  Markov Chain Monte Carlo (MCMC) algorithms ubiquitously employ complex
deterministic transformations to generate proposal points that are then
filtered by the Metropolis-Hastings-Green (MHG) test. However, the condition of
the target measure invariance puts restrictions on the design of these
transformations. In this paper, we first derive the acceptance test for the
stochastic Markov kernel considering arbitrary deterministic maps as proposal
generators. When applied to the transformations with orbits of period two
(involutions), the test reduces to the MHG test. Based on the derived test we
propose two practical algorithms: one operates by constructing periodic orbits
from any diffeomorphism, another on contractions of the state space (such as
optimization trajectories). Finally, we perform an empirical study
demonstrating the practical advantages of both kernels.
