set
  • Closed set where the dynamics is chaotic
  • Often has fractal structure and contains repelling periodic points
• Dichotomy between Fatou and Julia sets fundamental to complex dynamics
• Generalizations to higher dimensions and non-Archimedean fields active area of research

Periodic points classification

• Attracting periodic points: local contraction, contained in the Fatou set
  • Superattracting if the derivative vanishes
• Repelling periodic points: local expansion, contained in the Julia set
  • Dense in the Julia set for rational maps
• Indifferent (neutral) periodic points: neither attracting nor repelling
  • Parabolic points have rational rotation number
  • Siegel points have irrational rotation number satisfying Diophantine conditions
• Classification theorem (Fatou-Julia-Léau-Cremer) describes local dynamics near periodic points
• Arithmetic properties of periodic points connect dynamics to number theory

Attracting and repelling cycles

• Attracting cycles form the core of stable dynamics in the Fatou set
  • Basin of attraction consists of points converging to the cycle
  • Often associated with polynomial-like behavior
• Repelling cycles play a crucial role in chaotic dynamics
  • Form a dense subset of the Julia set
  • Used to construct symbolic dynamics and analyze topological entropy
• Saddle cycles exhibit both attracting and repelling directions
  • Important for understanding global dynamics in higher dimensions
• Homoclinic and heteroclinic orbits connect different cycles
  • Lead to complex dynamics and horseshoe-type constructions

Algebraic properties

• Algebraic aspects of dynamical systems in projective spaces connect to arithmetic geometry
• Provide tools for analyzing the long-term behavior of orbits and their arithmetic properties

Algebraic degree preservation

• Rational maps preserve algebraic degrees of subvarieties under preimages
• Degree of the image of a subvariety under a rational map bounded by product of degrees
• Allows for inductive arguments and degree estimates in dynamics
• Connects to intersection theory and enumerative geometry
• Examples:
  • Quadratic maps in $\mathbb{P}^2$ map lines to conics
  • Cremona involutions preserve the degree of curves

Height functions in dynamics

• Height functions measure arithmetic complexity of points in projective space
• Canonical height associated to a dynamical system
  • Satisfies $\hat{h}(f(P)) = d \cdot \hat{h}(P)$ for a degree $d$ map
  • Vanishes precisely on preperiodic points
• Northcott property ensures finiteness of points with bounded height and degree
• Used to study distribution of periodic points and orbits
• Connects dynamical systems to Diophantine approximation and arithmetic geometry

Arithmetic properties of orbits

• Orbit of a point under iteration encodes arithmetic information
• Density of orbits in projective space related to Galois theory and algebraic independence
• Preperiodic points have special arithmetic significance
  • Finite in number for maps over number fields (if $\deg f > 1$)
  • Distribution governed by equidistribution theorems
• Algebraic relations between orbit points studied via specialization and height theory
• Connections to arithmetic equidistribution and dynamical Mordell-Lang conjecture

Complex projective dynamics

• Study of holomorphic maps on complex projective spaces
• Combines techniques from several complex variables and algebraic geometry

Holomorphic maps on CP^n

• Holomorphic self-maps of $\mathbb{CP}^n$ given by homogeneous polynomials of the same degree
• Extend naturally to rational maps on $\mathbb{CP}^n$
• Local behavior near fixed points described by linearization theorems
• Global dynamics studied via normality criteria and pluripotential theory
• Examples:
  • Lattès maps arising from complex tori
  • Polynomial endomorphisms of $\mathbb{CP}^n$

Böttcher coordinates

• Local coordinates near superattracting fixed points of holomorphic maps
• Conjugate the map to the simple form $z \mapsto z^d$ in a neighborhood
• Crucial for understanding the structure of immediate basins of attraction
• Generalize to higher dimensions for superattracting fixed points
• Used to define dynamic rays and study landing properties

Green's functions in dynamics

• Plurisubharmonic functions associated to holomorphic maps on $\mathbb{CP}^n$
• Satisfy the functional equation $G(f(z)) = d \cdot G(z)$ for degree $d$ maps
• Measure the escape rate of orbits to infinity
• Level sets of Green's functions define dynamically interesting sets
  • Equicontinuity domains in Fatou set
  • Support of the measure of maximal entropy
• Connections to potential theory and harmonic analysis in several complex variables

Arithmetic aspects

• Arithmetic dynamics studies number-theoretic aspects of dynamical systems
• Combines techniques from algebraic geometry, number theory, and dynamical systems

Dynamical Mordell-Lang conjecture

• Generalizes the classical Mordell-Lang conjecture to dynamical settings
• States that the intersection of an orbit with a subvariety is finite or dense
• Proved in certain cases (Fakhruddin, Ghioca, Tucker)
• Connects orbit structure to algebraic geometry of the ambient space
• Applications to problems in arithmetic geometry and Diophantine equations

Dynamical Manin-Mumford problem

• Analogous to the classical Manin-Mumford conjecture for abelian varieties
• Concerns the distribution of preperiodic points in subvarieties
• Conjectured finiteness of preperiodic points contained in a proper subvariety
• Proved in some cases (toric varieties, abelian varieties)
• Connections to equidistribution theory and arithmetic intersection theory

Arithmetic equidistribution

• Study of limiting distributions of special points in dynamical systems
• Galois orbits of preperiodic points equidistribute with respect to canonical measures
• Height functions play a crucial role in formulating equidistribution results
• Applications to:
  • Density of rational preperiodic points
  • Distribution of CM points on modular curves
  • Arithmetic analogues of Fatou-Julia theory

Computational methods

• Numerical and symbolic techniques for studying projective dynamical systems
• Essential for exploring examples and generating conjectures

Iteration algorithms

• Efficient methods for iterating rational maps in projective coordinates
• Homogeneous coordinate representation to avoid loss of precision
• Adaptive precision techniques for maintaining accuracy during long orbits
• Parallel algorithms for exploring parameter spaces of dynamical systems
• Specialized methods for detecting periodic points and cycles

Symbolic dynamics in projective spaces

• Encoding of orbits using symbolic sequences
• Markov partitions for piecewise projective maps
• Kneading theory for one-dimensional projective dynamics
• Connections to coding theory and formal languages
• Applications to computing topological entropy and periodic points

Numerical approximation techniques

• Numerical methods for approximating invariant measures and attractors
• Discretization schemes for projective spaces (adaptive grids, spherical harmonics)
• Spectral methods for studying transfer operators in projective dynamics
• Monte Carlo techniques for estimating Lyapunov exponents and dimension
• Numerical continuation methods for following bifurcations in parameter space

Applications and examples

• Concrete instances of projective dynamical systems illustrating theoretical concepts
• Provide motivation for further research and connections to other areas of mathematics

Dynamics of Cremona maps

• Birational maps of the projective plane $\mathbb{P}^2$
• Include quadratic maps and their generalizations
• Study of dynamical degrees and complexity growth
• Connections to:
  • Algebraic geometry of surfaces
  • Group theory of Cremona groups
  • Classification of rational surfaces

Projective Hénon maps

• Generalizations of the classical Hénon map to projective spaces
• Form: $[x : y : z] \mapsto [y : z : P(x,y) - az]$ for a homogeneous polynomial $P$
• Exhibit complex dynamics with interesting Julia sets
• Studied for their:
  • Periodic points and cycles
  • Entropy and topological dynamics
  • Connections to automorphisms of affine spaces

Dynamics on K3 surfaces

• Study of automorphisms and rational maps on K3 surfaces
• Rich interplay between algebraic geometry and dynamical systems
• Examples include:
  • Kummer surfaces arising from abelian varieties
  • Quartic surfaces in $\mathbb{P}^3$
• Connections to:
  • Arithmetic of K3 surfaces
  • Mirror symmetry and string theory
  • Classification of algebraic surfaces

Current research directions

• Active areas of investigation in projective dynamics and arithmetic geometry
• Highlight open problems and recent developments in the field

Dynamical moduli spaces

• Spaces parametrizing dynamical systems with fixed degree and dimension
• Study of bifurcations and stability in families of maps
• Connections to:
  • Geometric invariant theory
  • Compactifications of moduli spaces
  • Arithmetic dynamics over function fields

Arithmetic dynamics on abelian varieties

• Study of endomorphisms and rational maps on abelian varieties
• Connections to:
  • Arithmetic geometry of abelian varieties
  • Heights and canonical measures
  • Dynamical analogues of the André-Oort conjecture

Dynamical degrees and complexity

• Asymptotic growth rates of degrees under iteration
• Algebraic and transcendental properties of dynamical degrees
• Connections to:
  • Birational geometry and the minimal model program
  • Ergodic theory and measure-theoretic entropy
  • Complexity theory of algebraic dynamical systems