Phase Transition of Logarithmic Capacity for the Uniform Gδ-Sets


We consider a family of dense Gδ subsets of [0, 1], defined as intersections of unions of small uniformly distributed intervals, and study their logarithmic capacity. Changing the speed at which the lengths of generating intervals decrease, we observe a sharp phase transition from full to zero capacity. Such a Gδ set can be considered as a toy model for the set of exceptional energies in the parametric version of the Furstenberg theorem on random matrix products. Our re-distribution construction can be considered as a generalization of a method applied by Ursell in his construction of a counter-example to a conjecture by Nevanlinna. Also, we propose a simple Cauchy-Schwartz inequality-based proof of related theorems by Lindeberg and by Erdös and Gillis.

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Victor Kleptsyn was partially supported by the project ANR Gromeov (ANR-19-CE40-0007), as well as by the Laboratory of Dynamical Systems and Applications NRU HSE, of the Ministry of science and higher education of the RF grant ag. No 075-15-2019-1931. Fernando Quintino was supported by NSF grants DMS-1855541 and DMS-1700143.

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Correspondence to Fernando Quintino.

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Kleptsyn, V., Quintino, F. Phase Transition of Logarithmic Capacity for the Uniform Gδ-Sets. Potential Anal (2021).

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  • Logarithmic capacity
  • Phase transition
  • Parametric Furstenberg theorem

Mathematics Subject Classification (2010)

  • Primary: 31A15 · 31C15
  • Secondary: 28A12