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We consider the Lie group of smooth diffeomorphisms Diff$(M)$ of a simple polytope $M$ in the euclidean space. Simple polytopes are special cases of manifolds with corners. The geometric setting allows to study in particular, the subgroup of face respecting diffeomorphisms and its Lie theoretic properties. We find a canonical Lie group structure for the quotient of the diffeomorphism by the subgroup Diff$^{\partial,id}(M)$ of maps that equal the identity on the boundary, turning the canonical quotient homomorphism Diff$(M)\rightarrow $Diff$(M)/$Diff$^{\partial,id}(M)$ into a smooth submersion. We also show that the identity component of the diffeomorphism group is generated by the exponential image, by proving general controllability results., Comment: 34 pages, v2: significantly improved the controllability result, corrected typos, rest of the results remain unchanged

One of the fundamental problems in shape analysis is to align curves or surfaces before computing geodesic distances between their shapes. Finding the optimal reparametrization realizing this alignment is a computationally demanding task, typically done by solving an optimization problem on the diffeomorphism group. In this paper, we propose an algorithm for constructing approximations of orientation-preserving diffeomorphisms by composition of elementary diffeomorphisms. The algorithm is implemented using PyTorch, and is applicable for both unparametrized curves and surfaces. Moreover, we show universal approximation properties for the constructed architectures, and obtain bounds for the Lipschitz constants of the resulting diffeomorphisms., Comment: 36 pages, 11 figures. Accepted by BIT Numerical Mathematics, not yet published

The concepts of the scale and tidy subgroups for an automorphism of a totally disconnected locally compact group were defined in seminal work by George A. Willis in the 1990s, and recently generalized to the case of endomorphisms (G. A. Willis, Math. Ann. 361 (2015), 403--442). We show that central facts concerning the scale, tidy subgroups, quotients, and contraction groups of automorphisms extend to the case of endomorphisms. In particular, we obtain results concerning the domain of attraction around an invariant closed subgroup., Comment: v3: 60 pages, this revised version will be accepted for publication in the Israel Journal of Mathematics