Piecewise-isometries are zero-entropy dynamical systems with very complex behaviour. In one-dimension they are intervalexchange transformations (IET), which are connected to diophantine arithmetic by the Boshernitzan and Carrol theorem: in any IET defined over a quadratic number field, the process of induction results, after scaling, in an eventually periodic sequence of IETs. This can be viewe...

A monomial Hénon mapping is defined as the wellknown two-dimensional Hénon map with the quadratic term replaced by a monomial. This paper introduces a conjecture about monomial Hénon mappings: Even Hénon mappings are chaotic and odd Hénon mappings are not chaotic in the first quadrant of the bifurcation parameter space. This conjecture is based on numerical simulations of this type of map.

For certain classes of holomorphic correspondences which are matings between Kleinian groups and polynomials, we prove the existence of pinching deformations, analogous to Maskit’s deformations of Kleinian groups which pinch loxodromic elements to parabolic elements. We apply our results to establish the existence of matings between quadratic maps and the modular group, for a large class of qua...

Factoring an integer is equivalent to express the integer as the difference of two squares. We test that for any odd modulus, in the corresponding ring of remainders, any element can be realized as the difference of two quadratic residues, and also that, for a fixed remainder value, the map assigning to each modulus the number of ways to express the remainder as difference of quadratic residues...

• A quadratic space is a pair (V,B) where V is a finite-dimensional vector space over a field K and B : V × V → K is a symmetric bilinear form. • For quadratic spaces (V,B) and (V ′, B′), a map τ : V → V ′ is an isometry iff τ is a linear isomorphism and ∀~v, ~ w ∈ V : B(~v, ~ w) = B′(τ(~v), τ(~ w)). In this case V and V ′ are isometric. • The quadratic form determined by B and basis β := {~v1,...

— Let E be an elliptic curve defined over a number field F and K/F a quadratic extension. For a point P ∈ E(F) that is a local trace for every completion of K/F, we find necessary and sufficient conditions for P to lie in the image of the global trace map. These conditions can then be used to determine whether a quadratic twist of E, as a genus one curve, has rational points. In the case of qua...

We study the geometric structure of the boundary of Herman rings in a model family of Blaschke products of degree 3. Shishikura’s quasiconformal surgery relates the Herman ring to the Siegel disk of a quadratic polynomial. By studying the regularity properties of the maps involved, we can transfer McMullen’s results on the fine local geometry of Siegel disks to the Herman ring setting.

Presumably, there are an infinity of scaling constants in 1D quadratic maps; therefore, it is meaningless to try to find all of them. However, some of these constants that have contributed to a better knowledge of the 1D quadratic maps have been published. In this work we illustrate some of the central features of the most important scaling constants and we introduce another one which has the n...

At this point we recall the following procedure for generating a r.v. X with PDF FX(x) from a uniform r.v. Y . As stated on page 125 in [3], “...given a uniform r.v. Y , the transformation X = F−1 X (Y ) will generate a r.v. with PDF FX(x)”. Note that F −1 X (y) denotes the inverse function, such that F−1 X (FX(x)) = x [4]. From the plot of FX(x) shown in Figure 1(b), it is apparent that FX(x) ...

We study the structure of the 2D bifurcation diagram for a two-parameter family of piecewise smooth unimodal maps f with one break point. Analysing the parameters of the normal form for the border-collision bifurcation of an attracting n-cycle of the map f, we describe the possible kinds of dynamics associated with such a bifurcation. Emergence and role of border-collision bifurcation curves in...