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label: paragraph


text: which can be checked via carefully managing ﬂoating point computations. Floating point checks of the hypotheses employ interval arithmetic in order to guarantee that we obtain mathematically rigorous results [74, 80, 83]. In order to formulate the suﬃcient conditions just discussed we implement a computer assisted a-posteriori scheme which has its roots in the the seminal work of [60, 32] on the Feigenbaum conjectures. We study the equation ◦ − f(P(θ)) = (g P)(θ) DP(θ)A θ = 0, u via a modiﬁed Newton-Kantorovich argument. More precisely, we show that the “Newton- like” operator − T(P) := P Af(P), is a contraction on a ball of radius r in the Banach space about the approximation solution P PMN. Here A is a problem dependent approximate inverse of Df(PMN), which we choose based on both numerical and analytic considerations. The argument is formalized in Section , with implementation discussed in Section . 1.3 Discussion In practice we learn that the argument outlined in Section succeeds or fails only after attempting the a-posteriori validation, and these attempts are computationally expensive. With this in mind we include the a-priori Theorem 3.2 in Section . The theorem says that, given some mild non-resonance conditions between the unstable eigenvalues (assumptions which are made precise in Deﬁnition 3.1) there exist choices of eigenvectors so that Equation (6) has a solution. The solution is unique up to the choice of the eigenvectors, but a-priori may parameterize only a small portion of the local unstable manifold near the equilibrium. Our Theorem is an inﬁnite dimensional generalization of the results in Section 10 of [14]. The proof of the Theorem 3.2 assures existence only for a small enough choice of the scalings of the unstable eigenvectors. On the other hand, in Section 3.1 we see that the size of the local manifold parameterized by P is determined in a rather explicit way by the size of these scalings. In applications we would like to choose the eigenvector scalings as large as possible, so that we learn more about the unstable manifold far from the equilibrium solution. This desire must be weighed against the fact that for larger choices of the scalings we risk loosing control of the convergence of the series. Viewed in this light, the tools of the present work provide a mathematically rigorous computer assisted method for pushing the existence results as far as possible in speciﬁc applications. Since the desired parameterization exists in a small enough neighborhood of the equilibrium solution by Theorem 3.2, our argument has a “continuation” ﬂavor (the continuation parameters being the eigenvector scalings, which in turn govern the size of the local unstable manifold in phase space). The novelty is that we care only about rigorous results at the end of the continuation. We argue in Sections and that expensive validation computations can be postponed until we are all but certain the computer aided proof will succeed. These considerations are also discussed at length for ﬁnite dimensional vector ﬁelds in [11]. Remark 1.5 (Extensions of the a-priori results). A word about the technical assumptions of Theorem 3.2. In addition to the non-resonance conditions postulated in Deﬁnition 3.1, Theorem 3.2 postulates that each of the unstable eigenvalues is real and that each has multiplicity one. Moreover we assume that the nonlinearity is given by an analytic function. We remark that these assumptions are technically convenient, and should not be inter- preted as fundamental restrictions. For example real invariant manifolds associated with 7

text: which

text: can

text: be

text: checked

text: via

text: carefully

text: managing

text: ﬂoating

text: point

text: computations.

text: Floating

text: point

text: checks

text: of

text: the

text: hypotheses

text: employ

text: interval

text: arithmetic

text: in

text: order

text: to

text: guarantee

text: that

text: we

text: obtain

text: mathematically

text: rigorous

text: results

text: In

text: order

text: to

text: formulate

text: the

text: suﬃcient

text: conditions

text: just

text: discussed

text: we

text: implement

text: a

text: computer

text: assisted

text: a-posteriori

text: scheme

text: which

text: has

text: its

text: roots

text: in

text: the

text: the

text: seminal

text: work

text: of

text: on

text: the

text: Feigenbaum

text: conjectures.

text: We

text: study

text: the

text: equation

text: f(P(θ))

text: (g

text: P)(θ)

text: DP(θ)A

text: θ

text: u

text: via

text: a

text: modiﬁed

text: Newton-Kantorovich

text: argument.

text: More

text: precisely,

text: we

text: show

text: that

text: the

text: “Newton-

text: like”

text: operator

text: T(P)

text: P

text: Af(P),

text: is

text: a

text: contraction

text: on

text: a

text: ball

text: of

text: radius

text: r

text: in

text: the

text: Banach

text: space

text: about

text: the

text: approximation

text: solution

text: P

text: PMN.

text: Here

text: A

text: is

text: a

text: problem

text: dependent

text: approximate

text: inverse

text: of

text: Df(PMN),

text: which

text: we

text: choose

text: based

text: on

text: both

text: numerical

text: and

text: analytic

text: considerations.

text: The

text: argument

text: is

text: formalized

text: in

text: Section

text: with

text: implementation

text: discussed

text: in

text: Section

text: Discussion

text: In

text: practice

text: we

text: learn

text: that

text: the

text: argument

text: outlined

text: in

text: Section

text: succeeds

text: or

text: fails

text: only

text: after

text: attempting

text: the

text: a-posteriori

text: validation,

text: and

text: these

text: attempts

text: are

text: computationally

text: expensive.

text: With

text: this

text: in

text: mind

text: we

text: include

text: the

text: a-priori

text: Theorem

text: in

text: Section

text: The

text: theorem

text: says

text: that,

text: given

text: some

text: mild

text: non-resonance

text: conditions

text: between

text: the

text: unstable

text: eigenvalues

text: (assumptions

text: which

text: are

text: made

text: precise

text: in

text: Deﬁnition

text: there

text: exist

text: choices

text: of

text: eigenvectors

text: so

text: that

text: Equation

text: has

text: a

text: solution.

text: The

text: solution

text: is

text: unique

text: up

text: to

text: the

text: choice

text: of

text: the

text: eigenvectors,

text: but

text: a-priori

text: may

text: parameterize

text: only

text: a

text: small

text: portion

text: of

text: the

text: local

text: unstable

text: manifold

text: near

text: the

text: equilibrium.

text: Our

text: Theorem

text: is

text: an

text: inﬁnite

text: dimensional

text: generalization

text: of

text: the

text: results

text: in

text: Section

text: of

text: The

text: proof

text: of

text: the

text: Theorem

text: assures

text: existence

text: only

text: for

text: a

text: small

text: enough

text: choice

text: of

text: the

text: scalings

text: of

text: the

text: unstable

text: eigenvectors.

text: On

text: the

text: other

text: hand,

text: in

text: Section

text: we

text: see

text: that

text: the

text: size

text: of

text: the

text: local

text: manifold

text: parameterized

text: by

text: P

text: is

text: determined

text: in

text: a

text: rather

text: explicit

text: way

text: by

text: the

text: size

text: of

text: these

text: scalings.

text: In

text: applications

text: we

text: would

text: like

text: to

text: choose

text: the

text: eigenvector

text: scalings

text: as

text: large

text: as

text: possible,

text: so

text: that

text: we

text: learn

text: more

text: about

text: the

text: unstable

text: manifold

text: far

text: from

text: the

text: equilibrium

text: solution.

text: This

text: desire

text: must

text: be

text: weighed

text: against

text: the

text: fact

text: that

text: for

text: larger

text: choices

text: of

text: the

text: scalings

text: we

text: risk

text: loosing

text: control

text: of

text: the

text: convergence

text: of

text: the

text: series.

text: Viewed

text: in

text: this

text: light,

text: the

text: tools

text: of

text: the

text: present

text: work

text: provide

text: a

text: mathematically

text: rigorous

text: computer

text: assisted

text: method

text: for

text: pushing

text: the

text: existence

text: results

text: as

text: far

text: as

text: possible

text: in

text: speciﬁc

text: applications.

text: Since

text: the

text: desired

text: parameterization

text: exists

text: in

text: a

text: small

text: enough

text: neighborhood

text: of

text: the

text: equilibrium

text: solution

text: by

text: Theorem

text: our

text: argument

text: has

text: a

text: “continuation”

text: ﬂavor

text: (the

text: continuation

text: parameters

text: being

text: the

text: eigenvector

text: scalings,

text: which

text: in

text: turn

text: govern

text: the

text: size

text: of

text: the

text: local

text: unstable

text: manifold

text: in

text: phase

text: space).

text: The

text: novelty

text: is

text: that

text: we

text: care

text: only

text: about

text: rigorous

text: results

text: at

text: the

text: end

text: of

text: the

text: continuation.

text: We

text: argue

text: in

text: Sections

text: and

text: that

text: expensive

text: validation

text: computations

text: can

text: be

text: postponed

text: until

text: we

text: are

text: all

text: but

text: certain

text: the

text: computer

text: aided

text: proof

text: will

text: succeed.

text: These

text: considerations

text: are

text: also

text: discussed

text: at

text: length

text: for

text: ﬁnite

text: dimensional

text: vector

text: ﬁelds

text: in

text: Remark

text: (Extensions

text: of

text: the

text: a-priori

text: results).

text: A

text: word

text: about

text: the

text: technical

text: assumptions

text: of

text: Theorem

text: In

text: addition

text: to

text: the

text: non-resonance

text: conditions

text: postulated

text: in

text: Deﬁnition

text: Theorem

text: postulates

text: that

text: each

text: of

text: the

text: unstable

text: eigenvalues

text: is

text: real

text: and

text: that

text: each

text: has

text: multiplicity

text: one.

text: Moreover

text: we

text: assume

text: that

text: the

text: nonlinearity

text: is

text: given

text: by

text: an

text: analytic

text: function.

text: We

text: remark

text: that

text: these

text: assumptions

text: are

text: technically

text: convenient,

text: and

text: should

text: not

text: be

text: inter-

text: preted

text: as

text: fundamental

text: restrictions.

text: For

text: example

text: real

text: invariant

text: manifolds

text: associated

text: with


text: 6 can capture more eﬃciently the incident power and fun- duced amount of hot material. This approach paves the nel it into the absorber volume. This absorbed power in way to a new class of THz and IR thermal emitters with a the presence of the antenna is then proportional to an controlled emission direction, spectrum and polarization. eﬀective absorption cross section denoted σ . In ad- ACKNOWLEDGMENTS ant dition, the antenna can be directional and frequency se- lective [32]. It follows from the generalized Kirchhoﬀ law We acknowledge the support of the Agence Nationale that if the absorption in the absorber volume is enhanced de la Recherche through the grant ANR-14-CE26-0023- then its thermal emission is enhanced. The total emit- 03 and the support of the ONERA through the project ted power can be increased by the same factor σant/σabs SONS. This work was supported by the US Department which can be larger than two orders of magnitude. Fur- of Energy, Oﬃce of Basic Energy Sciences, Division of thermore, the emission can be directional and frequency Materials Sciences and Engineering. It was performed, selective. We anticipate that a metasurface consisting of in part, at the Center for Integrated Nanotechnologies, a periodic array of subwavelength hot objects connected an Oﬃce of Science User Facility operated for the US to antennas could be optimized to behave as a blackbody Department of Energy (DOE) Oﬃce of Science. JJG is antenna with unity emissivity while using only a very re- a senior member of the Institut Universitaire de France. ∗ [24] A. Downes, P. Dumas, M.E. Welland, ”Measurement of high electron temperatures in single atom metal point contacts by [1] S.M. Rytov, Theory of electric ﬂuctuations and thermal radiatiolnig,hAtiermFiosrscioenC”,amApbprild.PgehyRs.eLseeattr.ch81C,e1n2t5e2r,(B20e0d2fo).rd MA, (1959). [25] M. Buret et al., ”Spontaneous hot-electron light emission from electron-fed optical antennas”, Nano Lett. 15, 5811 (2015). [2] L.D. Landau and E.M. Lifshitz, Statistical Physics, vol. 5, Pe[2rg6a] mMo.nFPrerietsasg,,OHx-fYordC,h(i1u9,6M9)..Steiner, V. Perebeinos, P. Avouris, ”Thermal infrared emission from biased graphene”, Nature [3] G.W. Kattawar and M. Eisner, ”Radiation from a homogeneous iNsoatnhoetremchanl osplohgeyre5”,, 4A9p7p(l2.O01p0t)..9, 2685 (1970). [4] J.J. Greﬀet and M. Nieto-Vesperinas, ”Field theory for gen[e2r7a]liYze.dDb.iKdiirmecteitonala.l, ”reBﬂreigcthitvivtiys:ibdleerliigvhattioenmiosfsioHnelfmrohmoltgzr’asphene”, Nature Nanotechnology 10, 676 (2015). reciprocity principle and Kirchhoﬀ’s law, ”J. Opt. Soc. Am. A[281]5T, .2L73a5ur(e1n9t9,8Y).. Todorov, A. Vasanelli, A. Delteil, C. Sirtori, I. Sagnes, G. Beadouin, ”Superradiant emission from a collective [5] W.C. Snyder, Z. Wan, X. Li, ”Thermodynamic constraints on reﬂeexcctiatnatcieonreicniparosceimtyicaonndduKcitrocrh”h,oPﬀh’sysla.Rwe”v,.LAeptpt.l.O11p5t.13877,402 (2015). 3464 (1998). [29] C.F. Bohren, D.R. Huﬀman, Absorption and Scattering of Light by Small Particles, Wiley, NY, (1983). [6] C. Luo, A. Narayanaswamy, G. Chen, J.D. Joannopoulos, ”Th[3er0m] aJl.Jr.aGdiraetﬀioent,frRo.mCpahrmotionnaitci,c”ryImstaaglse: faordmiraetcitocnailncunlaetairo-nﬁ”e,ld optics”, Progress in Surface Science 56, 133 (1997). Phys.Rev.Lett. 93, 213905 (2004). [31] L. Mandel, E.Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge, (1995). [7] D.L.C. Chan, M. Soljacic, J.D. Joannopoulos, ”Direct calcu[3la2t]ioJn. Aolfdtah,eJr.mMa.lReimcoi-sGsioanrcifao,rJt.hMr.eeL-odpimeze-nAslioonnsaoll,yGp.eBroiordemican, ”Optical antennas for nano-photonic applications”, Nanotech- photonic crystals slabs”, Phys. Rev. E. 74, 036615 (2006). nology 16, S230 (2005). [8] L.P. Wang, S. Basu and Z.M. Zhang, ”Direct and indirect methods for calculating thermal emission from layered structures with nonuniform temperatures”, J. of Heat Transfer 13, 072701 (2011). [9] N.J.Harrick, A.F. Turner, ”A thin ﬁlm optical cavity to induce absorption or thermal emission”, Appl.Opt. 9, 2111 (1970). [10] Z. Yu, N.P. Sergeant, T. Skauli, G. Zhang, H. Wang, S. Fan, ”Enhancing far-ﬁeld thermal emission with thermal extraction”, Nature Communications 4 1730 (2013). [11] S.-Y. Lin, J. Moreno, and J.G. Fleming, ”Three-dimensional photonic-crystal emitter for thermal photovoltaic power generation”, Appl.Phys.Lett. 83, 380 (2003). [12] T. Trupke, P. Wurfel, and M. A. Green, ”Comment on: Three-dimensional photonic-crystal emitter for thermal photovoltaic power generation”, Appl. Phys. Lett. 84, 1997 (2004). [13] J.G. Fleming, ”Addendum:Three-dimensional photonic-crystal emitter for thermal photovoltaic power generation”, Appl.Phys.Lett. 86, 249902 (2005). [14] I.S. Nefedov, L. A. Melnikov, ”Super-Planckian far-ﬁeld thermal emission from asymmetric hyperbolic metamaterials”, Appl.Phys.Lett. 105, 1610902 (2014). [15] S.I. Maslovski, C.R. Simovski, S.A. Tretyakov, ”Overcoming blackbody radiation limit in free space: metamaterial su- peremitter, ”arXiv:1412:4625v3 (2015). [16] S.E. Han, ”Theory of thermal emission from periodic structures”, Phys.Rev.B 80, 155108 (2009). [17] L. Tsang, E. Njoku, J.A. Kong, ”Microwave thermal emission from a stratiﬁed medium with nonuniform temperature distribution”, J. Appl.Phys. 46, 5127 (1975). [18] Bardati, F., and Solimini, D., ”On the emissivity of layered materials” IEEE Trans. Geosci. Remote Sens., GE-22 374 (1984). [19] K.P. Gaikovich, A.N. Reznik, V.L. Vaks, N.Y. Yurosova, ”New eﬀect in near-ﬁled thermal emission”, Phys.Rev.Lett.88, 104302 (2002). [20] S.E. Han, and D.J. Norris, ”Control of thermal emission by selective heating of periodic structures” Phys.Rev.Lett. 104, 043901 (2010). [21] J.J. Greﬀet , ”Controlled incandescence”, Nature 478, 191 (2011) [22] S. Vassant, I. Moldovan-Ledoyen, F. Marquier, F. Pardo, U. Gennser, A. Cavanna, J.L. Pelouard, J.J. Greﬀet, ”Electrical modulation of emissivity”, Appl.Phys.Lett.102, 081125 (2013). [23] T. Inoue, M. De Zoysa, T. Asano, S. Noda, ”Realization of dynamic thermal emission control”, Nature Materials 13, 928 (2014). ##LTLine## ##LTLine## ##LTLine##

text: can

text: capture

text: more

text: eﬃciently

text: the

text: incident

text: power

text: and

text: fun-

text: duced

text: amount

text: of

text: hot

text: material.

text: This

text: approach

text: paves

text: the

text: nel

text: it

text: into

text: the

text: absorber

text: volume.

text: This

text: absorbed

text: power

text: in

text: way

text: to

text: a

text: new

text: class

text: of

text: THz

text: and

text: IR

text: thermal

text: emitters

text: with

text: a

text: the

text: presence

text: of

text: the

text: antenna

text: is

text: then

text: proportional

text: to

text: an

text: controlled

text: emission

text: direction,

text: spectrum

text: and

text: polarization.

text: eﬀective

text: absorption

text: cross

text: section

text: denoted

text: σ

text: In

text: ad-

text: ACKNOWLEDGMENTS

text: ant

text: dition,

text: the

text: antenna

text: can

text: be

text: directional

text: and

text: frequency

text: se-

text: lective

text: It

text: follows

text: from

text: the

text: generalized

text: Kirchhoﬀ

text: law

text: We

text: acknowledge

text: the

text: support

text: of

text: the

text: Agence

text: Nationale

text: that

text: if

text: the

text: absorption

text: in

text: the

text: absorber

text: volume

text: is

text: enhanced

text: de

text: la

text: Recherche

text: through

text: the

text: grant

text: then

text: its

text: thermal

text: emission

text: is

text: enhanced.

text: The

text: total

text: emit-

text: and

text: the

text: support

text: of

text: the

text: ONERA

text: through

text: the

text: project

text: ted

text: power

text: can

text: be

text: increased

text: by

text: the

text: same

text: factor

text: σant/σabs

text: SONS.

text: This

text: work

text: was

text: supported

text: by

text: the

text: US

text: Department

text: which

text: can

text: be

text: larger

text: than

text: two

text: orders

text: of

text: magnitude.

text: Fur-

text: of

text: Energy,

text: Oﬃce

text: of

text: Basic

text: Energy

text: Sciences,

text: Division

text: of

text: thermore,

text: the

text: emission

text: can

text: be

text: directional

text: and

text: frequency

text: Materials

text: Sciences

text: and

text: Engineering.

text: It

text: was

text: performed,

text: selective.

text: We

text: anticipate

text: that

text: a

text: metasurface

text: consisting

text: of

text: in

text: part,

text: at

text: the

text: Center

text: for

text: Integrated

text: Nanotechnologies,

text: a

text: periodic

text: array

text: of

text: subwavelength

text: hot

text: objects

text: connected

text: an

text: Oﬃce

text: of

text: Science

text: User

text: Facility

text: operated

text: for

text: the

text: US

text: to

text: antennas

text: could

text: be

text: optimized

text: to

text: behave

text: as

text: a

text: blackbody

text: Department

text: of

text: Energy

text: (DOE)

text: Oﬃce

text: of

text: Science.

text: JJG

text: is

text: antenna

text: with

text: unity

text: emissivity

text: while

text: using

text: only

text: a

text: very

text: re-

text: a

text: senior

text: member

text: of

text: the

text: Institut

text: Universitaire

text: de

text: France.

text: A.

text: Downes,

text: P.

text: Dumas,

text: M.E.

text: Welland,

text: ”Measurement

text: of

text: high

text: electron

text: temperatures

text: in

text: single

text: atom

text: metal

text: point

text: contacts

text: by

text: S.M.

text: Rytov,

text: Theory

text: of

text: electric

text: ﬂuctuations

text: and

text: thermal

text: MA,

text: M.

text: Buret

text: et

text: al.,

text: ”Spontaneous

text: hot-electron

text: light

text: emission

text: from

text: electron-fed

text: optical

text: antennas”,

text: Nano

text: Lett.

text: L.D.

text: Landau

text: and

text: E.M.

text: Lifshitz,

text: Statistical

text: Physics,

text: vol.

text: V.

text: Perebeinos,

text: P.

text: Avouris,

text: ”Thermal

text: infrared

text: emission

text: from

text: biased

text: graphene”,

text: Nature

text: G.W.

text: Kattawar

text: and

text: M.

text: Eisner,

text: ”Radiation

text: from

text: a

text: homogeneous

text: iNsoatnhoetremchanl

text: J.J.

text: Greﬀet

text: and

text: M.

text: Nieto-Vesperinas,

text: ”Field

text: theory

text: for

text: Nature

text: Nanotechnology

text: reciprocity

text: principle

text: and

text: Kirchhoﬀ’s

text: law,

text: ”J.

text: Opt.

text: Soc.

text: Am.

text: Todorov,

text: A.

text: Vasanelli,

text: A.

text: Delteil,

text: C.

text: Sirtori,

text: I.

text: Sagnes,

text: G.

text: Beadouin,

text: ”Superradiant

text: emission

text: from

text: a

text: collective

text: W.C.

text: Snyder,

text: Z.

text: Wan,

text: X.

text: Li,

text: ”Thermodynamic

text: constraints

text: on

text: C.F.

text: Bohren,

text: D.R.

text: Huﬀman,

text: Absorption

text: and

text: Scattering

text: of

text: Light

text: by

text: Small

text: Particles,

text: Wiley,

text: NY,

text: C.

text: Luo,

text: A.

text: Narayanaswamy,

text: G.

text: Chen,

text: J.D.

text: Joannopoulos,

text: optics”,

text: Progress

text: in

text: Surface

text: Science

text: Phys.Rev.Lett.

text: L.

text: Mandel,

text: E.Wolf,

text: Optical

text: Coherence

text: and

text: Quantum

text: Optics,

text: Cambridge

text: University

text: Press,

text: Cambridge,

text: D.L.C.

text: Chan,

text: M.

text: Soljacic,

text: J.D.

text: Joannopoulos,

text: ”Direct

text: ”Optical

text: antennas

text: for

text: nano-photonic

text: applications”,

text: Nanotech-

text: photonic

text: crystals

text: slabs”,

text: Phys.

text: Rev.

text: E.

text: nology

text: L.P.

text: Wang,

text: S.

text: Basu

text: and

text: Z.M.

text: Zhang,

text: ”Direct

text: and

text: indirect

text: methods

text: for

text: calculating

text: thermal

text: emission

text: from

text: layered

text: structures

text: with

text: nonuniform

text: temperatures”,

text: J.

text: of

text: Heat

text: Transfer

text: N.J.Harrick,

text: A.F.

text: Turner,

text: ”A

text: thin

text: ﬁlm

text: optical

text: cavity

text: to

text: induce

text: absorption

text: or

text: thermal

text: emission”,

text: Appl.Opt.

text: Z.

text: Yu,

text: N.P.

text: Sergeant,

text: T.

text: Skauli,

text: G.

text: Zhang,

text: H.

text: Wang,

text: S.

text: Fan,

text: ”Enhancing

text: far-ﬁeld

text: thermal

text: emission

text: with

text: thermal

text: extraction”,

text: Nature

text: Communications

text: S.-Y.

text: Lin,

text: J.

text: Moreno,

text: and

text: J.G.

text: Fleming,

text: ”Three-dimensional

text: photonic-crystal

text: emitter

text: for

text: thermal

text: photovoltaic

text: power

text: generation”,

text: Appl.Phys.Lett.

text: T.

text: Trupke,

text: P.

text: Wurfel,

text: and

text: M.

text: A.

text: Green,

text: ”Comment

text: on:

text: Three-dimensional

text: photonic-crystal

text: emitter

text: for

text: thermal

text: photovoltaic

text: power

text: generation”,

text: Appl.

text: Phys.

text: Lett.

text: J.G.

text: Fleming,

text: ”Addendum:Three-dimensional

text: photonic-crystal

text: emitter

text: for

text: thermal

text: photovoltaic

text: power

text: generation”,

text: Appl.Phys.Lett.

text: I.S.

text: Nefedov,

text: L.

text: A.

text: Melnikov,

text: ”Super-Planckian

text: far-ﬁeld

text: thermal

text: emission

text: from

text: asymmetric

text: hyperbolic

text: metamaterials”,

text: Appl.Phys.Lett.

text: S.I.

text: Maslovski,

text: C.R.

text: Simovski,

text: S.A.

text: Tretyakov,

text: ”Overcoming

text: blackbody

text: radiation

text: limit

text: in

text: free

text: space:

text: metamaterial

text: su-

text: peremitter,

text: S.E.

text: Han,

text: ”Theory

text: of

text: thermal

text: emission

text: from

text: periodic

text: structures”,

text: Phys.Rev.B

text: L.

text: Tsang,

text: E.

text: Njoku,

text: J.A.

text: Kong,

text: ”Microwave

text: thermal

text: emission

text: from

text: a

text: stratiﬁed

text: medium

text: with

text: nonuniform

text: temperature

text: distribution”,

text: J.

text: Appl.Phys.

text: Bardati,

text: F.,

text: and

text: Solimini,

text: D.,

text: ”On

text: the

text: emissivity

text: of

text: layered

text: materials”

text: IEEE

text: Trans.

text: Geosci.

text: Remote

text: Sens.,

text: K.P.

text: Gaikovich,

text: A.N.

text: Reznik,

text: V.L.

text: Vaks,

text: N.Y.

text: Yurosova,

text: ”New

text: eﬀect

text: in

text: near-ﬁled

text: thermal

text: emission”,

text: S.E.

text: Han,

text: and

text: D.J.

text: Norris,

text: ”Control

text: of

text: thermal

text: emission

text: by

text: selective

text: heating

text: of

text: periodic

text: structures”

text: Phys.Rev.Lett.

text: J.J.

text: Greﬀet

text: ”Controlled

text: incandescence”,

text: Nature

text: S.

text: Vassant,

text: I.

text: Moldovan-Ledoyen,

text: F.

text: Marquier,

text: F.

text: Pardo,

text: U.

text: Gennser,

text: A.

text: Cavanna,

text: J.L.

text: Pelouard,

text: J.J.

text: Greﬀet,

text: ”Electrical

text: modulation

text: of

text: emissivity”,

text: T.

text: Inoue,

text: M.

text: De

text: Zoysa,

text: T.

text: Asano,

text: S.

text: Noda,

text: ”Realization

text: of

text: dynamic

text: thermal

text: emission

text: control”,

text: Nature

text: Materials

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 BENEDIKTE GRIMELAND C and m-cluster tilting objects of (H), this enables us to study the peri- m odic properties within the cluster categories and still obtain the necessary D information about periodicity in the bounded derived category b(H). The mutation classes of quivers of cluster tilted algebras of representation- ﬁnite hereditary algebras have been described in [21], [9], and [2]. Based on the symmetry properties of the quivers of the corresponding cluster tilted algebras, we classify the periodic properties of all 2-cluster tilting objects of representation-ﬁnite hereditary algebras as an application of the method described above. This gives a complete overview of all 2-cluster tilting ob- jects in all triangulated orbit categories of representation-ﬁnite hereditary algebras, with an easy way to determine all the indecomposable summands of the object. In regard to this, it is interesting to note two things. First, in many cases the combinatorics of type D is more involved than the combina- torics of type A. However this is not true for this case, the combinatorics of the proof for the 3-symmetric cases of type A is conceptuallly not any easier than the proofs of type D. Second, when considering cases of Dynkin type D , the parity of n aﬀects the results in a substantial way. n Earlier results on the classiﬁcation of orbit categories with a 2-cluster tilting object are given in [10], [7] and for a special case for type E in [18]. These results covers 2-cluster tilting objects in 2-Calabi Yau triangulated orbit categories, and the method used is to study orbit-categories of the 2- cluster category. As described above, our method can be applied regardless of the Calabi-Yau dimension of the category. Moreover, our method in itself provide a combinatorial means for ﬁnding explicit descriptions of any 2- cluster tilting object T in any orbit category in terms of its indecomposable summands, and hence also the quiver of the corresponding cluster tilted algebra. Furthermore, as noticed above, our method may be applied to any known m-cluster tilting object (m > 2) of a triangulated orbit category, to ﬁnd other orbit categories with m-cluster tilting objects arising from the same preimage in the bounded derived category. The paper is organized as follows: Section contains the basic deﬁni- tions from cluster tilting theory that will be used throughout the paper. In section we show how periodicity of cluster tilting subcategories in a trian- T gulated category determines cluster tilting subcategories in triangulated T orbit categories of . Using properties of the bounded derived category for representation-ﬁnite hereditary algebras we show that some properties of periodic cluster tilting subcategories of cluster categories carry over to the bounded derived category. Based on this we describe our main method, which is applied in later sections. Section give a short overview of the quivers of cluster tilted algebras of Dynkin type A and D, and also states results connected to quivers of cluster tilted algebras, that will be used in sections and . Then in section we use the method described in section , and classify for all 2-cluster tilting subcat- D egories of b(A ) which functors they are periodic under. The classiﬁcation n is given in terms of symmetric properties of the quivers of the corresponding C 2-cluster tilting subcategories in the cluster category (A ). In section we 2 n give the corresponding results for 2-cluster tilting subcategories of type D.

text: BENEDIKTE

text: GRIMELAND

text: C

text: and

text: m-cluster

text: tilting

text: objects

text: of

text: (H),

text: this

text: enables

text: us

text: to

text: study

text: the

text: peri-

text: m

text: odic

text: properties

text: within

text: the

text: cluster

text: categories

text: and

text: still

text: obtain

text: the

text: necessary

text: D

text: information

text: about

text: periodicity

text: in

text: the

text: bounded

text: derived

text: category

text: b(H).

text: The

text: mutation

text: classes

text: of

text: quivers

text: of

text: cluster

text: tilted

text: algebras

text: of

text: representation-

text: ﬁnite

text: hereditary

text: algebras

text: have

text: been

text: described

text: in

text: and

text: Based

text: on

text: the

text: symmetry

text: properties

text: of

text: the

text: quivers

text: of

text: the

text: corresponding

text: cluster

text: tilted

text: algebras,

text: we

text: classify

text: the

text: periodic

text: properties

text: of

text: all

text: tilting

text: objects

text: of

text: representation-ﬁnite

text: hereditary

text: algebras

text: as

text: an

text: application

text: of

text: the

text: method

text: described

text: above.

text: This

text: gives

text: a

text: complete

text: overview

text: of

text: all

text: tilting

text: ob-

text: jects

text: in

text: all

text: triangulated

text: orbit

text: categories

text: of

text: representation-ﬁnite

text: hereditary

text: algebras,

text: with

text: an

text: easy

text: way

text: to

text: determine

text: all

text: the

text: indecomposable

text: summands

text: of

text: the

text: object.

text: In

text: regard

text: to

text: this,

text: it

text: is

text: interesting

text: to

text: note

text: two

text: things.

text: First,

text: in

text: many

text: cases

text: the

text: combinatorics

text: of

text: type

text: D

text: is

text: more

text: involved

text: than

text: the

text: combina-

text: torics

text: of

text: type

text: A.

text: However

text: this

text: is

text: not

text: true

text: for

text: this

text: case,

text: the

text: combinatorics

text: of

text: the

text: proof

text: for

text: the

text: cases

text: of

text: type

text: A

text: is

text: conceptuallly

text: not

text: any

text: easier

text: than

text: the

text: proofs

text: of

text: type

text: D.

text: Second,

text: when

text: considering

text: cases

text: of

text: Dynkin

text: type

text: D

text: the

text: parity

text: of

text: n

text: aﬀects

text: the

text: results

text: in

text: a

text: substantial

text: way.

text: n

text: Earlier

text: results

text: on

text: the

text: classiﬁcation

text: of

text: orbit

text: categories

text: with

text: a

text: tilting

text: object

text: are

text: given

text: in

text: and

text: for

text: a

text: special

text: case

text: for

text: type

text: E

text: in

text: These

text: results

text: covers

text: tilting

text: objects

text: in

text: Yau

text: triangulated

text: orbit

text: categories,

text: and

text: the

text: method

text: used

text: is

text: to

text: study

text: orbit-categories

text: of

text: the

text: cluster

text: category.

text: As

text: described

text: above,

text: our

text: method

text: can

text: be

text: applied

text: regardless

text: of

text: the

text: Calabi-Yau

text: dimension

text: of

text: the

text: category.

text: Moreover,

text: our

text: method

text: in

text: itself

text: provide

text: a

text: combinatorial

text: means

text: for

text: ﬁnding

text: explicit

text: descriptions

text: of

text: any

text: cluster

text: tilting

text: object

text: T

text: in

text: any

text: orbit

text: category

text: in

text: terms

text: of

text: its

text: indecomposable

text: summands,

text: and

text: hence

text: also

text: the

text: quiver

text: of

text: the

text: corresponding

text: cluster

text: tilted

text: algebra.

text: Furthermore,

text: as

text: noticed

text: above,

text: our

text: method

text: may

text: be

text: applied

text: to

text: any

text: known

text: m-cluster

text: tilting

text: object

text: (m

text: of

text: a

text: triangulated

text: orbit

text: category,

text: to

text: ﬁnd

text: other

text: orbit

text: categories

text: with

text: m-cluster

text: tilting

text: objects

text: arising

text: from

text: the

text: same

text: preimage

text: in

text: the

text: bounded

text: derived

text: category.

text: The

text: paper

text: is

text: organized

text: as

text: follows:

text: Section

text: contains

text: the

text: basic

text: deﬁni-

text: tions

text: from

text: cluster

text: tilting

text: theory

text: that

text: will

text: be

text: used

text: throughout

text: the

text: paper.

text: In

text: section

text: we

text: show

text: how

text: periodicity

text: of

text: cluster

text: tilting

text: subcategories

text: in

text: a

text: trian-

text: T

text: gulated

text: category

text: determines

text: cluster

text: tilting

text: subcategories

text: in

text: triangulated

text: T

text: orbit

text: categories

text: of

text: Using

text: properties

text: of

text: the

text: bounded

text: derived

text: category

text: for

text: representation-ﬁnite

text: hereditary

text: algebras

text: we

text: show

text: that

text: some

text: properties

text: of

text: periodic

text: cluster

text: tilting

text: subcategories

text: of

text: cluster

text: categories

text: carry

text: over

text: to

text: the

text: bounded

text: derived

text: category.

text: Based

text: on

text: this

text: we

text: describe

text: our

text: main

text: method,

text: which

text: is

text: applied

text: in

text: later

text: sections.

text: Section

text: give

text: a

text: short

text: overview

text: of

text: the

text: quivers

text: of

text: cluster

text: tilted

text: algebras

text: of

text: Dynkin

text: type

text: A

text: and

text: D,

text: and

text: also

text: states

text: results

text: connected

text: to

text: quivers

text: of

text: cluster

text: tilted

text: algebras,

text: that

text: will

text: be

text: used

text: in

text: sections

text: and

text: Then

text: in

text: section

text: we

text: use

text: the

text: method

text: described

text: in

text: section

text: and

text: classify

text: for

text: all

text: tilting

text: subcat-

text: D

text: egories

text: of

text: b(A

text: which

text: functors

text: they

text: are

text: periodic

text: under.

text: The

text: classiﬁcation

text: n

text: is

text: given

text: in

text: terms

text: of

text: symmetric

text: properties

text: of

text: the

text: quivers

text: of

text: the

text: corresponding

text: C

text: tilting

text: subcategories

text: in

text: the

text: cluster

text: category

text: (A

text: In

text: section

text: we

text: n

text: give

text: the

text: corresponding

text: results

text: for

text: tilting

text: subcategories

text: of

text: type

text: D.


text: PERIODICITY OF CLUSTER TILTING OBJECTS 3 Type E is studied in section . Finally we consider the Euclidean and wild cases in section . 2. Background For more background on the following deﬁnitions see for example [14]. C T C T Deﬁnition 1. Let be a category and a full subcategory of . Then is called a functorially ﬁnite subcategory if it is both covariantly ﬁnite and ∈ C (cid:48) ∈ contravariantly ﬁnite. That is for each object X there are objects T, T T → (cid:48) → with morphisms and f : X T and g : T X such that for any object (cid:48)(cid:48) (cid:48)(cid:48) ∈ T (cid:48) (cid:48)(cid:48) −−H−o−m−C−(−f,−T−−)(cid:16) (cid:48)(cid:48) T there are epimorphisms HomC(T , T ) HomC(X, T ) (cid:48)(cid:48) (cid:48)(cid:48) −−H−o−m−C−(−T−,−g−)(cid:16) (cid:48)(cid:48) and HomC(T , T ) HomC(T , X). T Deﬁnition 2. A (cid:8)functorially ﬁnite subcateogory is calle(cid:9)d an r-cluster C tilting subcategory(cid:8)of if (cid:9) T ∈ C| T = X ExtiC( , X) = 0 for any 0 < i < r ∈ C| T = X ExtiC(X, ) = 0 for any 0 < i < r T If = add T for some object T , then T is called an r-cluster tilting object. Deﬁnition 3. We call a triangulated category locally ﬁnite if for an inde- composable object X there is only a ﬁnite number of isomorphism classes of (cid:54) indecomposable objects Y such that HomT (X, Y ) = 0. Note that deﬁnition 3 implies its own dual (see [22],[1]), i.e. for any indecomposable object Y there are only a ﬁnite number of isomorphism (cid:54) classes of indecomposable objects X such that HomT (X, Y ) = 0. Also note that any subcategory of a locally ﬁnite triangulated category is both contravariantly ﬁnite and covariantly ﬁnite. Next we will deﬁne what we mean by an orbit category. One can deﬁne orbit categories of all additive categories (see [16]), however we will only use the deﬁnition for triangulated categories and therefore give the deﬁnition in this context. See for example also [5]. T Deﬁnition 4. Let be a triangulated cateogory with an autoequivalence T → T O T T F : . The orbit cateogory ( ) has the same objects as . The F O T morphisms HomO(X, Y ) between two objects in ( ) are in bijection with ⊕ F the set HomT (X, F nY ). In later sections we will mostly consider orbit categories of the bounded D derived category b(H) where H is a representation-ﬁnite hereditary alge- C D − − bra. We deﬁne the m-cluster category of H to be (H) := b(H)/τ [m 1] ≥ m for m 2. In particular it is known ([16],[5]) that the m-cluster category D → C of an hereditary algebra is triangulated, and π : b(H) (H) is a tri- m m angle functor. Note that in a triangulated category with only ﬁnitely many isomorphism classes of indecomposable objects, a cluster tilting subcate- gory always gives rise to a cluster tilting object. We will often use the term cluster tilting object about a cluster tiling subcategory in such a setting. Furthermore, we note that the suspension functor in the bounded derived D category of a representation-ﬁnite hereditary algebra b(H) will be denoted

text: PERIODICITY

text: OF

text: CLUSTER

text: TILTING

text: OBJECTS

text: Type

text: E

text: is

text: studied

text: in

text: section

text: Finally

text: we

text: consider

text: the

text: Euclidean

text: and

text: wild

text: cases

text: in

text: section

label: section

text: Background

text: For

text: more

text: background

text: on

text: the

text: following

text: deﬁnitions

text: see

text: for

text: example

text: C

text: T

text: C

text: T

text: Deﬁnition

text: Let

text: be

text: a

text: category

text: and

text: a

text: full

text: subcategory

text: of

text: Then

text: is

text: called

text: a

text: functorially

text: ﬁnite

text: subcategory

text: if

text: it

text: is

text: both

text: covariantly

text: ﬁnite

text: and

text: C

text: contravariantly

text: ﬁnite.

text: That

text: is

text: for

text: each

text: object

text: X

text: there

text: are

text: objects

text: T,

text: T

text: T

text: with

text: morphisms

text: and

text: f

text: X

text: T

text: and

text: g

text: T

text: X

text: such

text: that

text: for

text: any

text: object

text: T

text: T

text: there

text: are

text: epimorphisms

text: HomC(T

text: T

text: HomC(X,

text: T

text: and

text: HomC(T

text: T

text: HomC(T

text: X).

text: T

text: Deﬁnition

text: A

text: ﬁnite

text: subcateogory

text: is

text: an

text: r-cluster

text: C

text: tilting

text: if

text: T

text: C|

text: T

text: X

text: ExtiC(

text: X)

text: for

text: any

text: i

text: r

text: C|

text: T

text: X

text: ExtiC(X,

text: for

text: any

text: i

text: r

text: T

text: If

text: add

text: T

text: for

text: some

text: object

text: T

text: then

text: T

text: is

text: called

text: an

text: r-cluster

text: tilting

text: object.

text: Deﬁnition

text: We

text: call

text: a

text: triangulated

text: category

text: locally

text: ﬁnite

text: if

text: for

text: an

text: inde-

text: composable

text: object

text: X

text: there

text: is

text: only

text: a

text: ﬁnite

text: number

text: of

text: isomorphism

text: classes

text: of

text: indecomposable

text: objects

text: Y

text: such

text: that

text: HomT

text: (X,

text: Y

text: Note

text: that

text: deﬁnition

text: implies

text: its

text: own

text: dual

text: (see

text: i.e.

text: for

text: any

text: indecomposable

text: object

text: Y

text: there

text: are

text: only

text: a

text: ﬁnite

text: number

text: of

text: isomorphism

text: classes

text: of

text: indecomposable

text: objects

text: X

text: such

text: that

text: HomT

text: (X,

text: Y

text: Also

text: note

text: that

text: any

text: subcategory

text: of

text: a

text: locally

text: ﬁnite

text: triangulated

text: category

text: is

text: both

text: contravariantly

text: ﬁnite

text: and

text: covariantly

text: ﬁnite.

text: Next

text: we

text: will

text: deﬁne

text: what

text: we

text: mean

text: by

text: an

text: orbit

text: category.

text: One

text: can

text: deﬁne

text: orbit

text: categories

text: of

text: all

text: additive

text: categories

text: (see

text: however

text: we

text: will

text: only

text: use

text: the

text: deﬁnition

text: for

text: triangulated

text: categories

text: and

text: therefore

text: give

text: the

text: deﬁnition

text: in

text: this

text: context.

text: See

text: for

text: example

text: also

text: T

text: Deﬁnition

text: Let

text: be

text: a

text: triangulated

text: cateogory

text: with

text: an

text: autoequivalence

text: T

text: T

text: O

text: T

text: T

text: F

text: The

text: orbit

text: cateogory

text: has

text: the

text: same

text: objects

text: as

text: The

text: F

text: O

text: T

text: morphisms

text: HomO(X,

text: Y

text: between

text: two

text: objects

text: in

text: are

text: in

text: bijection

text: with

text: F

text: the

text: set

text: HomT

text: (X,

text: F

text: nY

text: In

text: later

text: sections

text: we

text: will

text: mostly

text: consider

text: orbit

text: categories

text: of

text: the

text: bounded

text: D

text: derived

text: category

text: b(H)

text: where

text: H

text: is

text: a

text: representation-ﬁnite

text: hereditary

text: alge-

text: C

text: D

text: bra.

text: We

text: deﬁne

text: the

text: m-cluster

text: category

text: of

text: H

text: to

text: be

text: (H)

text: b(H)/τ

text: [m

text: m

text: for

text: m

text: In

text: particular

text: it

text: is

text: known

text: that

text: the

text: m-cluster

text: category

text: D

text: C

text: of

text: an

text: hereditary

text: algebra

text: is

text: triangulated,

text: and

text: π

text: b(H)

text: (H)

text: is

text: a

text: tri-

text: m

text: m

text: angle

text: functor.

text: Note

text: that

text: in

text: a

text: triangulated

text: category

text: with

text: only

text: ﬁnitely

text: many

text: isomorphism

text: classes

text: of

text: indecomposable

text: objects,

text: a

text: cluster

text: tilting

text: subcate-

text: gory

text: always

text: gives

text: rise

text: to

text: a

text: cluster

text: tilting

text: object.

text: We

text: will

text: often

text: use

text: the

text: term

text: cluster

text: tilting

text: object

text: about

text: a

text: cluster

text: tiling

text: subcategory

text: in

text: such

text: a

text: setting.

text: Furthermore,

text: we

text: note

text: that

text: the

text: suspension

text: functor

text: in

text: the

text: bounded

text: derived

text: D

text: category

text: of

text: a

text: representation-ﬁnite

text: hereditary

text: algebra

text: b(H)

text: will

text: be

text: denoted


text: PERIODICITY OF CLUSTER TILTING OBJECTS 25 • − ≥ ∈ Z τtn 1 [1] for odd n 5 and t • ≥ ∈ Z τ2t for all n 4 and t . • − ≥ ∈ Z τ2t 1 [1] for odd n 5 and t . O Proof. For even values of n there is only one type of AR-quiver of to F consider, which is cylindrical. However for odd n there are two possibilities, one is cylindrical and the other is cylindrical for β-objects with a Moebius- twist in the top two τ-orbits consisting of the α-objects. From the last part of theorem 43 it follows that for both even and odd n the 2-cluster tilting object where the corresponding quiver of EndC(T )op is an n-cycle is periodic under τ2. Since 2 is the smallest even natural number we get the last two parts of the corollary by applying lemma 6 and lemma 10. These cases also cover all categories having a 2-cluster tilting object due to the second part of theorem 43. The last part to consider is the ﬁrst part of theorem 43, which gives rise to the ﬁrst two parts of this corollary using lemma 6 and lemma 10. It is only necessary to consider the cases for odd n as all cases of even n is considered in the third part of the corollary. (cid:3) 7. 2-cluster tilting subcategories of type E In this section we focus on the cases, E , E and E . We treat each case 6 7 8 in its separate subsection . In each case we will use the properties of the AR-quiver of the 2-cluster category to study the periodic properties, as in the two previous sections. C The AR-quiver of (E ) is illustrated in ﬁgure 12. It has a Moebius 2 6 shape, the two outermost τ-orbits each contain 14 objects, whereas the two innermost τ-orbits contains only 7 objects each. Hence for any object X of C (E ) we have that τ14X = X. From this information we deduce that other 2 6 T T possible values of s such that τs = for a 2-cluster tilting subcategory of C (E ) are s = 2 or s = 7. 2 6 ◦ (cid:63) ◦ (cid:63) C Figure 12. The AR-quiver of (E ), the objects of same 2 6 shape on either side are identiﬁed. T Assume ﬁrst that s = 2. If there is a summand of in any given τ-orbit, T then this τ-orbit contains either 7 or 14 indecomposables from . This is T not possible as only has 6 indecomposable summands up to isomorphism. ##LTLine## ##LTLine##

text: PERIODICITY

text: OF

text: CLUSTER

text: TILTING

text: OBJECTS

label: list

text: Z

text: τtn

text: for

text: odd

text: n

text: and

text: t

text: Z

text: for

text: all

text: n

text: and

text: t

text: Z

text: for

text: odd

text: n

text: and

text: t

text: O

text: Proof.

text: For

text: even

text: values

text: of

text: n

text: there

text: is

text: only

text: one

text: type

text: of

text: AR-quiver

text: of

text: to

text: F

text: consider,

text: which

text: is

text: cylindrical.

text: However

text: for

text: odd

text: n

text: there

text: are

text: two

text: possibilities,

text: one

text: is

text: cylindrical

text: and

text: the

text: other

text: is

text: cylindrical

text: for

text: β-objects

text: with

text: a

text: Moebius-

text: twist

text: in

text: the

text: top

text: two

text: τ-orbits

text: consisting

text: of

text: the

text: α-objects.

text: From

text: the

text: last

text: part

text: of

text: theorem

text: it

text: follows

text: that

text: for

text: both

text: even

text: and

text: odd

text: n

text: the

text: tilting

text: object

text: where

text: the

text: corresponding

text: quiver

text: of

text: EndC(T

text: )op

text: is

text: an

text: n-cycle

text: is

text: periodic

text: under

text: Since

text: is

text: the

text: smallest

text: even

text: natural

text: number

text: we

text: get

text: the

text: last

text: two

text: parts

text: of

text: the

text: corollary

text: by

text: applying

text: lemma

text: and

text: lemma

text: These

text: cases

text: also

text: cover

text: all

text: categories

text: having

text: a

text: tilting

text: object

text: due

text: to

text: the

text: second

text: part

text: of

text: theorem

text: The

text: last

text: part

text: to

text: consider

text: is

text: the

text: ﬁrst

text: part

text: of

text: theorem

text: which

text: gives

text: rise

text: to

text: the

text: ﬁrst

text: two

text: parts

text: of

text: this

text: corollary

text: using

text: lemma

text: and

text: lemma

text: It

text: is

text: only

text: necessary

text: to

text: consider

text: the

text: cases

text: for

text: odd

text: n

text: as

text: all

text: cases

text: of

text: even

text: n

text: is

text: considered

text: in

text: the

text: third

text: part

text: of

text: the

text: corollary.

text: tilting

text: subcategories

text: of

text: type

text: E

text: In

text: this

text: section

text: we

text: focus

text: on

text: the

text: cases,

text: E

text: E

text: and

text: E

text: We

text: treat

text: each

text: case

text: in

text: its

text: separate

text: subsection

text: In

text: each

text: case

text: we

text: will

text: use

text: the

text: properties

text: of

text: the

text: AR-quiver

text: of

text: the

text: category

text: to

text: study

text: the

text: periodic

text: properties,

text: as

text: in

text: the

text: two

text: previous

text: sections.

text: C

text: The

text: AR-quiver

text: of

text: (E

text: is

text: illustrated

text: in

text: ﬁgure

text: It

text: has

text: a

text: Moebius

text: shape,

text: the

text: two

text: outermost

text: τ-orbits

text: each

text: contain

text: objects,

text: whereas

text: the

text: two

text: innermost

text: τ-orbits

text: contains

text: only

text: objects

text: each.

text: Hence

text: for

text: any

text: object

text: X

text: of

text: C

text: (E

text: we

text: have

text: that

text: X.

text: From

text: this

text: information

text: we

text: deduce

text: that

text: other

text: T

text: T

text: possible

text: values

text: of

text: s

text: such

text: that

text: τs

text: for

text: a

text: tilting

text: subcategory

text: of

text: C

text: (E

text: are

text: s

text: or

text: s

text: C

text: Figure

text: The

text: AR-quiver

text: of

text: (E

text: the

text: objects

text: of

text: same

text: shape

text: on

text: either

text: side

text: are

text: identiﬁed.

text: T

text: Assume

text: ﬁrst

text: that

text: s

text: If

text: there

text: is

text: a

text: summand

text: of

text: in

text: any

text: given

text: τ-orbit,

text: T

text: then

text: this

text: τ-orbit

text: contains

text: either

text: or

text: indecomposables

text: from

text: This

text: is

text: T

text: not

text: possible

text: as

text: only

text: has

text: indecomposable

text: summands

text: up

text: to

text: isomorphism.

text: ##LTLine##

text: ##LTLine##


text: PERIODICITY OF CLUSTER TILTING OBJECTS 29 (9) M (10) M (11) M (12) M The number of possible distributions of the 8 indecomposable summands are now signiﬁcantly reduced. The remaining possible distributions have all been checked, and the result is presented in table . Before proceeding to the table we present the necessary notation, as well as an example of the calculations. For the rest of the subsection we will call the bottom τ-orbit of the AR- quiver nr 1, the second bottom τ-orbit nr 2 and the top τ-orbit nr 8. We will now introduce a short notation for the distribution of summands be- tween these three τ-orbits. Denote the distribution of indecomposables by the vector (a, b, c), where a is the indecomposables in τ-orbit nr 1, b the indecomposables in τ-orbit nr 2 and c the indecomposables in τ-orbit nr 8. If a 2-cluster tilting object T has 2 indecomposables closed under τ8 in a τ-orbit, it will be denoted 2. If T has 4 indecomposables such that these are closed under τ4 it will be denoted 4. If there are 4 indecomposables in a ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: PERIODICITY

text: OF

text: CLUSTER

text: TILTING

text: OBJECTS

label: equation

text: M

text: M

text: M

text: M

text: The

text: number

text: of

text: possible

text: distributions

text: of

text: the

text: indecomposable

text: summands

text: are

text: now

text: signiﬁcantly

text: reduced.

text: The

text: remaining

text: possible

text: distributions

text: have

text: all

text: been

text: checked,

text: and

text: the

text: result

text: is

text: presented

text: in

text: table

text: Before

text: proceeding

text: to

text: the

text: table

text: we

text: present

text: the

text: necessary

text: notation,

text: as

text: well

text: as

text: an

text: example

text: of

text: the

text: calculations.

text: For

text: the

text: rest

text: of

text: the

text: subsection

text: we

text: will

text: call

text: the

text: bottom

text: τ-orbit

text: of

text: the

text: AR-

text: quiver

text: nr

text: the

text: second

text: bottom

text: τ-orbit

text: nr

text: and

text: the

text: top

text: τ-orbit

text: nr

text: We

text: will

text: now

text: introduce

text: a

text: short

text: notation

text: for

text: the

text: distribution

text: of

text: summands

text: be-

text: tween

text: these

text: three

text: τ-orbits.

text: Denote

text: the

text: distribution

text: of

text: indecomposables

text: by

text: the

text: vector

text: (a,

text: b,

text: c),

text: where

text: a

text: is

text: the

text: indecomposables

text: in

text: τ-orbit

text: nr

text: b

text: the

text: indecomposables

text: in

text: τ-orbit

text: nr

text: and

text: c

text: the

text: indecomposables

text: in

text: τ-orbit

text: nr

text: If

text: a

text: tilting

text: object

text: T

text: has

text: indecomposables

text: closed

text: under

text: τ8

text: in

text: a

text: τ-orbit,

text: it

text: will

text: be

text: denoted

text: If

text: T

text: has

text: indecomposables

text: such

text: that

text: these

text: are

text: closed

text: under

text: τ4

text: it

text: will

text: be

text: denoted

text: If

text: there

text: are

text: indecomposables

text: in

text: a

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text: 6 BENEDIKTE GRIMELAND ∈ T ∈ T Proof. Let T . Then for any T and for m > 1 we have that 1 − 2 HomD (T , T [m i]) = 0 for 0 < i < m. Using Serre duality we then b 1 2 obtain: ∼ − − S 0 = HomD (T , T [m i]) = D HomD (T [m i] , T ) b 1 2 ∼ b 2 1 S − = D HomD (T , ( T [ m]) [i]) ∼ b 2 1 S = D HomD (T , ( T ) [i]) b 2 m 1 S S T Hence the object T [m] = T is in add . Similarly one can show that S− 1 S− m 1 T (cid:3) the object 1T [m] = 1T also belongs to . 1 m 1 − This implies that we have a 1 1-correspondence between m-cluster tilting D subcategories in b(H) and m-cluster tilting subcategories of the m-cluster category: Corollary 9. Let H be a representation-ﬁnite hereditary algebra. Then D there is a 1-1 correspondence between m-cluster tilting subcategories of b(H) C D − − and m-cluster tilting subcategories of (H) = b(H)/τ [m 1]. m T C X Lemma 10. Let be an n-cluster tilting subcategory of (H), and let T D C → C m be the preimage of in b(H). If FC : (H) (H) is a functor of T mX m the form τs [t], and is FC-periodic then is periodic under the functor D → D FD : b(H) b(H) where FD = τs [t] . b b Proof. It is known from [5] and [16] that the m-cluster category is triangu- D → C ≥ lated and that π : b(H) (H) is a triangle functor for m 2. Since m m D π is a triangle functor it commutes with the suspension functors in b(H) m C and (H) (both denoted [1] by abuse of notation). m From proposition 1.3 in [5] we also have that π commutes with the m AR-translate in both categories. X ∈ X Let X be any object in . We need to show that FDX . We therefore ∈ T have that FC(π (X)) = π (FD(X)), and hence since FC(π (X)) we m ∈ X m m (cid:3) have that FD(X) . We will now describe the method that will be applied in sections , and , where we will continue the focus on representation-ﬁnite hereditary algebras. As we see from theorem 7, whether or not an m-cluster tilting subcategory X D of b(H) can be pushed down to an m-cluster tilting subcategory in an O X orbit category (H), depends on if is periodic under the functor F in D F b(H). Showing that a subcategory is F -periodic for a certain functor F will in the cases we will study require some counting, and is most easily carried out in a category with only ﬁnitely many isomorphism classes of C D indecomposable objects (i.e. in (H) rather than in in b(H)). However, m D by corollary 9 and lemma 10, for an m-cluster tilting subcategory of b(H) X with image π ( ) in the m-cluster category, we may study the periodicity of X m π ( ) under functors of the form τs [t] in the m-cluster category to obtain m X D information about the periodicity of under τs [t] in b(H). We will use this method to determine the period-properties under func- D tors of the form τs [t] for all 2-cluster tilting subcategories of b(H) for ∆ C a Dynkin diag−ram.∼Note that in 2-cluster categ∼ories 2(H), we have the isomorphism τ [1] = id as functors, and hence τ = [1]. Hence we only need to check for periodicity under one of the functors τs or [t]. In sections , and

text: BENEDIKTE

text: GRIMELAND

text: T

text: T

text: Proof.

text: Let

text: T

text: Then

text: for

text: any

text: T

text: and

text: for

text: m

text: we

text: have

text: that

text: HomD

text: (T

text: T

text: [m

text: i])

text: for

text: i

text: m.

text: Using

text: Serre

text: duality

text: we

text: then

text: b

text: obtain:

text: S

text: HomD

text: (T

text: T

text: [m

text: i])

text: D

text: HomD

text: (T

text: [m

text: i]

text: T

text: b

text: b

text: S

text: D

text: HomD

text: (T

text: T

text: m])

text: [i])

text: b

text: S

text: D

text: HomD

text: (T

text: T

text: [i])

text: b

text: m

text: S

text: S

text: T

text: Hence

text: the

text: object

text: T

text: [m]

text: T

text: is

text: in

text: add

text: Similarly

text: one

text: can

text: show

text: that

text: S−

text: S−

text: m

text: T

text: the

text: object

text: [m]

text: also

text: belongs

text: to

text: m

text: This

text: implies

text: that

text: we

text: have

text: a

text: between

text: m-cluster

text: tilting

text: D

text: subcategories

text: in

text: b(H)

text: and

text: m-cluster

text: tilting

text: subcategories

text: of

text: the

text: m-cluster

text: category:

text: Corollary

text: Let

text: H

text: be

text: a

text: representation-ﬁnite

text: hereditary

text: algebra.

text: Then

text: D

text: there

text: is

text: a

text: correspondence

text: between

text: m-cluster

text: tilting

text: subcategories

text: of

text: b(H)

text: C

text: D

text: and

text: m-cluster

text: tilting

text: subcategories

text: of

text: (H)

text: b(H)/τ

text: [m

text: m

text: T

text: C

text: X

text: Lemma

text: Let

text: be

text: an

text: n-cluster

text: tilting

text: subcategory

text: of

text: (H),

text: and

text: let

text: T

text: D

text: C

text: C

text: m

text: be

text: the

text: preimage

text: of

text: in

text: b(H).

text: If

text: FC

text: (H)

text: (H)

text: is

text: a

text: functor

text: of

text: T

text: mX

text: m

text: the

text: form

text: τs

text: [t],

text: and

text: is

text: FC-periodic

text: then

text: is

text: periodic

text: under

text: the

text: functor

text: D

text: D

text: FD

text: b(H)

text: b(H)

text: where

text: FD

text: τs

text: [t]

text: b

text: b

text: Proof.

text: It

text: is

text: known

text: from

text: and

text: that

text: the

text: m-cluster

text: category

text: is

text: triangu-

text: D

text: C

text: lated

text: and

text: that

text: π

text: b(H)

text: (H)

text: is

text: a

text: triangle

text: functor

text: for

text: m

text: Since

text: m

text: m

text: D

text: π

text: is

text: a

text: triangle

text: functor

text: it

text: commutes

text: with

text: the

text: suspension

text: functors

text: in

text: b(H)

text: m

text: C

text: and

text: (H)

text: (both

text: denoted

text: by

text: abuse

text: of

text: notation).

text: m

text: From

text: proposition

text: in

text: we

text: also

text: have

text: that

text: π

text: commutes

text: with

text: the

text: m

text: AR-translate

text: in

text: both

text: categories.

text: X

text: X

text: Let

text: X

text: be

text: any

text: object

text: in

text: We

text: need

text: to

text: show

text: that

text: FDX

text: We

text: therefore

text: T

text: have

text: that

text: FC(π

text: (X))

text: π

text: (FD(X)),

text: and

text: hence

text: since

text: FC(π

text: (X))

text: we

text: m

text: X

text: m

text: m

text: have

text: that

text: FD(X)

text: We

text: will

text: now

text: describe

text: the

text: method

text: that

text: will

text: be

text: applied

text: in

text: sections

text: and

text: where

text: we

text: will

text: continue

text: the

text: focus

text: on

text: representation-ﬁnite

text: hereditary

text: algebras.

text: As

text: we

text: see

text: from

text: theorem

text: whether

text: or

text: not

text: an

text: m-cluster

text: tilting

text: subcategory

text: X

text: D

text: of

text: b(H)

text: can

text: be

text: pushed

text: down

text: to

text: an

text: m-cluster

text: tilting

text: subcategory

text: in

text: an

text: O

text: X

text: orbit

text: category

text: (H),

text: depends

text: on

text: if

text: is

text: periodic

text: under

text: the

text: functor

text: F

text: in

text: D

text: F

text: b(H).

text: Showing

text: that

text: a

text: subcategory

text: is

text: F

text: -periodic

text: for

text: a

text: certain

text: functor

text: F

text: will

text: in

text: the

text: cases

text: we

text: will

text: study

text: require

text: some

text: counting,

text: and

text: is

text: most

text: easily

text: carried

text: out

text: in

text: a

text: category

text: with

text: only

text: ﬁnitely

text: many

text: isomorphism

text: classes

text: of

text: C

text: D

text: indecomposable

text: objects

text: (i.e.

text: in

text: (H)

text: rather

text: than

text: in

text: in

text: b(H)).

text: However,

text: m

text: D

text: by

text: corollary

text: and

text: lemma

text: for

text: an

text: m-cluster

text: tilting

text: subcategory

text: of

text: b(H)

text: X

text: with

text: image

text: π

text: in

text: the

text: m-cluster

text: category,

text: we

text: may

text: study

text: the

text: periodicity

text: of

text: X

text: m

text: π

text: under

text: functors

text: of

text: the

text: form

text: τs

text: [t]

text: in

text: the

text: m-cluster

text: category

text: to

text: obtain

text: m

text: X

text: D

text: information

text: about

text: the

text: periodicity

text: of

text: under

text: τs

text: [t]

text: in

text: b(H).

text: We

text: will

text: use

text: this

text: method

text: to

text: determine

text: the

text: period-properties

text: under

text: func-

text: D

text: tors

text: of

text: the

text: form

text: τs

text: [t]

text: for

text: all

text: tilting

text: subcategories

text: of

text: b(H)

text: for

text: C

text: a

text: Dynkin

text: diag−ram.∼Note

text: that

text: in

text: categ∼ories

text: we

text: have

text: the

text: isomorphism

text: τ

text: id

text: as

text: functors,

text: and

text: hence

text: τ

text: Hence

text: we

text: only

text: need

text: to

text: check

text: for

text: periodicity

text: under

text: one

text: of

text: the

text: functors

text: τs

text: or

text: [t].

text: In

text: sections

text: and


text: 10 BENEDIKTE GRIMELAND 5. 2-cluster tilting subcategories of type A In this section we will apply the method described in the last part of section to the cases where H is a path-algebra of type kA , where A is n n some orientation of the Dynkin diagram A . As discussed in section , it is n suﬃcient to study under which functors of the form τs the cluster tilting C ∈ Z objects of the cluster category (A ) are periodic, where s . We give 2 n a complete overview of the smallest possible positive value of s for each C cluster tilting object of (A ). Furthermore, we show that the periodicity 2 n of a 2-cluster tilting object depends on symmetric properties of the quiver of the corresponding cluster tilted algebra. First we need to recall some facts about the AR-quiver of the 2-cluster C C category of (A ). Recall that if n is even then all the τ-orbits in (A ) 2 n 2 n have n + 3 objects. If n = 2l + 1 is odd, then the innermost τ-orbit has l + 2 objects, and all the other τ-orbits have n + 3-objects. Hence for any C indecomposable object Z in (A ) we have τ(n+3)Z = Z (for n even and 2 n odd). We ﬁrst give a result narrowing down the possible values of s for which it is interesting to study periodicity of τs on the 2-cluster tilting subcategories. C Theorem 18. Let T be a 2-cluster tilting object in (A ) such that τsT = T 2 n for some 0 < s < n + 3. Then 1. either s = n+3, with n an odd number and T has exactly one sum- 2 i mand in the innermost τ-orbit 2. or s = n+3 and n is divisible by 3. 3 Proof. We will ﬁrst assume that all indecomposable summands of T lie in τ- C orbits in the AR-quiver of (A ) with n + 3 indecomposable objects. Since 2 n we have τn+3T = T and have assumed τsT = T , we must have that s is a factor of n + 3, i.e. n + 3 = sa . Then a = n+3 must be a factor of n, since s each τ-orbit containing one indecomposable summand of T must contain a multiple of a indecomposable summands of T , and T has n indecomposable ∈ N summands. That is (n + 3) = sa and n = ab for some b . Hence ∈ { } a 1, 3 . However by assumption s < n + 3, so that a > 1, therefore we have a = 3 and the second part follows. For the ﬁrst part, assume that there is at least one indecomposable sum- mand T of T that lie in a τ-orbit containing n+3 indecomposable objects i 2 n+3 (clearly this can only happen when n is odd). Then we have τ 2 Ti = Ti. Note that this is the minimal value of s such that τsT = T . This is most i i easily seen in a ﬁgure. Drawing the Ext-support of T will reveal that it has i extension to all other objects in the innermost τ-orbit. Hence the claim is (cid:3) proved. We will study the ﬁrst case of theorem 18 closer in subsection , and the second case in subsection . C In this subsection we will study 2-cluster tilting objects of (A ) which 2 n are covered by part 1 of theorem 18. It is clear from theorem 18 that if U is ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: BENEDIKTE

text: GRIMELAND

text: tilting

text: subcategories

text: of

text: type

text: A

text: In

text: this

text: section

text: we

text: will

text: apply

text: the

text: method

text: described

text: in

text: the

text: last

text: part

text: of

text: section

text: to

text: the

text: cases

text: where

text: H

text: is

text: a

text: path-algebra

text: of

text: type

text: kA

text: where

text: A

text: is

text: n

text: n

text: some

text: orientation

text: of

text: the

text: Dynkin

text: diagram

text: A

text: As

text: discussed

text: in

text: section

text: it

text: is

text: n

text: suﬃcient

text: to

text: study

text: under

text: which

text: functors

text: of

text: the

text: form

text: τs

text: the

text: cluster

text: tilting

text: C

text: Z

text: objects

text: of

text: the

text: cluster

text: category

text: (A

text: are

text: periodic,

text: where

text: s

text: We

text: give

text: n

text: a

text: complete

text: overview

text: of

text: the

text: smallest

text: possible

text: positive

text: value

text: of

text: s

text: for

text: each

text: C

text: cluster

text: tilting

text: object

text: of

text: (A

text: Furthermore,

text: we

text: show

text: that

text: the

text: periodicity

text: n

text: of

text: a

text: tilting

text: object

text: depends

text: on

text: symmetric

text: properties

text: of

text: the

text: quiver

text: of

text: the

text: corresponding

text: cluster

text: tilted

text: algebra.

text: First

text: we

text: need

text: to

text: recall

text: some

text: facts

text: about

text: the

text: AR-quiver

text: of

text: the

text: C

text: C

text: category

text: of

text: (A

text: Recall

text: that

text: if

text: n

text: is

text: even

text: then

text: all

text: the

text: τ-orbits

text: in

text: (A

text: n

text: n

text: have

text: n

text: objects.

text: If

text: n

text: is

text: odd,

text: then

text: the

text: innermost

text: τ-orbit

text: has

text: l

text: objects,

text: and

text: all

text: the

text: other

text: τ-orbits

text: have

text: n

text: Hence

text: for

text: any

text: C

text: indecomposable

text: object

text: Z

text: in

text: (A

text: we

text: have

text: Z

text: (for

text: n

text: even

text: and

text: n

text: odd).

text: We

text: ﬁrst

text: give

text: a

text: result

text: narrowing

text: down

text: the

text: possible

text: values

text: of

text: s

text: for

text: which

text: it

text: is

text: interesting

text: to

text: study

text: periodicity

text: of

text: τs

text: on

text: the

text: tilting

text: subcategories.

text: C

text: Theorem

text: Let

text: T

text: be

text: a

text: tilting

text: object

text: in

text: (A

text: such

text: that

text: τsT

text: T

text: n

text: for

text: some

text: s

text: n

text: Then

text: either

text: s

text: with

text: n

text: an

text: odd

text: number

text: and

text: T

text: has

text: exactly

text: one

text: sum-

text: i

text: mand

text: in

text: the

text: innermost

text: τ-orbit

text: or

text: s

text: and

text: n

text: is

text: divisible

text: by

text: Proof.

text: We

text: will

text: ﬁrst

text: assume

text: that

text: all

text: indecomposable

text: summands

text: of

text: T

text: lie

text: in

text: τ-

text: C

text: orbits

text: in

text: the

text: AR-quiver

text: of

text: (A

text: with

text: n

text: indecomposable

text: objects.

text: Since

text: n

text: we

text: have

text: T

text: and

text: have

text: assumed

text: τsT

text: T

text: we

text: must

text: have

text: that

text: s

text: is

text: a

text: factor

text: of

text: n

text: i.e.

text: n

text: sa

text: Then

text: a

text: must

text: be

text: a

text: factor

text: of

text: n,

text: since

text: s

text: each

text: τ-orbit

text: containing

text: one

text: indecomposable

text: summand

text: of

text: T

text: must

text: contain

text: a

text: multiple

text: of

text: a

text: indecomposable

text: summands

text: of

text: T

text: and

text: T

text: has

text: n

text: indecomposable

text: N

text: summands.

text: That

text: is

text: (n

text: sa

text: and

text: n

text: ab

text: for

text: some

text: b

text: Hence

text: a

text: However

text: by

text: assumption

text: s

text: n

text: so

text: that

text: a

text: therefore

text: we

text: have

text: a

text: and

text: the

text: second

text: part

text: follows.

text: For

text: the

text: ﬁrst

text: part,

text: assume

text: that

text: there

text: is

text: at

text: least

text: one

text: indecomposable

text: sum-

text: mand

text: T

text: of

text: T

text: that

text: lie

text: in

text: a

text: τ-orbit

text: containing

text: indecomposable

text: objects

text: i

text: (clearly

text: this

text: can

text: only

text: happen

text: when

text: n

text: is

text: odd).

text: Then

text: we

text: have

text: τ

text: Ti

text: Ti.

text: Note

text: that

text: this

text: is

text: the

text: minimal

text: value

text: of

text: s

text: such

text: that

text: τsT

text: T

text: This

text: is

text: most

text: i

text: i

text: easily

text: seen

text: in

text: a

text: ﬁgure.

text: Drawing

text: the

text: Ext-support

text: of

text: T

text: will

text: reveal

text: that

text: it

text: has

text: i

text: extension

text: to

text: all

text: other

text: objects

text: in

text: the

text: innermost

text: τ-orbit.

text: Hence

text: the

text: claim

text: is

text: proved.

text: We

text: will

text: study

text: the

text: ﬁrst

text: case

text: of

text: theorem

text: closer

text: in

text: subsection

text: and

text: the

text: second

text: case

text: in

text: subsection

text: C

text: In

text: this

text: subsection

text: we

text: will

text: study

text: tilting

text: objects

text: of

text: (A

text: which

text: n

text: are

text: covered

text: by

text: part

text: of

text: theorem

text: It

text: is

text: clear

text: from

text: theorem

text: that

text: if

text: U

text: is

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##