Typical representatives of free homotopy classes in multi-punctured plane

Maxim Arnold, Yuliy Baryshnikov, Yuriy Mileyko

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


We show that a uniform probability measure supported on a specific set of piecewise linear loops in a nontrivial free homotopy class in a multi-punctured plane is overwhelmingly concentrated around loops of minimal lengths. Our approach is based on extending Mogulskii's theorem to closed paths, which is a useful result of independent interest. In addition, we show that the above measure can be sampled using standard Markov Chain Monte Carlo techniques, thus providing a simple method for approximating shortest loops.

Original languageEnglish
Pages (from-to)623-659
Number of pages37
JournalJournal of Topology and Analysis
Issue number3
Publication statusPublished - Sept 1 2019

All Science Journal Classification (ASJC) codes

  • Analysis
  • Geometry and Topology


Dive into the research topics of 'Typical representatives of free homotopy classes in multi-punctured plane'. Together they form a unique fingerprint.

Cite this