For any affine variety equipped with coordinates, there is a surjective, continuous map from its Berkovich space to its tropicalisation. Exploiting torus actions, we develop techniques for finding an explicit, continuous section of this map. In particular, we prove that such a section exists for linear spaces, Grassmannians of planes (reproving a result due to Cueto, Häbich, and Werner), matrix...

We say that a continuous map / of a compact interval to itself is linear Markov if it is piecewise linear, and the set of all fk(x), where k > 0 and x is an endpoint of a linear piece, is finite. We provide an effective classification, up to topological conjugacy, for linear Markov maps and an effective procedure for determining whether such a map is transitive. We also consider expanding Marko...

We consider a piecewise continuous, piecewise monotone interval map and a weight of bounded variation, constant on homtervals and continuous at periodic points of the map. With these data we associate a sequence of weighted Milnor-Thurston kneading matrices, converging to a countable matrix with coeecients analytic functions. We show that the determinants of these matrices converge to the inver...

We discuss BFV deformation quantization [5] in the special case of a linear Hamiltonian torus action. In particular, we show that the Koszul complex on the moment map of an effective linear Hamiltonian torus action is acyclic. We rephrase the nonpositivity condition of [2] for linear Hamiltonian torus actions. It follows that reduced spaces of such actions admit continuous star products.

An N-dimensional real representation E of a finite group G is said to have the “Borsuk-Ulam Property” if any continuous G-map from the (N + 1)-fold join of G (an N-complex equipped with the diagonal G-action) to E has a zero. This happens iff the “Van Kampen characteristic class” of E is nonzero, so using standard computations one can explicitly characterize representations having the B-U prope...

The input output map of linear time-continuous systems is investigated. The class of all systems for which the input output map is given by the usual integral is investigated. A complete characterization of this class of systems is given in the paper.

We shall consider slightly more general problems. Namely, we shall investigate the existence of continuous: piecewise monotone, piecewise strictly monotone, and piecewise linear n-th roots of interval maps which have a continuous n-th root. Here by an n-th root of f we mean a map g such that f = g (g is the n-th iterate of g). A continuous map f : I → J , where I, J are closed intervals, is pie...

In the present article, we investigate a possibility of a real-valued map on the space of tuples of commuting trace-class self-adjoint operators, which behaves like the usual trace map on the space of trace-class linear operators. It turns out that such maps are related with continuous group homomorphisms from the Milnor’s K-group of the real numbers into the additive group of real numbers. Usi...