Aniq fanlar

Critical cusp cancellation and analytic response for a Bernoulli random interval map

cusp maps, transfer operators, invariant density, Lasota–Yorke inequality, analytic response, random dynamical systems.

Авторы

  • Davrbek Oltiboev V.I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, Uzbekistan, Узбекистан
  • Furqat Vafoyev National University of Uzbekistan named after M.Ulugbek, Tashkent, Uzbekistan, Узбекистан
Том 2 № 2.3 (2026): O'zMU xabarlari. Aniq fanlar
Статьи

Загрузки

We show that the critical cusp exponent β = −1/2 admits W 2,1 invariant densities in an
explicit family of expanding full-branch interval maps. The usual estimates exclude this exponent because
two zero-order coefficients in the second derivative of the transfer operator are individually non-integrable.
Their signed combination, however, contains the factor β(2β + 1) for a pure power cusp. We exploit its
cancellation at β = −1/2 to prove a W 2,1 Lasota–Yorke inequality and a spectral gap for the explicit family.
We also identify a separate obstruction caused by an interior cusp value. As an application, a small Bernoulli
probability perturbation between two fixed maps has a stationary density that depends real-analytically on
the perturbation parameter in W 2,1 , with explicit formulas for all response derivatives.