Related papers: Calculating algebraic entropies: an express method
We present a method for calculating the dynamical degree of a mapping with unconfined singularities. It is based on a method introduced by Halburd for the computation of the growth of the iterates of a rational mapping with confined…
We explain on a selection of mappings how the method introduced by Halburd and our simplified variant thereof, the so-called express method, can be used to calculate the dynamical degree of second-order rational mappings from nothing more…
We confront two integrability criteria for rational mappings. The first is the singularity confinement based on the requirement that every singularity, spontaneously appearing during the iteration of a mapping, disappear after some steps.…
We illustrate the use of the notion of derived recurrences introduced earlier to evaluate the algebraic entropy of self-maps of projective spaces. We in particular give an example, where a complete proof is still awaited, but where…
We present a new approach to singularity confinement which makes it an efficient and reliable discrete integrability detector. Our method is based on the full-deautonomisation procedure, which consists in analysing non-autonomous extensions…
In this paper we present a rigorous method for deciding whether a birational three point mapping that has the singularity confinement property is integrable or not, based only on the structure of its (confined) singularity patterns. We also…
We present a novel way to apply the singularity confinement property as a discrete integrability criterion. We shall use what we call a full deautonomisation approach, which consists in treating the free parameters in the mapping as…
We introduce a series of discrete mappings, which is considered to be an extension of the Hietarinta-Viallet mapping with one parameter. We obtain the algebraic entropy for this mapping by obtaining the recurrence relation for the degrees…
We examine the validity of the results obtained with the singularity confinement integrability criterion in the case of discrete Painlev\'e equations. The method used is based on the requirement of non-exponential growth of the homogeneous…
For any discrete time dynamical system with a rational evolution, we define an entropy, which is a global index of complexity for the evolution map. We analyze its basic properties and its relations to the singularities and the…
The purpose of this paper is to analyze the singularities of a well known benchmark problem ``Andrews' squeezing mechanism''. We show that for physically relevant parameter values this system admits singularities. The method is based on…
Typically, the entropy of an isolated system in equilibrium is calculated by counting the number of accessible microstates, or in more general cases by using the Gibbs formula. In irreversible processes entropy spontaneously increases and…
We investigate indeterminate points in discrete integrable system. They appear in singularity confinement phenomenon naturally. We develop a method to analyse indeterminate points of dynamical maps and using this method we clarify behaviour…
Bowen showed that a continuous expansive map with specification has a unique measure of maximal entropy. We show that the conclusion remains true under weaker non-uniform versions of these hypotheses. To this end, we introduce the notions…
For a class of piecewise hyperbolic maps in two dimensions, we propose a combinatorial definition of topological entropy by counting the maximal, open, connected components of the phase space on which iterates of the map are smooth. We…
We examine a family of 3-point mappings that include mappings solvable through linearization. The different origins of mappings of this type are examined: projective equations and Gambier systems. The integrable cases are obtained through…
The theory of degree growth and algebraic entropy plays a crucial role in the field of discrete integrable systems. However, a general method for calculating degree growth for lattice equations (partial difference equations) is not yet…
We present an algorithmic method for the calculation of the degrees of the iterates of birational mappings, based on Halburd's method for obtaining the degrees from the singularity structure of the mapping. The method uses only integer…
We present an analytical-numerical method providing robust upper estimates for the topological entropy or, more generally, uniform volume growth exponents of differentiable mappings. By introducing varying metrics, we simplify the analysis…
We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and,…