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We define Frobenius and monodromy operators on the de Rham cohomology of$K$-dagger spaces (rigid spaces with overconvergent structure sheaves) withstrictly semistable reduction $Y$, over a complete discrete valuation ring $K$of mixed characteristic. For this we introduce log rigid cohomology andgeneralize the so called Hyodo-Kato isomorphism to versions for non-proper $Y$,for non-perfect residue fields, for non-integrally defined coefficients, andfor the various strata of $Y$. We apply this to define and investigatecrystalline structure elements on the de Rham cohomology of Drinfel'd'ssymmetric space $X$ and its quotients. Our results are used in a critical wayin the recent proof of the monodromy-weight conjecture for quotients of $X$given by de Shalit.

Frobenius and monodromy operators in rigid analysis, and Drinfel'd's symmetric space