On Thurston's Euler class one conjecture

In 1976, Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that conversely, any integral second cohomology class with norm equal to one is the Euler class of a taut foliation. This is the first from a series of two papers that together give a negative answer to Thurston's conjecture. Here counterexamples have been constructed conditional on the fully marked surface theorem. In the second paper, joint with David Gabai, a proof of the fully marked surface theorem is given.Comment: 42 pages, 21 figures. The paper is split into two parts, and the appendix is appearing as a separate article joint with David Gabai. The results on taut foliations on sutured solid tori are generalised. A section on relative Euler class is added to address a possible oversight in the literature. Exposition is improved, and new open questions are raised. Final version to appear in Acta Mathematic

Lower bound for the Perron-Frobenius degrees of Perron numbers

Using an idea of Doug Lind, we give a lower bound for the Perron-Frobenius degree of a Perron number that is not totally-real. As an application, we prove that there are cubic Perron numbers whose Perron-Frobenius degrees are arbitrary large; a result known to Lind, McMullen and Thurston. A similar result is proved for biPerron numbers.Comment: To appear in Ergodic Theory and Dynamical Systems, 15 pages, 4 figure

Non-negative integral matrices with given spectral radius and controlled dimension

A celebrated theorem of Lind states that a positive real number is equal to the spectral radius of some integral primitive matrix, if and only if, it is a Perron algebraic integer. Given a Perron number $p$, we prove that there is an integral irreducible matrix with spectral radius $p$, and with dimension bounded above in terms of the algebraic degree, the ratio of the first two largest Galois conjugates, and arithmetic information about the ring of integers of its number field. This arithmetic information can be taken to be either the discriminant or the minimal Hermite-like thickness. Equivalently, given a Perron number $p$, there is an irreducible shift of finite type with entropy $\log(p)$ defined as an edge shift on a graph whose number of vertices is bounded above in terms of the aforementioned data.Comment: The referee's suggestions are incorporated; in particular an upper bound for the Perron--Frobenius degree is derived from the main result. See Theorem 1.6. To appear in Ergodic Theory and Dynamical System

Pseudo-Anosov maps with small stretch factors on punctured surfaces

Consider the problem of estimating the minimum entropy of pseudo-Anosov maps on a surface of genus $g$ with $n$ punctures. We determine the behaviour of this minimum number for a certain large subset of the $(g,n)$ plane, up to a multiplicative constant. In particular it has been shown that for fixed $n$, this minimum value behaves as $\frac{1}{g}$, proving what Penner speculated in 1991.Comment: To appear in Algebraic & Geometric Topology, 26 pages, 10 figure

LATE DEVONIAN-CARBONIFEROUS CONODONTS FROM EASTERN IRAN

Conodont data from acid-leaching 110 samples from two Late Devonian-Carboniferous areas in the Shotori Range (Tabas region) of eastern Iran are presented. At Howz-e-Dorah, a section (88 samples) commencing high in the Bahram Formation (Givetian-early Frasnian) extended through the Shishtu Formation (Frasnian, Early hassi Zone or older, to latest Tournaisian, anchoralis-latus Zone) and the Sardar Formation (earliest Visean, texanus Zone, to late Namurian, sinuatus-corrugatus-sulcatus Zone) and into the Jamal Formation (Permian). Four less exhaustively sampled sections (22 samples) show the Kale Sardar area to be tectonically more complicated than the Howz-e-Dorah area. Useful marker horizons in the Howz-e-Dorah section, well constrained by conodont data, are: the early Frasnian (no older than Early hassi Zone) biostromal beds of the Shishtu Formation, an early Famennian (Late triangularis to Early crepida) interval of oolitic limestone, a cyclothem sequence straddling the Early Carboniferous-Late Carboniferous boundary, and an Early Permian interval of siliceous sand ("the white quartzite" of previous authors). Additionally, several iron-rich horizons, readily traceable from locality to locality, are well constrained by conodont ages. Eighty-five conodont species/subspecies are documented representing 24 genera.. Two new species, Polygnathus capollocki and Polygnathus ratebi and one new subspecies, Icriodus alternatus mawsonae are described.&nbsp

The fully marked surface theorem

In his seminal 1976 paper Bill Thurston observed that a closed leaf S of a foliation has Euler characteristic equal, up to sign, to the Euler class of the foliation evaluated on [S], the homology class represented by S. The main result of this paper is a converse for taut foliations: if the Euler class of a taut foliation $\mathcal{F}$ evaluated on [S] equals up to sign the Euler characteristic of S and the underlying manifold is hyperbolic, then there exists another taut foliation $\mathcal{F'}$ such that $S$ is homologous to a union of leaves and such that the plane field of $\mathcal{F'}$ is homotopic to that of $\mathcal{F}$. In particular, $\mathcal{F}$ and $\mathcal{F'}$ have the same Euler class. In the same paper Thurston proved that taut foliations on closed hyperbolic 3-manifolds have Euler class of norm at most one, and conjectured that, conversely, any integral cohomology class with norm equal to one is the Euler class of a taut foliation. This is the second of two papers that together give a negative answer to Thurston's conjecture. In the first paper, counterexamples were constructed assuming the main result of this paper.Comment: 36 pages, 16 figures. Portions of this work previously appeared as an appendix to arXiv:1603.03822, but has evolved into its own work and has been accepted for publication separately. Final version to appear in Acta Mathematic

Thurston norm and Euler classes of tight contact structures

Bill Thurston proved that taut foliations of hyperbolic 3-manifolds have Euler classes of norm at most one, and conjectured that any integral second cohomology class of norm equal to one is realised as the Euler class of some taut foliation. Recent work of the second author, joint with David Gabai, has produced counterexamples to this conjecture. Since tight contact structures exist whenever taut foliations do and their Euler classes also have norm at most one, it is natural to ask whether the Euler class one conjecture might still be true for tight contact structures. In this short note, we show that the previously constructed counterexamples for Euler classes of taut foliations in [Yaz20] are in fact realised as Euler classes of tight contact structures. This shows that the Euler class one conjecture has a chance of being true for tight contact structures.Comment: 12 pages, 4 figure

A New Amplification Regime for Traveling Wave Tubes with Third Order Modal Degeneracy

Engineering of the eigenmode dispersion of slow-wave structures (SWSs) to achieve desired modal characteristics, is an effective approach to enhance the performance of high power traveling wave tube (TWT) amplifiers or oscillators. We investigate here for the first time a new synchronization regime in TWTs based on SWSs operating near a third order degeneracy condition in their dispersion. This special three-eigenmode synchronization is associated with a stationary inflection point (SIP) that is manifested by the coalescence of three Floquet-Bloch eigenmodes in the SWS. We demonstrate the special features of "cold" (without electron beam) periodic SWSs with SIP modeled as coupled transmission lines (CTLs) and investigate resonances of SWSs of finite length. We also show that by tuning parameters of a periodic SWS one can achieve an SIP with nearly ideal flat dispersion relationship with zero group velocity or a slightly slanted one with a very small (positive or negative) group velocity leading to different operating schemes. When the SIP structure is synchronized with the electron beam potential benefits for amplification include (i) gain enhancement, (ii) gain-bandwidth product improvement, and (iii) higher power efficiency, when compared to conventional Pierce-like TWTs. The proposed theory paves the way for a new approach for potential improvements in gain, power efficiency and gain-bandwidth product in high power microwave amplifiers.Comment: 15 pages, 11 figure