# The Arithmetic of Dynamical Systems

@inproceedings{Silverman2007TheAO, title={The Arithmetic of Dynamical Systems}, author={Joseph H. Silverman}, year={2007} }

* Provides an entry for graduate students into an active field of research
* Each chapter includes exercises, examples, and figures
* Will become a standard reference for researchers in the field
* Contains descriptions of many known results and conjectures, together with an extensive bibliography
This book provides an introduction to the relatively new discipline of arithmetic dynamics. Whereas classical discrete dynamics is the study of iteration of self-maps of the complex plane or real…

## 577 Citations

Chapter IX: Some Problems of Arithmetic Origin in Rational Dynamics

- MathematicsLecture Notes in Mathematics
- 2020

These are lecture notes from a course in arithmetic dynamics given in Grenoble in June 2017. The main purpose of this text is to explain how arithmetic equidistribution theory can be used in the…

ARITHMETIC DYNAMICS OF QUADRATIC POLYNOMIALS AND DYNAMICAL UNITS

- Mathematics
- 2011

Title of dissertation: ARITHMETIC DYNAMICS OF QUADRATIC POLYNOMIALS AND DYNAMICAL UNITS Chatchawan Panraksa, Doctor of Philosophy, 2011 Dissertation directed by: Professor Lawrence C. Washington…

Difference fields and descent in algebraic dynamics. I

- MathematicsJournal of the Institute of Mathematics of Jussieu
- 2008

Abstract We draw a connection between the model-theoretic notions of modularity (or one-basedness), orthogonality and internality, as applied to difference fields, and questions of descent in in…

On recent results of ergodic property for p -adic dynamical systems

- Mathematics
- 2014

Theory of dynamical systems in fields of p-adic numbers is an important part of algebraic and arithmetic dynamics. The study of p-adic dynamical systems is motivated by their applications in various…

Potential Theory and Dynamics on the Berkovich Projective Line

- Mathematics
- 2010

The purpose of this book is to develop the foundations of potential theory and rational dynamics on the Berkovich projective line over an arbitrary complete, algebraically closed non-Archimedean…

The arithmetic Hodge-index theorem and rigidity of algebraic dynamical systems over function fields

- Mathematics
- 2019

Author(s): Carney, Alexander | Advisor(s): Yuan, Xinyi | Abstract: In one of the fundamental results of Arakelov’s arithmetic intersection theory, Faltings and Hriljac (independently) proved the…

Dynamical moduli spaces and elliptic curves

- Mathematics
- 2018

— In these notes, we present a connection between the complex dynamics of a family of rational functions ft : P1 → P1, parameterized by t in a Riemann surface X, and the arithmetic dynamics of ft on…

The Cycle Structure of Unicritical Polynomials

- MathematicsInternational Mathematics Research Notices
- 2018

A polynomial with integer coefficients yields a family of dynamical systems indexed by primes as follows: for any prime $p$, reduce its coefficients mod $p$ and consider its action on the field $…

Examples of dynamical degree equals arithmetic degree

- Mathematics
- 2012

Let f : X --> X be a dominant rational map of a projective variety defined over a number field. An important geometric-dynamical invariant of f is its (first) dynamical degree d_f= lim…

Heights on moduli space for post-critically finite dynamical systems

- Mathematics
- 2011

The purpose of this Research In Teams event was to consider the arithmetic properties of post-critically finite (PCF) rational maps. In the study of complex holomorphic dynamics, it is a general…

## References

SHOWING 1-10 OF 53 REFERENCES

S-integer dynamical systems: periodic points.

- Mathematics
- 1997

We associate via duality a dynamical system to each pair (RS,x), where RS is the ring of S-integers in an A-field k, and x is an element of RS\{0}. These dynamical systems include the circle doubling…

Homoclinic points of algebraic ℤ^{}-actions

- Mathematics
- 1999

An algebraic zd-action is an action of Zd by (continuous) automorphisms of a compact abelian group. The dynamics of a single group automorphism have been investigated in great detail over the past…

Advanced Topics in the Arithmetic of Elliptic Curves

- Mathematics
- 1994

In The Arithmetic of Elliptic Curves, the author presented the basic theory culminating in two fundamental global results, the Mordell-Weil theorem on the finite generation of the group of rational…

Almost all $S$-integer dynamical systems have many periodic points

- MathematicsErgodic Theory and Dynamical Systems
- 1998

We show that for almost every ergodic $S$-integer dynamical system the radius of convergence of the dynamical zeta function is no larger than $\exp(-\frac{1}{2}h_{\rm top})<1$. In the arithmetic case…

Dynamics in One Complex Variable: Introductory Lectures

- Mathematics
- 2000

These notes study the dynamics of iterated holomorphic mappings from a Riemann surface to itself, concentrating on the classical case of rational maps of the Riemann sphere. They are based on…

Institute for Mathematical Physics Expansive Subdynamics for Algebraic Z D {actions Expansive Subdynamics for Algebraic Z D -actions

- Mathematics
- 2000

A general framework for investigating topological actions of Z d on compact metric spaces was proposed by Boyle and Lind in terms of expansive behavior along lower dimensional subspaces of R d. Here…

Institute for Mathematical Physics Homoclinic Points of Algebraic Z D {actions Homoclinic Points of Algebraic Z D -actions

- Mathematics
- 1996

Let be an action of Z d by continuous automorphisms of a compact abelian group X. A point x in X is called homoclinic for if n x ! 0X as knk ! 1. We study the set (X) of homoclinic points for , which…

Algebra

- MathematicsNature
- 1941

THIS is a text–book intended primarily for undergraduates. It is designed to give a broad basis of knowledge comprising such theories and theorems in those parts of algebra which are mentioned in the…

Basic Number Theory

- Mathematics
- 1997

The first goal of algebraic number theory is the generalization of the theorem on the unique representation of natural numbers as products of prime numbers to algebraic numbers. Gauss considered the…

Expansive Subdynamics

- Mathematics
- 1996

This paper provides a framework for studying the dynamics of commuting homeomorphisms. Let α be a continuous action of Zd on an infinite compact metric space. For each subspace V of Rd we introduce a…