Singularity of the Spectrum for Smooth Area-Preserving Flows in Genus Two and Translation Surfaces Well Approximated by Cylinders

Chaika, Jon; Frączek, Krzysztof; Kanigowski, Adam; Ulcigrai, Corinna

doi:10.1007/s00220-020-03895-x

Cited by 12 publications

(5 citation statements)

References 40 publications

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“…zero) class of locally Hamiltonian flows in U min for which the Poincaré section can be chosen to be a selfsimilar interval exchange transformation 9 and restrict the observable f to belong to an infinite dimensional (but finite codimension g) space. For extensions of flows in this special class, though, we could provide a complete description of the ergodic behavior and prove a dichotomy between ergodicity and reducibility.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…For some flows in genus one with a degenerate singularity (sometimes known as Kochergin flows), Forni, Fayad and Kanigowski could recently, prove in [12] that the spectrum is countably Lebesgue. The first typical spectral result for surfaces of higher genus, namely g ≥ 2 was recently proved by Chaika, Kanigowski and the authors, who showed in [9] that a typical locally Hamiltonian flow on a genus two surface with two isomorphic simple saddles has purely singular spectrum.…”

Section: Definitions Background Materials and Reductionsmentioning

confidence: 99%

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On the asymptotic growth of Birkhoff integrals for locally Hamiltonian flows and ergodicity of their extensions

Frączek¹,

Ulcigrai²

2021

Preprint

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We consider smooth area-preserving flows (also known as locally Hamiltonian flows) on surfaces of genus g ≥ 1 and study ergodic integrals of smooth observables along the flow trajectories. We show that these integrals display a power deviation spectrum and describe the cocycles that lead the pure power behaviour, giving a new proof of results by Forni (Annals 2002) and Bufetov (Annals 2014) and generalizing them to observables which are non-zero at fixed points. This in particular completes the proof of the original formulation of the Kontsevitch-Zorich conjecture. Our proof is based on building suitable correction operators for cocycles with logarithmic singularities over a full measure set of interval exchange transformations (IETs), in the spirit of Marmi-Moussa-Yoccoz work on piecewise smooth cocycles over IETs. In the case of symmetric singularities, exploiting former work of the second author (Annals 2011), we prove a tightness result for a finite codimension class of observables. We then apply the latter result to prove the existence of ergodic infinite extensions for a full measure set of locally Hamiltonian flows with non-degenerate saddles in any genus g ≥ 2.2000 Mathematics Subject Classification. 37E35, 37A40, 37A10, 37C83. 1 More precisely, referring to the decomposition described in § 2.1.1, we call Arnold flow the restriction to a minimal component obtained by removing the center and the disk filled by periodic orbits around it (called island), which, as Arnold shows in [2], is always bounded by a saddle loop.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Definitions Background Materials and Reductionsmentioning

confidence: 99%

On the asymptotic growth of Birkhoff integrals for locally Hamiltonian flows and ergodicity of their extensions

Frączek¹,

Ulcigrai²

2021

Preprint

Self Cite

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“…Thus it remains to check the Condition 1 in Proposition 1.4. For this purpose, we consider the following result which follows from [5,Lemma 4.6]. Lemma 3.11.…”

Section: Proof Of the Skew-products Ergodicitymentioning

confidence: 99%

Ergodicity of explicit logarithmic cocycles over IETs

Berk¹,

Trujillo²,

Ulcigrai³

2022

Preprint

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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.

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“…On the other hand, understanding more subtle spectral properties of locally Hamiltonian flows seems to be still in its infancy. Only recently the first results have appeared for the singular spectrum in [3] and the countable Lebesgue spectrum in [6].…”

Section: 11mentioning

confidence: 99%

New phenomena in deviation of Birkhoff integrals for locally Hamiltonian flows

Frączek¹,

Kim²

2021

Preprint

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We consider smooth locally Hamiltonian flows on compact surfaces of genus g ≥ 2 to prove their deviation of Birkhoff integrals for smooth observables. Our work generalizes results of Forni [10] and Bufetov [2] which prove the existence of deviation spectrum of Birkhoff integrals for observables whose jets vanish at sufficiently high order around fixed points of the flow. They showed that ergodic integrals can display a power spectrum of behaviours with exactly g positive exponents related to the positive Lyapunov exponents of the cocycle so-called Kontsevich-Zorich, a renormalization cocycle over the Teichmüller flow on a stratum of the moduli space of translation surfaces.Our paper extends the study of the spectrum of deviations of ergodic integrals beyond the case of observables whose jets vanish at sufficiently high order around fixed points. We prove the existence of some extra terms in deviation spectrum related to non-vanishing of the derivatives of observables at fixed points. The proof of this new phenomenon is based on tools developed in the recent work of the first author and Ulcigrai [8] for locally Hamiltonian flows having only (simple) non-degenerate saddles. In full generality, due to the occurrence of (multiple) degenerate saddles, we introduce new methods of handling functions with polynomial singularities over a full measure set of IETs.

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Singularity of the Spectrum for Smooth Area-Preserving Flows in Genus Two and Translation Surfaces Well Approximated by Cylinders

Cited by 12 publications

References 40 publications

On the asymptotic growth of Birkhoff integrals for locally Hamiltonian flows and ergodicity of their extensions

On the asymptotic growth of Birkhoff integrals for locally Hamiltonian flows and ergodicity of their extensions

Ergodicity of explicit logarithmic cocycles over IETs

New phenomena in deviation of Birkhoff integrals for locally Hamiltonian flows

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