Abstract
In this paper, local entropy via the preimage structure for noninvertible maps is considered. For a topological dynamical system and its factor, several relative and local versions of preimage entropies with respect to a Borel cover from topological and measure-theoretic viewpoints are introduced and investigated. The relationships among these quantities are discussed, in particular, variational principles (or variational inequalities) are established. Moreover, it is shown that each local version of these relative preimage entropies coincides with its corresponding global version for a system with uniform separation of preimages.
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