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We consider finite-state Markov chains that can be naturally decomposed into smaller "projection" and "restriction" chains. Possibly this decomposition will be inductive, in that the restriction chains will be smaller copies of the initial chain. We provide expressions for Poincare (resp. log-Sobolev) constants of the initial Markov chain in terms of Poincare (resp. log-Sobolev) constants of the projection and restriction chains, together with further a parameter. In the case of the Poincaredoi:10.1214/105051604000000639 fatcat:ngikmzgluba6xhudwtesdc3txy