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Differentiable conjugacies link dynamical systems that share properties such as the stability multipliers of corresponding orbits. It provides a stronger classification than topological conjugacy, which only requires qualitative similarity.…
By associating to a curve C of genus g=2k and a pencil of degree d=k+1 the so-called trace curve (resp. the reduced trace curve) we define a rational map from the Hurwitz space of admissible covers of genus g=2k and degree d=k+1 to a moduli…
We give new upper bounds for the number of nonconstant holomorphic maps depending only on the genus. Our estimates improve previously known bounds. The proof is based on the study of pullbacks of holomorphic differentials, together with…
The limit of energies of a sequence of harmonic maps as their annular domains approach the boundary of moduli space depends upon the boundary point approached. The infinite energy case is associated with limits of images containing ruled…
Recently continuous rational maps between real algebraic varieties have attracted the attention of several researchers. In this paper we continue the investigation of approximation properties of continuous rational maps with values in…
Any affine map on the (n+1)-dimensional Euclidean space gives rise to a natural map on the n-dimensional sphere whose dynamical aspects are not so well-studied in the literature. We explore the dynamical aspects of these maps by…
In this paper we give an explanation of a number of observations relating to degree growth of birational mappings of the plane and their deautonomisation by singularity confinement. These observations are of a link between two a priori…
We consider families of mappings with moduli inequalities, having different definition domains. Under some additional assumptions we have proved that such families are uniformly equicontinuous. We have considered four main cases: when…
One proves a far-reaching upper bound for the degree of a generically finite rational map between projective varieties over a base field of arbitrary characteristic. The bound is expressed as a product of certain degrees that appear…
In this paper, we consider boundary extensions of two classes of mappings between metric measure spaces. These two mapping classes extend in particular the well-studied geometric mappings such as quasiregular mappings with integrable…
Experiments observing the liquid surface in a vertically oscillating container have indicated that modeling the dynamics of such systems require maps that admit states at infinity. In this paper we investigate the bifurcations in such a…
We consider moduli spaces of dynamical systems of correspondences over the projective line as a generalization of moduli spaces of dynamical systems of endomorphisms on the projective line. We obtain the rationality of the moduli spaces.…
In these lecture notes, we present a connection between the complex dynamics of a family of rational functions $f_t: \mathbb{P}^1\to \mathbb{P}^1$, parameterized by $t$ in a Riemann surface $X$, and the arithmetic dynamics of $f_t$ on…
In the late 1980's Sullivan initiated a programme to prove quasisymmetric rigidity in one-dimensional dynamics: interval or circle maps that are topologically conjugate are quasisymmetrically conjugate (provided some obvious necessary…
The topological interpretation of modal logics provides descriptive languages and proof systems for reasoning about points of topological spaces. Recent work has been devoted to model checking of spatial logics on discrete spatial…
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 if $f$ and $g$ are postcritically finite rational maps whose Julia sets $\mathcal{J}(f), \mathcal{J}(g)$, respectively, are Sierpi\'nski carpets, and if $\xi$ is a quasiregular map of the Riemann sphere $\widehat{\mathbb{C}}$…
Working in positive characteristic, we show how one can use information about the dimension of moduli spaces of rational curves on a Fano variety $X$ over $\mathbb{F}_q$ to obtain strong estimates for the number of $\mathbb{F}_q(t)$-points…
We proved the so called complex bounds for multimodal, infinitely renormalizable analytic maps with bounded combinatorics: deep renormalizations have polynomial-like extensions with definite modulus. The complex bounds is the first step to…
An important question is to describe topological conjugacy classes of dynamical systems. Here we show that within the space of real analytic one-dimensional maps with critical points of prescribed order, the conjugacy class of a map is a…