Geodesic
Scaling entropy sequence: invariance and examples
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIV, Tome 432 (2015), pp. 128-161

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A scaling entropy sequence of an automorphism is an entropy-type metric invariant suggested by A. M. Vershik. We confirm his conjecture that it does not depend on the choice of a semimetric. This means that it is indeed a metric invariant. We also calculate this invariant for several classical dynamical systems. 
@article{ZNSL_2015_432_a8,
     author = {P. B. Zatitskiy},
     title = {Scaling entropy sequence: invariance and examples},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {128--161},
     publisher = {mathdoc},
     volume = {432},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2015_432_a8/}
}
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P. B. Zatitskiy. Scaling entropy sequence: invariance and examples. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XXIV, Tome 432 (2015), pp. 128-161. http://geodesic.mathdoc.fr/item/ZNSL_2015_432_a8/