Ergodicity of explicit logarithmic cocycles over IETs

2022/10/28 by P. Berk, Przemysław Berk, Frank Trujillo +6 · 2 citations
Mathematics · Computer Science · #Dynamical Systems (math.DS) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2210.16343

Abstract

We prove ergodicity in a class of skew-product extensions of interval exchange transformations given by cocycles with logarithmic singularities. This, in particular, gives explicit examples of ergodic ℝ-extensions of minimal locally Hamiltonian flows with non-degenerate saddles in genus two. More generally, given any symmetric irreducible permutation, we show that for almost every choice of lengths vector, the skew-product built over the IET with the given permutation and lengths vector given by a cocycle, with symmetric, logarithmic singularities, which is odd when restricted to each continuity subinterval is ergodic.

Citations

• Recurrence of Co-Cycles and Random Walks
• Ergodicity of cylinder flows arising from irregularities of distribution
• Non-ergodic ℤ -periodic billiards and infinite translation surfaces
• Ergodicity of skew products over linearly recurrent IETs

Cited by

• Coexistence of two contrasting recurrence properties of certain non-integrable cocycles
• On the ergodicity of anti-symmetric skew products with singularities and its applications

Related

• On the ergodicity of anti-symmetric skew products with singularities and its applications
• Ergodic properties of infinite extension of symmetric interval exchange transformations
• Cocycles over interval exchange transformations and multivalued Hamiltonian flows
• On the ergodicity of infinite antisymmetric extensions of symmetric IETs
• Skew-product systems over infinite interval exchange transformations