@article{20839,
  abstract     = {For every couple of Hausdorff functions ψ and φ verifying some mild assumptions, there exists a compact subset K of the Baire space such that the φ-Hausdorff measure and the ψ-packing measure on K are both finite and positive. Such examples are then embedded in any infinite dimensional Banach space to answer positively a question of Fan on the existence of metric spaces with arbitrary scales.},
  author       = {Helfter, Mathieu},
  issn         = {2308-1317},
  journal      = {Journal of Fractal Geometry},
  publisher    = {EMS Press},
  title        = {{Sets with arbitrary Hausdorff and packing scales in infinite dimensional Banach spaces}},
  doi          = {10.4171/jfg/177},
  year         = {2025},
}

@article{19496,
  abstract     = {We introduce the notions of scale for sets and measures on metric space by generalizing the usual notions of dimension. Several versions of scales are introduced such as Hausdorff, packing, box, local and quantization. They are defined for different growth, allowing a refined study of infinite dimensional spaces. We prove general theorems comparing the different versions of scales. They are applied to describe geometries of ergodic decompositions, of the Wiener measure and from functional spaces. The first application solves a problem of Berger on the notions of emergence (2020); the second lies in the geometry of the Wiener measure and extends the work of Dereich–Lifshits (2005); the last refines Kolmogorov–Tikhomirov (1958) study on finitely differentiable functions.},
  author       = {Helfter, Mathieu},
  issn         = {1432-1823},
  journal      = {Mathematische Zeitschrift},
  publisher    = {Springer Nature},
  title        = {{Scales}},
  doi          = {10.1007/s00209-025-03719-5},
  volume       = {310},
  year         = {2025},
}

@article{20838,
  abstract     = {For any 1 ≤ r ≤ ∞, we show that every diffeomorphism of a manifold of the form
R/Z × M is a total renormalization of a Cr-close to identity map. In other words, for
every diffeomorphism f of R/Z×M, there exists a map g arbitrarily close to identity
such that the first return map of g to a domain is conjugate to f and moreover the
orbit of this domain is equal to R/Z×M. This enables us to localize near the identity
the existence of many properties in dynamical systems, such as being Bernoulli for a
smooth volume form.},
  author       = {Berger, Pierre and Gourmelon, Nicolaz and Helfter, Mathieu},
  issn         = {1432-1297},
  journal      = {Inventiones mathematicae},
  number       = {2},
  pages        = {431--468},
  publisher    = {Springer Nature},
  title        = {{Every diffeomorphism is a total renormalization of a close to identity map}},
  doi          = {10.1007/s00222-024-01305-w},
  volume       = {239},
  year         = {2024},
}