A rational map with infinitely many points of distinct arithmetic degrees

  title={A rational map with infinitely many points of distinct arithmetic degrees},
  author={John Lesieutre and Matthew Satriano},
  journal={Ergodic Theory and Dynamical Systems},
  pages={3051 - 3055}
Let be a dominant rational self-map of a smooth projective variety defined over $\overline{\mathbb{Q}}$ . For each point $P\in X(\overline{\mathbb{Q}})$ whose forward $f$ -orbit is well defined, Silverman introduced the arithmetic degree $\unicode[STIX]{x1D6FC}_{f}(P)$ , which measures the growth rate of the heights of the points $f^{n}(P)$ . Kawaguchi and Silverman conjectured that $\unicode[STIX]{x1D6FC}_{f}(P)$ is well defined and that, as $P$ varies, the set of values obtained by $\unicode… Expand
2 Citations
Current trends and open problems in arithmetic dynamics
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 inExpand
A transcendental dynamical degree
We give an example of a dominant rational selfmap of the projective plane whose dynamical degree is a transcendental number.


Dynamical degree, arithmetic entropy, and canonical heights for dominant rational self-maps of projective space
  • J. Silverman
  • Mathematics
  • Ergodic Theory and Dynamical Systems
  • 2012
Abstract Let φ:ℙN⤏ℙN be a dominant rational map. The dynamical degree of φ is the quantity δφ=lim (deg φn)1/n. When φ is defined over ${\bar {{\mathbb {Q}}}}$, we define the arithmetic degree of aExpand
Examples of dynamical degree equals arithmetic degree
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= limExpand
On the dynamical and arithmetic degrees of rational self-maps of algebraic varieties
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.Expand
Degrees of iterates of rational maps on normal projective varieties
Let X be a normal projective variety defined over an algebraically closed field of arbitrary characteristic. We study the sequence of intermediate degrees of the iterates of a dominant rationalExpand
Dynamical canonical heights for Jordan blocks, arithmetic degrees of orbits, and nef canonical heights on abelian varieties
Let f : X --> X be an endomorphism of a normal projective variety defined over a global field K, and let D_0,D_1,D_2,... be divisor classes that form a Jordan block with eigenvalue b for the actionExpand
Arithmetic degrees and dynamical degrees of endomorphisms on surfaces
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 arithmeticExpand
Projective Surface Automorphisms of Positive Topological Entropy from an Arithmetic Viewpoint
Let <i>X</i> be a smooth projective surface over a number field <i>K</i>([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i"/] C), and <i>f</i>: <i>X</i> → <i>X</i> anExpand
The g-periodic subvarieties for an automorphism g of positive entropy on a compact Kahler manifold
For a compact Kahler manifold X and a strongly primitive automorphism g of positive entropy, it is shown that X has at most rank NS(X) of g-periodic prime divisors B_i (i.e., g^s(B_i) = B_i for someExpand
Dynamics of Rational Surface Automorphisms: Linear Fractional Recurrences
We consider the family fa,b(x,y)=(y,(y+a)/(x+b)) of birational maps of the plane and the parameter values (a,b) for which fa,b gives an automorphism of a rational surface. In particular, we findExpand
Periodicities in linear fractional recurrences: Degree growth of birational surface maps
We consider the set of all 2-step recurrences (difference equations) that are given by linear fractional maps. These give birational maps of the plane. We determine the degree growth of theseExpand