We study one-dimensional algebraic families of pairs given by a polynomial with a marked point. We prove an "unlikely intersection" statement for such pairs thereby exhibiting strong rigidity features for these pairs. We infer from this result the dynamical André-Oort conjecture for curves in the moduli space of polynomials, by describing one-dimensional families in this parameter space containing infinitely many post-critically finite parameters.
We discuss analogies between the etale site of arithmetic schemes and the algebraic topology of dynamical systems. The emphasis is on Lefschetz numbers. We also discuss similarities between infinite primes in arithmetic and fixed points of dynamical systems.
We classify all post-critically finite unicritical polynomials defined over the maximal totally real algebraic extension of ${\mathbb Q}$. Two auxiliary results used in the proof of this result may be of some independent interest. The first is a recursion formula for the $n$-diameter of an interval, which uses properties of Jacobi polynomials. The second is a numerical criterion which allows one to the give a bound on the degree of any algebraic integer having all of its complex embeddings in a real interval of length less than $4$.
We show that the dynamical degree of an (i.i.d) random sequence of dominant, rational self-maps on projective space is almost surely constant. We then apply this result to height growth and height counting problems in random orbits.
Let $n\geq 2$, and let $f$ be a polynomial of degree at least 2 with coefficients in a number field or a characteristic 0 function field $K$. We present two arithmetic applications of a recent theorem of Medvedev-Scanlon to the dynamics of the map $(f,...,f):\ (\bP^1_{K})^n\longrightarrow (\bP^1_K)^n$, namely the dynamical analogues of the Hasse principle and the Bombieri-Masser-Zannier height bound theorem. In particular, we prove that the Hasse principle holds when we intersect an orbit and a preperiodic subvariety, and that points in the intersection of a curve with the union of all periodi…
Let $f$ be a polynomial with integer coefficients such that $f(n)$ positive for any positive integer $n$. We consider diverging sequences $\{ y_n\}$ given by $y_0 = b$ and $y_{n+1} = f^{y_n}(a)$ with positive integers $a$ and $b$. We show such a sequence converges in $\widehat{\mathbb{Z}}$ and the limit is independent of $b$, if and only if $f$ does not become a permutation of length $p$ on $\mathbb{Z}/p\mathbb{Z}$ for any prime number $p$. We also show that $b'$-adic asymptotic approximations of the equation $f^y(a) = y$ holds in $\mathbb{N}$ for some bases $b'$.
For a dominant rational self-map on a smooth projective variety defined over a number field, Kawaguchi and Silverman conjectured that the (first) dynamical degree is equal to the arithmetic degree at a rational point whose forward orbit is well-defined and Zariski dense. We prove this conjecture for surjective endomorphisms on smooth projective surfaces. For surjective endomorphisms on any smooth projective varieties, we show the existence of rational points whose arithmetic degrees are equal to the dynamical degree. Moreover, we prove that there exists a Zariski dense set of rational points h…
For a dominant rational self-map on a smooth projective variety defined over a number field, Shu Kawaguchi and Joseph H. Silverman conjectured that the dynamical degree is equal to the arithmetic degree at a rational point whose forward orbit is well-defined and Zariski dense. We give some examples of self-maps on product varieties and rational points on them for which the Kawaguchi-Silverman conjecture holds.
To a given Pisot unit $β$ we associate a finite abelian group whose size appears to be equal to the discriminant of $β$. We call it the Pisot group and find its representation in the two-sided $β$-compactum in the case of $β$ satisfying the relation $Fin(β)=\Bbb Z[β]\cap[0,1)$. As a motivation for the definition, we show that the Pisot group is the kernel of some important arithmetic coding of the toral automorphism given by the companion matrix naturally associated with $β$.
Consider an homogeneous space under a locally compact group G and a lattice in G. Then the lattice naturally acts on the homogeneous space. Looking at a dense orbit, one may wonder how to describe its repartition. One then adopt a dynamical point of view and compare the asymptotic distribution of points in the orbits with the natural measure on the space. In the setting of Lie groups and their homogeneous spaces, several results showed an equidistribution of points in the orbits using Ratner's rigidity of polynomial dynamics in homogeneous spaces. We address here this problem in the setting of…
We proved that any Lorentz transformation of 2-torus is Anosov automorphism. One completely describes admissible parameters of Lorentz transformations and their arithmetical properties. One proved that an admissible speed light parameter has a countable spectra accumulating to this parameter.
We study arithmetic degree of a dominant rational self-map on a smooth projective variety over a function field of characteristic zero. We see that the notion of arithmetic degree and some related problems over function fields are interpreted into geometric ones. We give another proof of the theorem that the arithmetic degree at any point is smaller than or equal to the dynamical degree. We give a sufficient condition for an arithmetic degree to coincide with the dynamical degree, and prove that any self-map has so many points whose arithmetic degrees are equal to the dynamical degree. We stud…
Let $ϕ:X\dashrightarrow X$ be a dominant rational map of a smooth variety and let $x\in X$, all defined over $\bar{\mathbb Q}$. The dynamical degree $δ(ϕ)$ measures the geometric complexity of the iterates of $ϕ$, and the arithmetic degree $α(ϕ,x)$ measures the arithmetic complexity of the forward $ϕ$-orbit of $x$. It is known that $α(ϕ,x)\leδ(ϕ)$, and it is conjectured that if the $ϕ$-orbit of $x$ is Zariski dense in $X$, then $α(ϕ,x)=δ(ϕ)$, i.e., arithmetic complexity equals geometric complexity. In this note we prove this conjecture in the case that $X$ is an abelian variety, extending earl…
Arithmetic $k$-Polynomial Dynamical Localization for $k$ Power-Law Quasi-Periodic Long-Range Operators on $\ell^2(\mathbb{Z}^d)$
School of Mathematical Sciences, Soochow University（苏州大学数学科学学院）；Department of Mathematics, Nanjing Audit University（南京审计大学数学系）；Department of Mathematics, Texas A&M University（德克萨斯农工大学数学系）
Let F : P^N --> P^N be a dominant rational map. The dynamical degree of F is the quantity d_F = lim (deg F^n)^(1/n). When F is defined over a number field, we define the arithmetic degree of an algebraic point P to be a_F(P) = limsup h(F^n(P))^(1/n) and the canonical height of P to be h_F(P) = limsup h(F^n(P))/n^k d_F^n for an appropriately chosen integer k = k_F. In this article we prove some elementary relations and make some deep conjectures relating d_F, a_F(P), and h_F(P). We prove our conjectures for monomial maps.
Let f : X --> X be a dominant rational map of a projective variety defined over a global field, let d_f be the dynamical degree of f, and let h_X be a Weil height on X relative to an ample divisor. We prove that h_X(f^n(P)) << (d_f + e)^n h_X(P), where the implied constant depends only on X, h_X, f, and e. As applications, we prove a fundamental inequality a_f(P) \le d_f for the upper arithmetic degree and we construct canonical heights for (nef) divisors. We conjecture that a_f(P) = d_f whenever the orbit of P is Zariski dense, and we describe some cases for which we can prove our conjecture.
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 SpecRadius((f^n)^*)^{1/n}. For algebraic points P of X whose forward orbits are well-defined, there is an analogous (upper) arithmetic degree a_f(P) = limsup h_X(f^n(P))^{1/n}, where h_X is an ample Weil height on X. In an earlier paper, we proved the fundamental inequality a_f(P) \le d_f and conjectured that a_f(P) = d_f whenever the orbit of P is Zariski dense. In this paper we show that the conjecture is true…
We generalized several results for the arithmetic dynamics of monomial maps, including Silverman's conjectures on height growth, dynamical Mordell-Lang conjecture, and dynamical Manin-Mumford conjecture. These results are originally known for monomial maps on algebraic tori. We extend the results to arbitrary toric varieties.
We study the dynamics of the unicritical polynomial family $f_{d,c}(z)=z^d+c\in \mathbb{C}[z]$. The $c$-values for which $f_{d,c}$ has a strictly preperiodic postcritical orbit are called Misiurewicz parameters, and they are the roots of Misiurewicz polynomials. The arithmetic properties of these special parameters have found applications in both arithmetic and complex dynamics. In this paper, we investigate some new such properties. In particular, when $d$ is a prime power and $c$ is a Misiurewicz parameter, we prove certain arithmetic relations between the points in the postcritical orbit of…
We study dynamical systems arising from word maps on simple groups. We develop a geometric method based on the classical trace map for investigating periodic points of such systems. These results lead to a new approach to the search of Engel-like sequences of words in two variables which characterize finite solvable groups. They also give rise to some new phenomena and concepts in the arithmetic of dynamical systems.
Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry, and partly from $p$-adic analogues of theorems and conjectures in classical complex dynamics. In this article we survey some of the motivating problems and some of the recent progress in the field of arithmetic dynamics.
