Let φ : X → X be a map on an projective variety. It is known that whenever φ∗ : Pic(X) ⊗ R → Pic(X) ⊗ R has an eigenvalue α > 1, we can build a canonical measure, a canonical height and a canonical metric associated to φ. In the present work, we establish the following fact: if two commuting maps φ, ψ : X → X satisfy these conditions, for eigenvalues α and β and the same eigenvector L, then the...

Fixed points are also called as invariant points. Invariant point theorems very essential tools in solving problems arising different branches of mathematical analysis. In the present paper, we establish three unique common using two self-mappings, four self-mappings and six bicomplex valued metric space. first theorem, generate a theorem for by weaker conditions such weakly compatible, general...

On a space of stable maps, the psi classes are modified by subtracting certain boundary divisors. The top products of modified psi classes, usual psi classes, and classes pulled back along the evaluation maps are called twisted descendants; it is shown that in genus 0, they admit a complete recursion and are determined by the Gromov-Witten invariants. One motivation for this construction is tha...