Related papers: On the complex dynamics of birational surface maps…

Jonsson and Reschke showed that birational selfmaps on projective surface defined over a number field satisfy the energy condition of Bedford and Diller so their ergodic properties are very well understood. Under suitable hypotheses on the…

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…

The dynamical degree $\lambda(f)$ of a birational transformation $f$ measures the exponential growth rate of the degree of the formulae that define the $n$-th iterate of $f$. We study the set of all dynamical degrees of all birational…

Bedford asked if there exists a birational self map $f$ of the complex projective plane such that for any automorphism $A$ of the complex projective plane $A\circ f$ is not conjugate to an automorphism. Blanc gave such a $f$ of degree $6$…

The dynamics of a rational surface map $f : X \dashrightarrow X$ are easier to analyse when $f$ is `algebraically stable'. Here we investigate when and how this condition can be achieved by conjugating $f$ with a birational change of…

Our aim is to illustrate how one can effectively apply the basic ideas and notions of topological entropy and dynamical degrees, together with recent progress of minimal model theory in higher dimension, for an explicit study of birational…

Since the end of the XIXth century, we know that each birational map of the complex projective plane is the product of a finite number of quadratic birational maps of the projective plane; this motivates our work which essentially deals…

We give some explicit upper bounds on the effective birationality of the canonical or anti-canonical system for a singular surface. In particular, we show that for any surface $X$ with $\epsilon$-lc singularity and the canonical divisor…

We prove that for a dominant rational self-map $f$ on a quasi-projective variety defined over $\overline{\mathbb{Q}}$, there is a point whose $f$-orbit is well-defined and its arithmetic degree is arbitrarily close to the first dynamical…

We define arithmetical and dynamical degrees for dynamical systems with several rational maps on projective varieties, study their properties and relations, and prove the existence of a canonical height function associated with divisorial…

We classify birational maps of projective smooth surfaces whose non-critical periodic points are Zariski dense. In particular, we show that if the first dynamical degree is greater than one, then the periodic points are Zariski dense.

We construct a family of birational maps acting on two dimensional projective varieties, for which the growth of the degrees of the iterates is cubic. It is known that this growth can be bounded, linear, quadratic or exponential for such…

This work dynamically classifies a 9-parametric family of birational maps f : C2 -> C2. From the sequence of the degrees dn of the iterates of f, we find the dynamical degree delta(f) of f. We identify when dn grows periodically, linearly,…

Consider a cohomologically hyperbolic birational self-map defined over the algebraic numbers, for example, a birational self-map in dimension two with the first dynamical degree greater than one, or in dimension three with the first and the…

We prove that dynamical degrees of rational self-maps on projective varieties can be interpreted as spectral radii of naturally defined operators on suitable Banach spaces. Generalizing Shokurov's notion of b-divisors, we consider the space…

We give examples of birational selfmaps of $\mathbb{P}^d, d \geq 3$, whose dynamical degree is a transcendental number. This contradicts a conjecture by Bellon and Viallet. The proof uses a combination of techniques from algebraic dynamics…

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 these birational maps. We find the all the maps in…

This text deals with birationnal diffeomorphisms of real algebraic surfaces which have simple real dynamics and rich complex dynamics. We give an example of such a transformation on P^1xP^1, then we show that this situation is exceptional…

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 f be a rational mapping of a space X . The complexity of (f,X) as a dynamical system is measured by the dynamical degrees $\delta_p(f)$, $1\le p\le {\rm dim}(X)$. We give the definition of the dynamical degrees show how they are…