text: IEEE TRANSACTIONS ON AUTOMATIC CONTROL 6 costly, and any approximation, strictly speaking, requires robustness analysis. 0 Secondly, it is demonstrated in Section that our linear-quadratic bounds (5) and (54)–(56) may prohibit one from declaring global −1 stabilization even when it is likely to be achieved. In fact, such restrictive bounds are not necessary. Their purpose is to simplify −2 1.5 stability analysis by making the case “almost linear”. The simpli- ﬁcation is evinced in the exponential rate of decay (64) which is −3 u generally not expected from nonlinear systems. A possible way to KL −4 1 avoid assumptions (5) and (54)–(56) and to achieve class rate of x1 decay in (64) is to follow the methodology of [6]. −5 π ACKNOWLEDGMENT The author thanks Prof. M. Krstic for suggesting a number of 0.5 −6 substantial improvements to the presentation. π 4 REFERENCES −7 [1] G. Goebel, U. Mu¨nz, and F. Allgo¨wer, “Stabilization of linear systems 0 with distributed input delay,” in 2010 American Control Conference, −8 0 1 2 3 4 5 6 7 8 9 10 t Marriott Waterfront, Baltimore, MD, USA, 2010, pp. 5800–5805. [2] F. Mazenc, S.-I. Niculescu, and M. Bekaik, “Stabilization of time- −9 varying nonlinear systems with distributed input delay by feedback of 0 1 2 3 4 5 6 7 8 9 10 plant’s state,” IEEE Trans. Autom. Control, vol. 58, no. 1, pp. 264–269, t Jan. 2013. Fig. 1: Simulation of (89) under the feedback (58), (104) with initial [3] A. Manitius and A. W. Olbrot, “Finite spectrum assignment problem for conditions x (0) ∈ {π/2, π, 3π/2}, x (0) = π/2, u ≡ 1. systems with delays,” IEEE Trans. Autom. Control, vol. AC-24, no. 4, 1 2 0 pp. 541–552, Aug. 1979. [4] W. H. Kwon and A. E. Pearson, “Feedback stabilization of linear systems with delayed control,” IEEE Trans. Autom. Control, vol. AC-25, no. 2, pp. 266–269, Apr. 1980. v (y) = yTV y (98) [5] Z. Artstein, “Linear systems with delayed control: a reduction,” IEEE 0 Trans. Autom. Control, vol. AC-27, no. 4, pp. 869–879, Aug. 1982. [6] M. Krstic, “Input delay compensation for forward complete and strict- with some positive deﬁnite V . The requirement of Corollary 1 is feedforward nonlinear systems,” IEEE Trans. Autom. Control, vol. 5, no. 2, pp. 287–303, Feb. 2010. then [7] A. Ponomarev, “Reduction-based robustness analysis of linear predic- − ≤ − tor feedback for distributed input delays,” IEEE Trans. Autom. Con- 2fT(y)V y 4k(V y)TBBTV y w (y). (99) trol, to be published, doi:10.1109/TAC.2015.2437520, available online: 0 http://arxiv.org/abs/1505.02402. [8] M. Krstic, “Lyapunov tools for predictor feedbacks for delay systems: It is convenient to decompose f(y) as inverse optimality and robustness to delay mismatch,” Automatica, vol. 4, no. 11, pp. 2930–2935, Nov. 2008. [9] N. Bekiaris-Liberis and M. Krstic, “Lyapunov stability of linear predictor f(y) = F(y)y, (100) feedback for distributed input delays,” IEEE Trans. Autom. Control, (cid:18) (cid:19) vol. 56, no. 3, pp. 655–660, Mar. 2011. [10] F. Mazenc, S.-I. Niculescu, and M. Krstic, “Lyapunov–Krasovskii func- where tionals and application to input delay compensation for linear time- 0 1 invariant systems,” Automatica, vol. 48, no. 7, pp. 1317–1323, Jul. 2012. F(y) = , (101) [11] W. Michiels and S.-I. Niculescu, “On the delay sensitivity of Smith α(y) 0 Predictors,” Int. J. Syst. Sci., vol. 34, no. 8–9, pp. 543–551, Jul. 2003. α(y) = sin y1 ∈ [−0.22, 1], (102) [12] M. Jankovic, “Forwarding, backstepping, and ﬁnite spectrum assignment y for time delay systems,” Automatica, vol. 45, no. 1, pp. 2–9, Jan. 2009. 1 [13] I. Karafyllis and M. Krstic, “Delay-robustness of linear predictor feed- then (99) will follow from negative deﬁniteness of the matrix back without restriction on delay rate,” Automatica, vol. 49, no. 6, pp. 1761–1767, Jun. 2013. − − − V 1FT(y) + F(y)V 1 4kBBT (103) [14] D. Bresch-Pietri and M. Krstic, “Delay-adaptive predictor feedback for systems with unknown long actuator delay,” IEEE Trans. Autom. Control, vol. 55, no. 9, pp. 2106–2112, Sep. 2010. for all y ∈ Rn, all B ∈ B, and some k. We found computationally [15] D. Bresch-Pietri, J. Chauvin, and N. Petit, “Adaptive control scheme for uncertain time-delay systems,” Automatica, vol. 48, no. 8, pp. 1536– that for h = π/4 it is valid to take V = I and k = 1, so the feedback 1552, Aug. 2012. function suggested by Corollary 1 is [16] Z. Palmor, “Stability properties of Smith dead-time compensator con- troller,” Int. J. Control, vol. 32, no. 6, pp. 937–949, 1980. − κ(y, ϕ) = BT(y, ϕ)y. (104) [17] S. Mondie´, M. Dambrine, and O. Santos, “Approximation of control laws with distributed delays: a necessary condition for stability,” Kybernetika, vol. 38, no. 5, pp. 541–551, 2002. Fig. 1 shows the simulation results of (89) under the feedback (58), [18] S. Mondie´ and W. Michiels, “A safe implementation for ﬁnite spectrum assignment: robustness analysis,” in Proc. 42nd IEEE Conf. Decision (104) for different initial conditions. Euler’s approximation with time and Control, vol. 5, Dec. 2003, pp. 4569–4574. step of 0.01 was used for calculating x(t), β(s), and ξ(s). [19] S. Evesque, A. M. Annaswamy, S. Niculescu, and A. P. Dowling, VII. CONCLUSIONS “Adaptive control of a class of time-delay systems,” J. Dyn. Sys., Meas., Control, vol. 125, no. 2, pp. 186–193, Jun. 2003. Let us pose some problems for a future discussion. [20] V. L. Kharitonov, “An extension of the prediction scheme to the case of Firstly, concerning practical implementation of the proposed feed- systems with both input and state delay,” Automatica, vol. 50, no. 1, pp. back, it should be mentioned that solving nonlinear equations may be 211–217, Jan. 2014. ##LTFigure## ##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: IEEE

text: TRANSACTIONS

text: ON

text: AUTOMATIC

text: CONTROL

text: costly,

text: and

text: any

text: approximation,

text: strictly

text: speaking,

text: requires

text: robustness

text: analysis.

text: Secondly,

text: it

text: is

text: demonstrated

text: in

text: Section

text: that

text: our

text: linear-quadratic

text: bounds

text: and

text: may

text: prohibit

text: one

text: from

text: declaring

text: global

text: −1

text: stabilization

text: even

text: when

text: it

text: is

text: likely

text: to

text: be

text: achieved.

text: In

text: fact,

text: such

text: restrictive

text: bounds

text: are

text: not

text: necessary.

text: Their

text: purpose

text: is

text: to

text: simplify

text: −2

text: stability

text: analysis

text: by

text: making

text: the

text: case

text: “almost

text: linear”.

text: The

text: simpli-

text: ﬁcation

text: is

text: evinced

text: in

text: the

text: exponential

text: rate

text: of

text: decay

text: which

text: is

text: −3

text: u

text: generally

text: not

text: expected

text: from

text: nonlinear

text: systems.

text: A

text: possible

text: way

text: to

text: KL

text: −4

text: avoid

text: assumptions

text: and

text: and

text: to

text: achieve

text: class

text: rate

text: of

text: decay

text: in

text: is

text: to

text: follow

text: the

text: methodology

text: of

text: −5

text: π

text: ACKNOWLEDGMENT

text: The

text: author

text: thanks

text: Prof.

text: M.

text: Krstic

text: for

text: suggesting

text: a

text: number

text: of

text: −6

text: substantial

text: improvements

text: to

text: the

text: presentation.

text: π

text: REFERENCES

text: −7

text: G.

text: Goebel,

text: U.

text: Mu¨nz,

text: and

text: F.

text: Allgo¨wer,

text: “Stabilization

text: of

text: linear

text: systems

text: with

text: distributed

text: input

text: delay,”

text: in

text: American

text: Control

text: Conference,

text: −8

text: t

text: Marriott

text: Waterfront,

text: Baltimore,

text: MD,

text: USA,

text: pp.

text: F.

text: Mazenc,

text: S.-I.

text: Niculescu,

text: and

text: M.

text: Bekaik,

text: “Stabilization

text: of

text: time-

text: −9

text: varying

text: nonlinear

text: systems

text: with

text: distributed

text: input

text: delay

text: by

text: feedback

text: of

text: plant’s

text: state,”

text: IEEE

text: Trans.

text: Autom.

text: Control,

text: vol.

text: no.

text: pp.

text: t

text: Jan.

text: Fig.

text: Simulation

text: of

text: under

text: the

text: feedback

text: with

text: initial

text: A.

text: Manitius

text: and

text: A.

text: W.

text: Olbrot,

text: “Finite

text: spectrum

text: assignment

text: problem

text: for

text: conditions

text: x

text: π,

text: x

text: u

text: systems

text: with

text: delays,”

text: IEEE

text: Trans.

text: Autom.

text: Control,

text: vol.

text: no.

text: pp.

text: Aug.

text: W.

text: H.

text: Kwon

text: and

text: A.

text: E.

text: Pearson,

text: “Feedback

text: stabilization

text: of

text: linear

text: systems

text: with

text: delayed

text: control,”

text: IEEE

text: Trans.

text: Autom.

text: Control,

text: vol.

text: no.

text: pp.

text: Apr.

text: v

text: (y)

text: yTV

text: y

text: Z.

text: Artstein,

text: “Linear

text: systems

text: with

text: delayed

text: control:

text: a

text: reduction,”

text: IEEE

text: Trans.

text: Autom.

text: Control,

text: vol.

text: no.

text: pp.

text: Aug.

text: M.

text: Krstic,

text: “Input

text: delay

text: compensation

text: for

text: forward

text: complete

text: and

text: strict-

text: with

text: some

text: positive

text: deﬁnite

text: V

text: The

text: requirement

text: of

text: Corollary

text: is

text: feedforward

text: nonlinear

text: systems,”

text: IEEE

text: Trans.

text: Autom.

text: Control,

text: vol.

text: no.

text: pp.

text: Feb.

text: then

text: A.

text: Ponomarev,

text: “Reduction-based

text: robustness

text: analysis

text: of

text: linear

text: predic-

text: tor

text: feedback

text: for

text: distributed

text: input

text: delays,”

text: IEEE

text: Trans.

text: Autom.

text: Con-

text: y

text: y)TBBTV

text: y

text: w

text: (y).

text: trol,

text: to

text: be

text: published,

text: available

text: online:

text: M.

text: Krstic,

text: “Lyapunov

text: tools

text: for

text: predictor

text: feedbacks

text: for

text: delay

text: systems:

text: It

text: is

text: convenient

text: to

text: decompose

text: f(y)

text: as

text: inverse

text: optimality

text: and

text: robustness

text: to

text: delay

text: mismatch,”

text: Automatica,

text: vol.

text: no.

text: pp.

text: Nov.

text: N.

text: Bekiaris-Liberis

text: and

text: M.

text: Krstic,

text: “Lyapunov

text: stability

text: of

text: linear

text: predictor

text: f(y)

text: F(y)y,

text: feedback

text: for

text: distributed

text: input

text: delays,”

text: IEEE

text: Trans.

text: Autom.

text: Control,

text: vol.

text: no.

text: pp.

text: Mar.

text: F.

text: Mazenc,

text: S.-I.

text: Niculescu,

text: and

text: M.

text: Krstic,

text: “Lyapunov–Krasovskii

text: func-

text: where

text: tionals

text: and

text: application

text: to

text: input

text: delay

text: compensation

text: for

text: linear

text: time-

text: invariant

text: systems,”

text: Automatica,

text: vol.

text: no.

text: pp.

text: Jul.

text: F(y)

text: W.

text: Michiels

text: and

text: S.-I.

text: Niculescu,

text: “On

text: the

text: delay

text: sensitivity

text: of

text: Smith

text: α(y)

text: Predictors,”

text: Int.

text: J.

text: Syst.

text: Sci.,

text: vol.

text: no.

text: pp.

text: Jul.

text: α(y)

text: sin

text: M.

text: Jankovic,

text: “Forwarding,

text: backstepping,

text: and

text: ﬁnite

text: spectrum

text: assignment

text: y

text: for

text: time

text: delay

text: systems,”

text: Automatica,

text: vol.

text: no.

text: pp.

text: Jan.

text: I.

text: Karafyllis

text: and

text: M.

text: Krstic,

text: “Delay-robustness

text: of

text: linear

text: predictor

text: feed-

text: then

text: will

text: follow

text: from

text: negative

text: deﬁniteness

text: of

text: the

text: matrix

text: back

text: without

text: restriction

text: on

text: delay

text: rate,”

text: Automatica,

text: vol.

text: no.

text: pp.

text: Jun.

text: V

text: F(y)V

text: D.

text: Bresch-Pietri

text: and

text: M.

text: Krstic,

text: “Delay-adaptive

text: predictor

text: feedback

text: for

text: systems

text: with

text: unknown

text: long

text: actuator

text: delay,”

text: IEEE

text: Trans.

text: Autom.

text: Control,

text: vol.

text: no.

text: pp.

text: Sep.

text: for

text: all

text: y

text: Rn,

text: all

text: B

text: B,

text: and

text: some

text: k.

text: We

text: found

text: computationally

text: D.

text: Bresch-Pietri,

text: J.

text: Chauvin,

text: and

text: N.

text: Petit,

text: “Adaptive

text: control

text: scheme

text: for

text: uncertain

text: time-delay

text: systems,”

text: Automatica,

text: vol.

text: no.

text: pp.

text: that

text: for

text: h

text: it

text: is

text: valid

text: to

text: take

text: V

text: I

text: and

text: k

text: so

text: the

text: feedback

text: Aug.

text: function

text: suggested

text: by

text: Corollary

text: is

text: Z.

text: Palmor,

text: “Stability

text: properties

text: of

text: Smith

text: dead-time

text: compensator

text: con-

text: troller,”

text: Int.

text: J.

text: Control,

text: vol.

text: no.

text: pp.

text: κ(y,

text: ϕ)

text: BT(y,

text: ϕ)y.

text: S.

text: Mondie´,

text: M.

text: Dambrine,

text: and

text: O.

text: Santos,

text: “Approximation

text: of

text: control

text: laws

text: with

text: distributed

text: delays:

text: a

text: necessary

text: condition

text: for

text: stability,”

text: Kybernetika,

text: vol.

text: no.

text: pp.

text: Fig.

text: shows

text: the

text: simulation

text: results

text: of

text: under

text: the

text: feedback

text: S.

text: Mondie´

text: and

text: W.

text: Michiels,

text: “A

text: safe

text: implementation

text: for

text: ﬁnite

text: spectrum

text: assignment:

text: robustness

text: analysis,”

text: in

text: Proc.

text: IEEE

text: Conf.

text: Decision

text: for

text: different

text: initial

text: conditions.

text: Euler’s

text: approximation

text: with

text: time

text: and

text: Control,

text: vol.

text: Dec.

text: pp.

text: step

text: of

text: was

text: used

text: for

text: calculating

text: x(t),

text: β(s),

text: and

text: ξ(s).

text: S.

text: Evesque,

text: A.

text: M.

text: Annaswamy,

text: S.

text: Niculescu,

text: and

text: A.

text: P.

text: Dowling,

text: VII.

text: CONCLUSIONS

text: “Adaptive

text: control

text: of

text: a

text: class

text: of

text: time-delay

text: systems,”

text: J.

text: Dyn.

text: Sys.,

text: Meas.,

text: Control,

text: vol.

text: no.

text: pp.

text: Jun.

text: Let

text: us

text: pose

text: some

text: problems

text: for

text: a

text: future

text: discussion.

text: V.

text: L.

text: Kharitonov,

text: “An

text: extension

text: of

text: the

text: prediction

text: scheme

text: to

text: the

text: case

text: of

text: Firstly,

text: concerning

text: practical

text: implementation

text: of

text: the

text: proposed

text: feed-

text: systems

text: with

text: both

text: input

text: and

text: state

text: delay,”

text: Automatica,

text: vol.

text: no.

text: pp.

text: back,

text: it

text: should

text: be

text: mentioned

text: that

text: solving

text: nonlinear

text: equations

text: may

text: be

text: Jan.

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text: IEEE TRANSACTIONS ON AUTOMATIC CONTROL 7 [21] ——, “Predictor based stabilization of neutral type systems with input delay,” Automatica, vol. 52, no. 2, pp. 125–134, Feb. 2015. [22] D. Bresch-Pietri and M. Krstic, “Delay-adaptive control for nonlinear systems,” IEEE Trans. Autom. Control, vol. 59, no. 5, pp. 1203–1218, May 2014. [23] N. Bekiaris-Liberis and M. Krstic, “Compensation of state-dependent input delay for nonlinear systems,” IEEE Trans. Autom. Control, vol. 58, no. 2, pp. 275–289, Feb. 2013. [24] M. Jankovic, R. Sepulchre, and P. V. Kokotovic, “Constructive Lyapunov stabilization of nonlinear cascade systems,” IEEE Trans. Autom. Control, vol. 41, no. 12, pp. 1723–1735, Dec. 1996.

text: IEEE

label: paragraph

text: TRANSACTIONS

text: ON

text: AUTOMATIC

text: CONTROL

label: reference

text: “Predictor

text: based

text: stabilization

text: of

text: neutral

text: type

text: systems

text: with

text: input

text: delay,”

text: Automatica,

text: vol.

text: no.

text: pp.

text: Feb.

text: D.

text: Bresch-Pietri

text: and

text: M.

text: Krstic,

text: “Delay-adaptive

text: control

text: for

text: nonlinear

text: systems,”

text: IEEE

text: Trans.

text: Autom.

text: Control,

text: vol.

text: no.

text: pp.

text: May

text: N.

text: Bekiaris-Liberis

text: and

text: M.

text: Krstic,

text: “Compensation

text: of

text: state-dependent

text: input

text: delay

text: for

text: nonlinear

text: systems,”

text: IEEE

text: Trans.

text: Autom.

text: Control,

text: vol.

text: no.

text: pp.

text: Feb.

text: M.

text: Jankovic,

text: R.

text: Sepulchre,

text: and

text: P.

text: V.

text: Kokotovic,

text: “Constructive

text: Lyapunov

text: stabilization

text: of

text: nonlinear

text: cascade

text: systems,”

text: IEEE

text: Trans.

text: Autom.

text: Control,

text: vol.

text: no.

text: pp.

text: Dec.


text: (cid:48) ≤ ≤ with γ(0) = x and γ(d) = x . Let N be the largest integer less than Cd , and set ai = γ(iC) for 0 i N. Then, (cid:48) ≥ (cid:48) − dY(p(x), p(x )) dY(p(a0), p(aN))+dY(p(aN), p(x )) K − ≥ (cid:48) − N∑1 − dY(p(aN),x ) K + (dY(p(ai), p(ai+1)) K) i=0 ≥ − · K +K N ≥ K · (cid:48) − C dX(x,x ) 2K. → Since p: X Y is Lipschitz by assumption, this completes the proof. 4 The co-surface graph CS F This section introduces the co-surface graph CS of the free group and develops its basic properties. F First we deﬁne CS and discuss its relationship to other Out( )–graphs appearing in the literature. Then in § we use the theory of alignment preserving maps to calculate the boundary of CS. Finally, in § we show that a subgroup Γ ≤ Out(F) qi-embeds into CS if and only if it is purely atoroidal and qi-embeds into F. Recall that an element a ∈ F is primitive is a belongs to a free basis for F; this is an invariant of the conjugacy class α of a and so we also call α primitive. The primitive loop graph of F is the simplicial graph PL whose vertices are the primitive conjugacy classes of F, and where two conjugacy classes α,β are joined F by an edge if and only if they have representatives that are jointly part of a free basis of . We equip PL with the path metric d in which each edge has length 1. As each primitive element generates a cyclic free factor of F, there is natuPrLal inclusion map PL0 → F. It is straightforward to check that this inclusion is 2–bilipschitz → and 1–dense and therefore admits a 4–quasi-isometry coarse inverse which we denote D : F PL. We also → ∈ have the coarse projection π : X PL deﬁned by sending G X to the set of embedded closed loops on PL ◦ G; this projection coarsely agrees with the composition D π . F The primitive loop graph measures, in a sense, how algebraically complicated primitive conjugacy classes are with respect to each other. The co-surface graph, on the other hand, is designed to measure how topolog- ically complicated primitive conjugacy classes are with respect to each other: Deﬁnition 4.1 (Co-surface graph CS). The co-surface graph CS of the free group F is the simplicial graph whose vertices are conjugacy classes of primitive elements, and where two vertices α and β are joined by ∼ an edge if there is a once-punctured surface S and an isomorphism π1(S) = F with respect to which α and β may both be represented by simple closed curves on S. π CS π el PL X PL CS D π (cid:48) F el F Figure 2: The co-surface graph and its friends. All rightward pointing arrows are coarsely Lipschitz, and D F is a quasi-isometry. Coarsely speaking, all maps are Out( )–equivariant and the diagram commutes. In other words, each once-punctured surface S with π1(S) ∼= F determines a subset CS ⊂ PL0 consisting of those primitive conjugacy classes that correspond to (nonseparating) simple closed curves on S. As our ⊂ graph CS is obtained by collapsing each C PL to a set of diameter 1, it records the geometry of the S 11 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: with

text: x

text: and

text: γ(d)

text: x

text: Let

text: N

text: be

text: the

text: largest

text: integer

text: less

text: than

text: Cd

text: and

text: set

text: ai

text: γ(iC)

text: for

text: i

text: N.

text: Then,

text: dY(p(x),

text: p(x

text: p(aN))+dY(p(aN),

text: p(x

text: K

text: dY(p(aN),x

text: K

text: (dY(p(ai),

text: K)

text: K

text: +K

text: N

text: K

text: C

text: dX(x,x

text: Since

text: p:

text: X

text: Y

text: is

text: Lipschitz

text: by

text: assumption,

text: this

text: completes

text: the

text: proof.

label: section

text: The

text: co-surface

text: graph

text: CS

text: F

text: This

text: section

text: introduces

text: the

text: co-surface

text: graph

text: CS

text: of

text: the

text: free

text: group

text: and

text: develops

text: its

text: basic

text: properties.

text: F

text: First

text: we

text: deﬁne

text: CS

text: and

text: discuss

text: its

text: relationship

text: to

text: other

text: Out(

text: )–graphs

text: appearing

text: in

text: the

text: literature.

text: Then

text: in

text: we

text: use

text: the

text: theory

text: of

text: alignment

text: preserving

text: maps

text: to

text: calculate

text: the

text: boundary

text: of

text: CS.

text: Finally,

text: in

text: we

text: show

text: that

text: a

text: subgroup

text: Γ

text: Out(F)

text: qi-embeds

text: into

text: CS

text: if

text: and

text: only

text: if

text: it

text: is

text: purely

text: atoroidal

text: and

text: qi-embeds

text: into

text: F.

text: Recall

text: that

text: an

text: element

text: a

text: F

text: is

text: primitive

text: is

text: a

text: belongs

text: to

text: a

text: free

text: basis

text: for

text: F;

text: this

text: is

text: an

text: invariant

text: of

text: the

text: conjugacy

text: class

text: α

text: of

text: a

text: and

text: so

text: we

text: also

text: call

text: α

text: primitive.

text: The

text: primitive

text: loop

text: graph

text: of

text: F

text: is

text: the

text: simplicial

text: graph

text: PL

text: whose

text: vertices

text: are

text: the

text: primitive

text: conjugacy

text: classes

text: of

text: F,

text: and

text: where

text: two

text: conjugacy

text: classes

text: α,β

text: are

text: joined

text: F

text: by

text: an

text: edge

text: if

text: and

text: only

text: if

text: they

text: have

text: representatives

text: that

text: are

text: jointly

text: part

text: of

text: a

text: free

text: basis

text: of

text: We

text: equip

text: PL

text: with

text: the

text: path

text: metric

text: d

text: in

text: which

text: each

text: edge

text: has

text: length

text: As

text: each

text: primitive

text: element

text: generates

text: a

text: cyclic

text: free

text: factor

text: of

text: F,

text: there

text: is

text: natuPrLal

text: inclusion

text: map

text: F.

text: It

text: is

text: straightforward

text: to

text: check

text: that

text: this

text: inclusion

text: is

text: and

text: and

text: therefore

text: admits

text: a

text: coarse

text: inverse

text: which

text: we

text: denote

text: D

text: F

text: PL.

text: We

text: also

text: have

text: the

text: coarse

text: projection

text: π

text: X

text: PL

text: deﬁned

text: by

text: sending

text: G

text: X

text: to

text: the

text: set

text: of

text: embedded

text: closed

text: loops

text: on

text: PL

text: G;

text: this

text: projection

text: coarsely

text: agrees

text: with

text: the

text: composition

text: D

text: π

text: F

text: The

text: primitive

text: loop

text: graph

text: measures,

text: in

text: a

text: sense,

text: how

text: algebraically

text: complicated

text: primitive

text: conjugacy

text: classes

text: are

text: with

text: respect

text: to

text: each

text: other.

text: The

text: co-surface

text: graph,

text: on

text: the

text: other

text: hand,

text: is

text: designed

text: to

text: measure

text: how

text: topolog-

text: ically

text: complicated

text: primitive

text: conjugacy

text: classes

text: are

text: with

text: respect

text: to

text: each

text: other:

text: Deﬁnition

text: (Co-surface

text: graph

text: CS).

text: The

text: co-surface

text: graph

text: CS

text: of

text: the

text: free

text: group

text: F

text: is

text: the

text: simplicial

text: graph

text: whose

text: vertices

text: are

text: conjugacy

text: classes

text: of

text: primitive

text: elements,

text: and

text: where

text: two

text: vertices

text: α

text: and

text: β

text: are

text: joined

text: by

text: an

text: edge

text: if

text: there

text: is

text: a

text: once-punctured

text: surface

text: S

text: and

text: an

text: isomorphism

text: F

text: with

text: respect

text: to

text: which

text: α

text: and

text: β

text: may

text: both

text: be

text: represented

text: by

text: simple

text: closed

text: curves

text: on

text: S.

text: π

text: CS

text: π

text: el

text: PL

text: X

text: PL

text: CS

text: D

text: π

text: F

text: el

text: F

text: Figure

text: The

text: co-surface

text: graph

text: and

text: its

text: friends.

text: All

text: rightward

text: pointing

text: arrows

text: are

text: coarsely

text: Lipschitz,

text: and

text: D

text: F

text: is

text: a

text: quasi-isometry.

text: Coarsely

text: speaking,

text: all

text: maps

text: are

text: Out(

text: )–equivariant

text: and

text: the

text: diagram

text: commutes.

text: In

text: other

text: words,

text: each

text: once-punctured

text: surface

text: S

text: with

text: F

text: determines

text: a

text: subset

text: CS

text: consisting

text: of

text: those

text: primitive

text: conjugacy

text: classes

text: that

text: correspond

text: to

text: (nonseparating)

text: simple

text: closed

text: curves

text: on

text: S.

text: As

text: our

text: graph

text: CS

text: is

text: obtained

text: by

text: collapsing

text: each

text: C

text: PL

text: to

text: a

text: set

text: of

text: diameter

text: it

text: records

text: the

text: geometry

text: of

text: the

text: S

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∈ F → | | Let a be such that a is optimally stretch∈ed by f : G1 G2, i.e. (cid:96)(a G2) = Lip(f)(cid:96)(a G1), and let k > 0 be the smallest positive integer such that ak H. Then using our observation above | | | edX(G1,G2) = (cid:96)(a|G2) = (cid:96)(ak|G2) ≤ sup (cid:96)(h|G2) = edX(H)(H|G1,H|G2). (cid:96)(a G1) (cid:96)(ak G1) h∈H (cid:96)(h G1) ≤ ∗ ∗ ∗ → Hence, we also hav∗e dX(G1,G2) dX(H)(i (G1),i (G2)) show{ing}that i : X X(H) is an isometry. To show that i maps folding paths to folding paths, let G ∈ be a folding path in X with cor- { → } t t [0,L] responding folding maps φ : G G (see § for the deﬁnitions of folding maps and folding paths). Set G˜t = H|Gt = i∗(Gt) witsht covsering tm{sa<pt pt :| G˜t →→ G|t. W} e claim that {G˜t} is a folding path in X(H) with corresponding lifted folding maps φ˜ : H G H G . Indeed, that each of these lifts is a local st s t s<t Lip(φ˜st)–homo◦thety with◦Lip(φ˜st) = dX(H)(G˜s,G˜t) follows from what we have already shown. What’s more, the equality pt φ˜st = φst ps implies that the illegal turn structure that φ˜st induces on G˜s is exactly the lift (via p ) of the illegal turn structure that φ induces on G . From this it follows that each φ˜ is a folding map and, s st { s } st → by the uniqueness of folding paths [BF2, §2], that G˜ is the folding path determined by φ˜ : G˜ G˜ . t 0L 0 L 5.2 Proof of Proposition 5.2 ∈ Combining our work in the previous sections, we now turn to the proof of Proposition 5.2. Let R X be ∈ ∈ a rose with a petal representing α CS0. Let α˜ CS0(H) be primitive conjugacy class of H represented by | ∈ ∈ | an embedded loop of H R covering this petal of R. Note that α πCS(R) and α˜ πCS(H R), where πCS is the projection from outer space to the corresponding co-surface graph. ⊂ Fix a ﬁnite generating set S˜ of ΓH with projection S Γ . By abuse of notation, we identify ΓH and Γ H H with the corres(cid:55)→pond·ing Cayle(cid:55)→y gr·aph|s Cay(ΓH,S) and Cay(ΓH,S¯), which are geodesic metri→c spaces. The orbits maps g˜ g˜ α˜ and g˜ g˜ H R then extend equivariantly to continuous maps ψ : ΓH CS(H) and → → Ψ: ΓH X(H), and similarly for ΓH. Note also that the projection ΓH ΓH is a qua→si-isometry. Proof of Proposition 5.2. By Lemma 3.4, it sufﬁces to show that the orbit map ψ : ΓH CS(H) is alignment preserving and metrically proper. The strategy is to relate the corresponding orbit in X with folding paths via Theorem 2.6 and to use that these folding paths lift to X(H) by Proposition 5.5. Note that since Γ qi-embeds into CS, Γ is hyperbolic. Further, as Γ is ﬁnite index in Γ, Γ , and hence ΓH, is also hyperbolic. H H → In details, let (g˜−,g˜0,g˜+) be an ali→gned tr≤iple in ΓH and let γ˜: I ΓH be a geodesic passing through g˜0 with γ˜(I±) = g˜±. The projection γ : I ΓH Γ is then a uniform quasigeodesic a{nd s}o maps to a uniform quasigeodesic in CS by assumption. Hence Theorem 2.6 provides a folding path G ∈ in X that has t t→[0,L] → uniformly bounded symmetric Hausdorff distance from the image of the composition I Γ X. Since ∗ → H i : X X(H) is a ΓH–equivariant isometric embedding (Proposition 5.5), the same holds for the image of → → ∗ | · | · | · I | ΓH X(H) and the folding path i (Gt) = H Gt. In particular, the three points g˜−{H|R,}g˜0 H R, and g˜+ H R all lie within uniforml→y bounded symmetric Hausdorff distance of the geodesic H Gt· in X(·H). Si·nce the projection πCS : X(H) CS(H) is alignment preservin→g, it follows that the triple (g˜− α˜ ,g˜0 α˜ ,g˜+ α˜ ) is uniformly aligned in CS(H). Therefore the orbit map ΓH CS(H) is alignment preserving. → It remains to show that ΓH CS(H) is metrically proper. If this were not the case, then there is a ∈ | | → · ≤ ≥ ∪ sequence of xi ΓH with xi ∞ but dCS(H)(α˜ ,xi α→˜ ) ∈D for some D 0. By compactness of ΓH ∂ΓH, after pas→sing to a subsequence we may assume that xi q ∂ΓH = ∂ΓH = ∂Γ. Since we have already shown that ΓH→ CS(H)∪is coarsely alignment p·reserving, use Claim 4.11 to obtain a geodesic (g˜i)i≥0 in ΓH such that g˜i q in ΓH ∂ΓH but dCS(H)(α˜ ,g˜i α·˜ ) is uniformly bounded. · Now pass to a subsequence such that gi R converges to T in X. Since (gi α)i≥0 is a quasige·odesic in CS, we see combining Theorem 2.9 and Theorem 4.8 that the tree T is free and arational and that g α converges ∪ | · · | i to the equivalence class of T in CS ∂CS. Hence, (H gi R = g˜i H R)i≥0 conve·rge|s in X(H·) to the tree TH. By Corollary 5.4, TH is free and arat·iona→l. Hence, by Lemma 4.7, πCS(H)(g˜i H R) = g˜i α˜ is unbounded in CS(H). In p→articular, dCS(H)(α˜ ,g˜i α˜ ) ∞ contradicting the conclusion of the previous paragraph. This shows that ΓH CS(H) is metrically proper and completes the proof. 18 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: F

text: Let

text: a

text: be

text: such

text: that

text: a

text: is

text: optimally

text: stretch∈ed

text: by

text: f

text: i.e.

text: and

text: let

text: k

text: be

text: the

text: smallest

text: positive

text: integer

text: such

text: that

text: ak

text: H.

text: Then

text: using

text: our

text: observation

text: above

text: sup

text: h∈H

text: Hence,

text: we

text: also

text: hav∗e

text: dX(H)(i

text: show{ing}that

text: i

text: X

text: X(H)

text: is

text: an

text: isometry.

text: To

text: show

text: that

text: i

text: maps

text: folding

text: paths

text: to

text: folding

text: paths,

text: let

text: G

text: be

text: a

text: folding

text: path

text: in

text: X

text: with

text: cor-

text: t

text: t

text: responding

text: folding

text: maps

text: φ

text: G

text: G

text: (see

text: for

text: the

text: deﬁnitions

text: of

text: folding

text: maps

text: and

text: folding

text: paths).

text: Set

text: G˜t

text: H|Gt

text: i∗(Gt)

text: witsht

text: covsering

text: tm{sa<pt

text: pt

text: G˜t

text: G|t.

text: W}

text: e

text: claim

text: that

text: {G˜t}

text: is

text: a

text: folding

text: path

text: in

text: X(H)

text: with

text: corresponding

text: lifted

text: folding

text: maps

text: φ˜

text: H

text: G

text: H

text: G

text: Indeed,

text: that

text: each

text: of

text: these

text: lifts

text: is

text: a

text: local

text: st

text: s

text: t

text: s<t

text: Lip(φ˜st)–homo◦thety

text: with◦Lip(φ˜st)

text: dX(H)(G˜s,G˜t)

text: follows

text: from

text: what

text: we

text: have

text: already

text: shown.

text: What’s

text: more,

text: the

text: equality

text: pt

text: φ˜st

text: φst

text: ps

text: implies

text: that

text: the

text: illegal

text: turn

text: structure

text: that

text: φ˜st

text: induces

text: on

text: G˜s

text: is

text: exactly

text: the

text: lift

text: (via

text: p

text: of

text: the

text: illegal

text: turn

text: structure

text: that

text: φ

text: induces

text: on

text: G

text: From

text: this

text: it

text: follows

text: that

text: each

text: φ˜

text: is

text: a

text: folding

text: map

text: and,

text: s

text: st

text: s

text: st

text: by

text: the

text: uniqueness

text: of

text: folding

text: paths

text: that

text: G˜

text: is

text: the

text: folding

text: path

text: determined

text: by

text: φ˜

text: G˜

text: G˜

text: t

text: L

text: Proof

text: of

text: Proposition

text: Combining

text: our

text: work

text: in

text: the

text: previous

text: sections,

text: we

text: now

text: turn

text: to

text: the

text: proof

text: of

text: Proposition

text: Let

text: R

text: X

text: be

text: a

text: rose

text: with

text: a

text: petal

text: representing

text: α

text: Let

text: α˜

text: be

text: primitive

text: conjugacy

text: class

text: of

text: H

text: represented

text: by

text: an

text: embedded

text: loop

text: of

text: H

text: R

text: covering

text: this

text: petal

text: of

text: R.

text: Note

text: that

text: α

text: πCS(R)

text: and

text: α˜

text: πCS(H

text: R),

text: where

text: πCS

text: is

text: the

text: projection

text: from

text: outer

text: space

text: to

text: the

text: corresponding

text: co-surface

text: graph.

text: Fix

text: a

text: ﬁnite

text: generating

text: set

text: S˜

text: of

text: ΓH

text: with

text: projection

text: S

text: Γ

text: By

text: abuse

text: of

text: notation,

text: we

text: identify

text: ΓH

text: and

text: Γ

text: H

text: H

text: with

text: the

text: gr·aph|s

text: Cay(ΓH,S)

text: and

text: Cay(ΓH,S¯),

text: which

text: are

text: geodesic

text: metri→c

text: spaces.

text: The

text: orbits

text: maps

text: g˜

text: g˜

text: α˜

text: and

text: g˜

text: g˜

text: H

text: R

text: then

text: extend

text: equivariantly

text: to

text: continuous

text: maps

text: ψ

text: ΓH

text: CS(H)

text: and

text: Ψ:

text: ΓH

text: X(H),

text: and

text: similarly

text: for

text: ΓH.

text: Note

text: also

text: that

text: the

text: projection

text: ΓH

text: ΓH

text: is

text: a

text: qua→si-isometry.

text: Proof

text: of

text: Proposition

text: By

text: Lemma

text: it

text: sufﬁces

text: to

text: show

text: that

text: the

text: orbit

text: map

text: ψ

text: ΓH

text: CS(H)

text: is

text: alignment

text: preserving

text: and

text: metrically

text: proper.

text: The

text: strategy

text: is

text: to

text: relate

text: the

text: corresponding

text: orbit

text: in

text: X

text: with

text: folding

text: paths

text: via

text: Theorem

text: and

text: to

text: use

text: that

text: these

text: folding

text: paths

text: lift

text: to

text: X(H)

text: by

text: Proposition

text: Note

text: that

text: since

text: Γ

text: qi-embeds

text: into

text: CS,

text: Γ

text: is

text: hyperbolic.

text: Further,

text: as

text: Γ

text: is

text: ﬁnite

text: index

text: in

text: Γ,

text: Γ

text: and

text: hence

text: ΓH,

text: is

text: also

text: hyperbolic.

text: H

text: H

text: In

text: details,

text: let

text: be

text: an

text: ali→gned

text: tr≤iple

text: in

text: ΓH

text: and

text: let

text: γ˜:

text: I

text: ΓH

text: be

text: a

text: geodesic

text: passing

text: through

text: with

text: γ˜(I±)

text: g˜±.

text: The

text: projection

text: γ

text: I

text: ΓH

text: Γ

text: is

text: then

text: a

text: uniform

text: quasigeodesic

text: a{nd

text: s}o

text: maps

text: to

text: a

text: uniform

text: quasigeodesic

text: in

text: CS

text: by

text: assumption.

text: Hence

text: Theorem

text: provides

text: a

text: folding

text: path

text: G

text: in

text: X

text: that

text: has

text: t

text: uniformly

text: bounded

text: symmetric

text: Hausdorff

text: distance

text: from

text: the

text: image

text: of

text: the

text: composition

text: I

text: Γ

text: X.

text: Since

text: H

text: i

text: X

text: X(H)

text: is

text: a

text: ΓH–equivariant

text: isometric

text: embedding

text: (Proposition

text: the

text: same

text: holds

text: for

text: the

text: image

text: of

text: I

text: ΓH

text: X(H)

text: and

text: the

text: folding

text: path

text: i

text: (Gt)

text: H

text: Gt.

text: In

text: particular,

text: the

text: three

text: points

text: H

text: R,

text: and

text: g˜+

text: H

text: R

text: all

text: lie

text: within

text: uniforml→y

text: bounded

text: symmetric

text: Hausdorff

text: distance

text: of

text: the

text: geodesic

text: H

text: Gt·

text: in

text: X(·H).

text: Si·nce

text: the

text: projection

text: πCS

text: X(H)

text: CS(H)

text: is

text: alignment

text: preservin→g,

text: it

text: follows

text: that

text: the

text: triple

text: (g˜−

text: α˜

text: α˜

text: ,g˜+

text: α˜

text: is

text: uniformly

text: aligned

text: in

text: CS(H).

text: Therefore

text: the

text: orbit

text: map

text: ΓH

text: CS(H)

text: is

text: alignment

text: preserving.

text: It

text: remains

text: to

text: show

text: that

text: ΓH

text: CS(H)

text: is

text: metrically

text: proper.

text: If

text: this

text: were

text: not

text: the

text: case,

text: then

text: there

text: is

text: a

text: sequence

text: of

text: xi

text: ΓH

text: with

text: xi

text: but

text: dCS(H)(α˜

text: ,xi

text: α→˜

text: ∈D

text: for

text: some

text: D

text: By

text: compactness

text: of

text: ΓH

text: ∂ΓH,

text: after

text: pas→sing

text: to

text: a

text: subsequence

text: we

text: may

text: assume

text: that

text: xi

text: q

text: ∂ΓH

text: ∂ΓH

text: ∂Γ.

text: Since

text: we

text: have

text: already

text: shown

text: that

text: ΓH→

text: CS(H)∪is

text: coarsely

text: alignment

text: p·reserving,

text: use

text: Claim

text: to

text: obtain

text: a

text: geodesic

text: in

text: ΓH

text: such

text: that

text: g˜i

text: q

text: in

text: ΓH

text: ∂ΓH

text: but

text: dCS(H)(α˜

text: ,g˜i

text: α·˜

text: is

text: uniformly

text: bounded.

text: Now

text: pass

text: to

text: a

text: subsequence

text: such

text: that

text: gi

text: R

text: converges

text: to

text: T

text: in

text: X.

text: Since

text: (gi

text: is

text: a

text: quasige·odesic

text: in

text: CS,

text: we

text: see

text: combining

text: Theorem

text: and

text: Theorem

text: that

text: the

text: tree

text: T

text: is

text: free

text: and

text: arational

text: and

text: that

text: g

text: α

text: converges

text: i

text: to

text: the

text: equivalence

text: class

text: of

text: T

text: in

text: CS

text: ∂CS.

text: Hence,

text: (H

text: gi

text: R

text: g˜i

text: H

text: conve·rge|s

text: in

text: X(H·)

text: to

text: the

text: tree

text: TH.

text: By

text: Corollary

text: TH

text: is

text: free

text: and

text: arat·iona→l.

text: Hence,

text: by

text: Lemma

text: πCS(H)(g˜i

text: H

text: R)

text: g˜i

text: α˜

text: is

text: unbounded

text: in

text: CS(H).

text: In

text: p→articular,

text: dCS(H)(α˜

text: ,g˜i

text: α˜

text: contradicting

text: the

text: conclusion

text: of

text: the

text: previous

text: paragraph.

text: This

text: shows

text: that

text: ΓH

text: CS(H)

text: is

text: metrically

text: proper

text: and

text: completes

text: the

text: proof.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ∈ Ch∈oose h Γ a(cid:48)rbitrar(cid:48)ily, a∈nd let g = g0,...,gm = h be a geodesic from g to h in Γ. Let us write →σi := σ(gi) Tgi and σi := σ (gi) Tgi for the value of the two K¯≤–qi-≤sections in the ﬁber Tgi of p: E B over gi. (cid:48)Ch≥oose any vertices gi,gj along ou−r geodesic with i j i + 2N, and suppose temporarily that dbtehdgaeisg(rieσesipiX,ˆsaσstohi|of)m[Fˆz|ei.,WxDz(cid:48)i∈]0i.tihXnˆRrTseegosciptaihesllaclttfaritbbooem=ltehdσi§sib−tbhy1a(cid:48)aσstthii(cid:48)sxTe,g∈ct(cid:12)(cid:12)ihy=Feˆc−leiipslceacml1yl(cid:48)(cye(cid:12)(cid:12)glniritc)eaidcilusl=ycaerσdesiid−mwu1ocpσerlidi(cid:48)dc∈.iiabSFlˆeottmfrzeFˆaiey=wwniσhothiotasbtnheydeeczpdy(cid:48)irgc=oelpiscσeaari(cid:48)ltrxliey,esslraoetbhdtehaulatcet(cid:107)edtdihbb.e(cid:107)yHXgˆtoe=hwoede|efivbrsee|iXrceˆ, and that ib Xˆ differs fro∈m dgi(Fσi,σi ) =∈σi 1,Fσi Xˆ by at most 1. Choosing a lift g˜i Aut( ) of gi Out( ), the action g˜i on E restricts to a simplicial automorphism fWTrhoiemthrerTfeo1srpet(cid:48)oe, c⊂sTtingticoetthhtahitseriledasbepneeltceiﬁtdsceatdhtigeoenep,datghtehe [lezalibe,emz(cid:48)il]ieninsgtaga˜fni(udbn)dth∈aumsFegnaitcvatelssdooanmnTaigdiinetnhotefiﬁtshcaeamtaiexoinwsaooyff itTbh1ain=t iCbCaaaycy(tF(sˆFˆ,oX,nˆXˆ)C,)iawtyfi(otFhˆll,oTXwˆg)is.. that [zi,zi] Tgi is a fundamental domain of the axis for g˜i(b) acting on Tgi. Letting β and α denote t·he conjugacy c(cid:48)la∈sses of b and a, respectively, it follows that g˜i(β) = gi(β) is 1–alm·ost contained in α at ≤gi R (since σi,σi Agi(a) by construction and all edges(cid:0)in the universa(cid:1)l cover of gi R have length 1/rk(F) 1). Since | · | (cid:107) (cid:107) ≥ (cid:48) − ≥ (cid:48) ≥ (cid:96)(gi(β) gi R) = (cid:96)(β R) = rk(1F) ib Xˆ rk(1F) dgi(σi,σi ) 1 2rk1(F)dgi(σi,σi ) L0 (cid:48) ≥ by the assumption dgi(σi,σi ) D0, we may apply Lemma 6.10 to conclude | · ≥ − | · ≥ − (cid:48) (cid:96)(gi(β) gj R) Cλ j i(cid:96)(gi(β) gi R) 2rkC(F)λ j idgi(σi,σi ). ≤ − ∈ ∈ For each ≤i < p≤ j, set sp = gp−11gp S and le·t··tp S˜ be(cid:48) the c(cid:48) hose·n··lift of sp in the generating set W of EΓ. For i··· p j let us also deﬁne zp = ziti+1 tp and zp = ziti+1 tp, both of which(cid:48) are p(cid:48)oints over gp = gisi+1 sp. Since gi,g(cid:0)i+1,...,gj (cid:1)is a(cid:16)g(cid:48)eo(cid:48)desic i−n(cid:17)Γ≤, it follows that zi,...,zj and zi,...,zj are both geodesics in EΓ and thus that dE (zi,zj) = dE (zi,zj) = j i 2N. Observe now that z−j 1z(cid:48)j = (ziti+1 ···tj)−1 z(cid:48)iti+1 ···tj = t−j 1 ···ti−+11 z−i 1z(cid:48)i (ti+1 ···tj) = ϕibϕ−1 = iϕ(b) ∈ Fˆ, where ϕ is the speciﬁc li(cid:48)ft ϕ = t−j 1 ···ti−+11 ∈ Aut(F) of g−j 1gi = s−j 1 ···s−i+11. In particular, we see that the distance between zj(cid:12)(cid:12)and (cid:12)(cid:12)zj in t(cid:13)(cid:13)he ﬁb(cid:13)(cid:13)er Tgj satisﬁes dgj(zj,z(cid:48)j) = iϕ(b) Xˆ ≥ iϕ(b) Xˆ = rk(F)(cid:96)(ϕ(β)|R) = rk(F)(cid:96)(gi(β)|gj ·R) ≥ C2 λ j−idgi(σi,σi(cid:48)). (10) (cid:48) (cid:48) Let us(cid:48)now(cid:48) compare this distance−to dgj(σ≤j,σj). Since σ,σ are K¯–qi-sections, the quantitie(cid:48)s d(cid:48)E (≤σi,σj) and dE (σi ,σj) are bounded by K¯(j i) + K¯ 2K¯N + K¯. Since we also have dE (σi,zi),dE (σi ,zi) 1 by construction, the triangle inequality thus gives ≤ dE (zj,σj) dE (zj,zi)+dE (zi,σi)+dE (σi,σj) ≤ ≤ 2N +1+2K¯N +K¯ 6K¯N, (cid:48) (cid:48) ≤ → and similarly, dE (zj,σj) 6K¯N. By metric properness of the graph bundle E B, it follows that (cid:48) (cid:48) ≤ (cid:48) ≥ ≥ − dgj−(z≥j,σj),dgj(zj,σj) f(6K¯N). Combining with (10) and using dgi(σi,σi ) D0 8C 1 f(6K¯N) and λ j i 1, we conclude that (cid:48) ≥ (cid:48) − − (cid:48) (cid:48) dgj(σj,σj) d(cid:0)gj(zj,zj)(cid:1) dgj(zj,σj) dgj(zj,σj) ≥ − (cid:48) − C2 λ j idgi(σi,σi ) 2f(6K¯N) ≥ − − (cid:48) ≥ − (cid:48) λ j i 12 C2 dgi(σi,σi ) C4 λ j idgi(σi,σi ). 27 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Ch∈oose

text: h

text: Γ

font: LEBFRQ+StandardSymL

text: a∈nd

text: let

text: g

text: h

text: be

text: a

text: geodesic

text: from

text: g

text: to

text: h

text: in

text: Γ.

text: Let

text: us

text: write

text: →σi

text: σ(gi)

text: Tgi

text: and

text: σi

text: σ

text: (gi)

text: Tgi

text: for

text: the

text: value

text: of

text: the

text: two

text: K¯≤–qi-≤sections

text: in

text: the

text: ﬁber

text: Tgi

text: of

text: p:

text: E

text: B

text: over

text: gi.

text: any

text: vertices

text: gi,gj

text: along

text: ou−r

text: geodesic

text: with

text: i

text: j

text: i

text: and

text: suppose

text: temporarily

text: that

text: and

text: that

text: ib

text: Xˆ

text: differs

text: fro∈m

text: dgi(Fσi,σi

text: =∈σi

text: Xˆ

text: by

text: at

text: most

text: Choosing

text: a

text: lift

text: g˜i

text: Aut(

text: of

text: gi

text: Out(

text: the

text: action

text: g˜i

text: on

text: E

text: restricts

text: to

text: a

text: simplicial

text: automorphism

text: that

text: [zi,zi]

text: Tgi

text: is

text: a

text: fundamental

text: domain

text: of

text: the

text: axis

text: for

text: g˜i(b)

text: acting

text: on

text: Tgi.

text: Letting

text: β

text: and

text: α

text: denote

text: t·he

text: conjugacy

text: of

text: b

text: and

text: a,

text: respectively,

text: it

text: follows

text: that

text: g˜i(β)

text: gi(β)

text: is

text: contained

text: in

text: α

text: at

text: ≤gi

text: R

text: (since

text: σi,σi

text: Agi(a)

text: by

text: construction

text: and

text: all

text: the

text: cover

text: of

text: gi

text: R

text: have

text: length

text: Since

text: gi

text: R)

text: R)

text: ib

text: Xˆ

text: dgi(σi,σi

text: by

text: the

text: assumption

text: dgi(σi,σi

text: we

text: may

text: apply

text: Lemma

text: to

text: conclude

text: gj

text: R)

text: Cλ

text: j

text: gi

text: R)

text: j

text: idgi(σi,σi

text: For

text: each

text: ≤i

text: p≤

text: j,

text: set

text: sp

text: S

text: and

text: le·t··tp

text: S˜

text: the

text: hose·n··lift

text: of

text: sp

text: in

text: the

text: generating

text: set

text: W

text: of

text: EΓ.

text: For

text: i···

text: p

text: j

text: let

text: us

text: also

text: deﬁne

text: zp

text: tp

text: and

text: zp

text: tp,

text: both

text: of

text: are

text: over

text: gp

text: sp.

text: Since

text: it

text: follows

text: that

text: zi,...,zj

text: and

text: zi,...,zj

text: are

text: both

text: geodesics

text: in

text: EΓ

text: and

text: thus

text: that

text: dE

text: (zi,zj)

text: dE

text: (zi,zj)

text: j

text: i

text: Observe

text: now

text: that

text: z−j

text: ···tj)−1

text: ···tj

text: t−j

text: z−i

text: ···tj)

text: ϕibϕ−1

text: iϕ(b)

text: Fˆ,

text: where

text: ϕ

text: is

text: the

text: speciﬁc

text: ϕ

text: t−j

text: Aut(F)

text: of

text: g−j

text: s−j

text: In

text: particular,

text: we

text: see

text: that

text: the

text: distance

text: between

text: in

text: Tgj

text: satisﬁes

text: iϕ(b)

text: Xˆ

text: iϕ(b)

text: Xˆ

text: ·R)

text: λ

text: Let

text: compare

text: this

text: distance−to

text: dgj(σ≤j,σj).

text: Since

text: σ,σ

text: are

text: K¯–qi-sections,

text: the

text: (≤σi,σj)

text: and

text: dE

text: (σi

text: ,σj)

text: are

text: bounded

text: by

text: K¯(j

text: i)

text: K¯

text: K¯.

text: Since

text: we

text: also

text: have

text: dE

text: (σi,zi),dE

text: (σi

text: ,zi)

text: by

text: construction,

text: the

text: triangle

text: inequality

text: thus

text: gives

text: dE

text: (zj,σj)

text: dE

text: (zj,zi)+dE

text: (zi,σi)+dE

text: (σi,σj)

text: +K¯

text: and

text: similarly,

text: dE

text: (zj,σj)

text: By

text: metric

text: properness

text: of

text: the

text: graph

text: bundle

text: E

text: B,

text: it

text: follows

text: that

text: dgj−(z≥j,σj),dgj(zj,σj)

text: Combining

text: with

text: and

text: using

text: dgi(σi,σi

text: and

text: λ

text: j

text: i

text: we

text: conclude

text: that

text: dgj(σj,σj)

text: dgj(zj,σj)

text: dgj(zj,σj)

text: λ

text: j

text: idgi(σi,σi

text: λ

text: j

text: i

text: dgi(σi,σi

text: λ

text: j

text: idgi(σi,σi

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [KL1] Ilya Kapovich and Martin Lustig. Geometric intersection number and analogues of the curve com- plex for free groups. Geom. Topol., 13(3):1805–1833, 2009. [KL2] Ilya Kapovich and Martin Lustig. Intersection form, laminations and currents on free groups. Geom. Funct. Anal., 19(5):1426–1467, 2010. [KL3] Richard P. Kent, IV and Christopher J. Leininger. Shadows of mapping class groups: capturing convex cocompactness. Geom. Funct. Anal., 18(4):1270–1325, 2008. [KL4] Richard P. Kent, IV and Christopher J. Leininger. A geometric criterion to be pseudo-Anosov. Michigan Math. J., 63(2):227–251, 2014. [Kob] Tsuyoshi Kobayashi. Heights of simple loops and pseudo-anosov homeomorphisms. In Braids, volume 78, pages 327–338. Amer. Math. Soc., Providence, RI, Contemp. Math., 1988. [KR] Ilya Kapovich and Kasra Raﬁ. On the hyperbolicity of the free splitting and free factor complexes. Groups Geom. Dyn., 8(2):391–414, 2014. [Man] Brian Mann. Some hyperbolic Out(F )-graphs and nonunique ergodicity of very small F -trees. N N ProQuest LLC, Ann Arbor, MI, 2014. Thesis (Ph.D.)–The University of Utah. [Mit1] Mahan Mitra. Ending laminations for hyperbolic group extensions. Geom. Funct. Anal., 7(2):379– 402, 1997. [Mit2] Mahan Mitra. Cannon-Thurston maps for hyperbolic group extensions. Topology, 37(3):527–538, 1998. [Mit3] Mahan Mitra. On a theorem of Scott and Swarup. Proc. Amer. Math. Soc, 127(6):1625–1631, 1999. [MM] Howard A. Masur and Yair N. Minsky. Geometry of the complex of curves. I. Hyperbolicity. Invent. Math., 138(1):103–149, 1999. [Mos] Lee Mosher. Hyperbolic extensions of groups. J. Pure Appl. Algebra, 110(3):305–314, 1996. [MR1] Brian Mann and Patrick Reynolds. In preparation. [MR2] Mahan Mj and Kasra Raﬁ. Algebraic ending laminations and quasiconvexity. Preprint arXiv:1506.08036v2, 2015. [MS] Mahan Mj and Pranab Sardar. A combination theorem for metric bundles. Geom. Funct. Anal., 22(6):1636–1707, 2012. [Rey1] Patrick Reynolds. On indecomposable trees in the boundary of outer space. Geometriae Dedicata, 153(1):59–71, 2011. [Rey2] Patrick Reynolds. Reducing systems for very small trees. Preprint arXiv:1211.3378, 2012. [SS] Peter G. Scott and Gadde A. Swarup. Geometric ﬁniteness of certain Kleinian groups. Proc. Amer. Math. Soc, 109(3):765–768, 1990. Department of Mathematics Department of Mathematics Vanderbilt University Yale University 1326 Stevenson Center 10 Hillhouse Ave Nashville, TN 37240 , U.S.A New Haven, CT 06520, U.S.A E-mail: spencer.dowdall@vanderbilt.edu E-mail: s.taylor@yale.edu 33

text: Ilya

text: Kapovich

text: and

text: Martin

text: Lustig.

text: Geometric

text: intersection

text: number

text: and

text: analogues

text: of

text: the

text: curve

text: com-

text: plex

text: for

text: free

text: groups.

text: Geom.

text: Topol.,

text: Ilya

text: Kapovich

text: and

text: Martin

text: Lustig.

text: Intersection

text: form,

text: laminations

text: and

text: currents

text: on

text: free

text: groups.

text: Geom.

text: Funct.

text: Anal.,

text: Richard

text: P.

text: Kent,

text: IV

text: and

text: Christopher

text: J.

text: Leininger.

text: Shadows

text: of

text: mapping

text: class

text: groups:

text: capturing

text: convex

text: cocompactness.

text: Geom.

text: Funct.

text: Anal.,

text: Richard

text: P.

text: Kent,

text: IV

text: and

text: Christopher

text: J.

text: Leininger.

text: A

text: geometric

text: criterion

text: to

text: be

text: pseudo-Anosov.

text: Michigan

text: Math.

text: J.,

text: [Kob]

text: Tsuyoshi

text: Kobayashi.

text: Heights

text: of

text: simple

text: loops

text: and

text: pseudo-anosov

text: homeomorphisms.

text: In

text: Braids,

text: volume

text: pages

text: Amer.

text: Math.

text: Soc.,

text: Providence,

text: RI,

text: Contemp.

text: Math.,

text: [KR]

text: Ilya

text: Kapovich

text: and

text: Kasra

text: Raﬁ.

text: On

text: the

text: hyperbolicity

text: of

text: the

text: free

text: splitting

text: and

text: free

text: factor

text: complexes.

text: Groups

text: Geom.

text: Dyn.,

text: [Man]

text: Brian

text: Mann.

text: Some

text: hyperbolic

text: Out(F

text: )-graphs

text: and

text: nonunique

text: ergodicity

text: of

text: very

text: small

text: F

text: -trees.

text: N

text: N

text: ProQuest

text: LLC,

text: Ann

text: Arbor,

text: MI,

text: Thesis

text: (Ph.D.)–The

text: University

text: of

text: Utah.

text: Mahan

text: Mitra.

text: Ending

text: laminations

text: for

text: hyperbolic

text: group

text: extensions.

text: Geom.

text: Funct.

text: Anal.,

text: Mahan

text: Mitra.

text: Cannon-Thurston

text: maps

text: for

text: hyperbolic

text: group

text: extensions.

text: Topology,

text: Mahan

text: Mitra.

text: On

text: a

text: theorem

text: of

text: Scott

text: and

text: Swarup.

text: Proc.

text: Amer.

text: Math.

text: Soc,

text: [MM]

text: Howard

text: A.

text: Masur

text: and

text: Yair

text: N.

text: Minsky.

text: Geometry

text: of

text: the

text: complex

text: of

text: curves.

text: I.

text: Hyperbolicity.

text: Invent.

text: Math.,

text: [Mos]

text: Lee

text: Mosher.

text: Hyperbolic

text: extensions

text: of

text: groups.

text: J.

text: Pure

text: Appl.

text: Algebra,

text: Brian

text: Mann

text: and

text: Patrick

text: Reynolds.

text: In

text: preparation.

text: Mahan

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text: bundles.

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text: Patrick

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text: indecomposable

text: trees

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text: outer

text: space.

text: Geometriae

text: Dedicata,

text: Patrick

text: Reynolds.

text: Reducing

text: systems

text: for

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text: trees.

text: Preprint

text: [SS]

text: Peter

text: G.

text: Scott

text: and

text: Gadde

text: A.

text: Swarup.

text: Geometric

text: ﬁniteness

text: of

text: certain

text: Kleinian

text: groups.

text: Proc.

text: Amer.

text: Math.

text: Soc,

text: Department

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text: Mathematics

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text: TN

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text: Haven,

text: CT

text: U.S.A

text: E-mail:

text: spencer.dowdall@vanderbilt.edu

text: E-mail:

text: s.taylor@yale.edu


text: Draft version May 28, 2020 Preprint typeset using LATEX style emulateapj v. 01/23/15 ASSESSMENT OF SYSTEMATIC CHROMATIC ERRORS THAT IMPACT SUB–1% PHOTOMETRIC PRECISION IN LARGE-AREA SKY SURVEYS T. S. Li1, D. L. DePoy1, J. L. Marshall1, D. Tucker2, R. Kessler3, J. Annis2, G. M. Bernstein4, S. Boada1, D. L. Burke5,6, D. A. Finley2, D. J. James7, S. Kent2, H. Lin2, J. Marriner2, N. Mondrik1, D. Nagasawa1, E. S. Rykoff6,5, D. Scolnic3, A. R. Walker7, W. Wester2, T. M. C. Abbott7, S. Allam2, A. Benoit-Le´vy8,9,10, E. Bertin8,10, D. Brooks9, D. Capozzi11, A. Carnero Rosell12,13, M. Carrasco Kind14,15, J. Carretero16,17, M. Crocce16, C. E. Cunha6, C. B. D’Andrea11,18, L. N. da Costa12,13, S. Desai19,20, H. T. Diehl2, P. Doel9, B. Flaugher2, P. Fosalba16, J. Frieman2,3, E. Gaztanaga16, D. A. Goldstein21,22, D. Gruen6,23,5,24, R. A. Gruendl14,15, G. Gutierrez2, K. Honscheid25,26, K. Kuehn27, N. Kuropatkin2, M. A. G. Maia12,13, P. Melchior28, C. J. Miller29,30, R. Miquel31,17, J. J. Mohr19,20,23, E. Neilsen2, R. C. Nichol11, B. Nord2, R. Ogando12,13, A. A. Plazas32, A. K. Romer33, A. Roodman6,5, M. Sako4, E. Sanchez34, V. Scarpine2, M. Schubnell30, I. Sevilla-Noarbe34,14, R. C. Smith7, M. Soares-Santos2, F. Sobreira2,12, E. Suchyta4, G. Tarle30, D. Thomas11, V. Vikram35 (The DES Collaboration) Draft version May 28, 2020 ABSTRACT Meeting the science goals for many current and future ground-based optical large-area sky surveys requires that the calibrated broadband photometry is both stable in time and uniform over the sky to 1% precision or better. Past and current surveys have achieved photometric precision of 1–2% by calibrating the survey’s stellar photometry with repeated measurements of a large number of stars observed in multiple epochs. The calibration techniques employed by these surveys only consider the relative frame-by-frame photometric zeropoint oﬀset and the focal plane position- dependent illumination corrections, which are independent of the source color. However, variations in the wavelength dependence of the atmospheric transmission and the instrumental throughput induce source color-dependent systematic errors. These systematic errors must also be considered to achieve the most precise photometric measurements. In this paper, we examine such systematic chromatic errors using photometry from the Dark Energy Survey (DES) as an example. We ﬁrst deﬁne a natural magnitude system for DES and calculate the systematic errors on stellar magnitudes, when the atmospheric transmission and instrumental throughput deviate from the natural system. We conclude that the systematic chromatic errors caused by the change of airmass in each exposure, the change of the precipitable water vapor and aerosol in the atmosphere over time, and the non-uniformity of instrumental throughput over the focal plane, can be up to 2% in some bandpasses. We then compare the calculated systematic chromatic errors with the observed DES data. For the test sample data, we correct these errors using measurements of the atmospheric transmission and instrumental throughput from auxiliary calibration systems. The residual after correction is less than 0.3%. Moreover, we calculate such systematic chromatic errors for Type Ia supernovae and elliptical galaxies and ﬁnd that the chromatic errors for non-stellar objects are redshift-dependent and can be larger than those for stars at certain redshifts. Keywords: 1 George P. and Cynthia Woods Mitchell Institute for Funda- Rua Gal. Jos´e Cristino 77, Rio de Janeiro, RJ - 20921-400, mental Physics and Astronomy, and Department of Physics and Brazil Astronomy, Texas A&M University, College Station, TX 77843, 13 Observat´orio Nacional, Rua Gal. Jos´e Cristino 77, Rio de USA Janeiro, RJ - 20921-400, Brazil 2 Fermi National Accelerator Laboratory, P. O. Box 500, 14 Department of Astronomy, University of Illinois, 1002 W. Batavia, IL 60510, USA Green Street, Urbana, IL 61801, USA 3 Kavli Institute for Cosmological Physics, University of 15 National Center for Supercomputing Applications, 1205 Chicago, Chicago, IL 60637, USA West Clark St., Urbana, IL 61801, USA 4 Department of Physics and Astronomy, University of Penn- 16 Institut de Ci`encies de l’Espai, IEEC-CSIC, Campus UAB, sylvania, Philadelphia, PA 19104, USA Carrer de Can Magrans, s/n, 08193 Bellaterra, Barcelona, Spain 5 SLAC National Accelerator Laboratory, Menlo Park, CA 17 Institut de F´ısica d’Altes Energies (IFAE), The Barcelona 94025, USA Institute of Science and Technology, Campus UAB, 08193 Bel- 6 Kavli Institute for Particle Astrophysics & Cosmology, P. O. laterra (Barcelona) Spain Box 2450, Stanford University, Stanford, CA 94305, USA 18 School of Physics and Astronomy, University of Southamp- 7 Cerro Tololo Inter-American Observatory, National Optical ton, Southampton, SO17 1BJ, UK Astronomy Observatory, Casilla 603, La Serena, Chile 19 Excellence Cluster Universe, Boltzmannstr. 2, 85748 Garch- 8 CNRS, UMR 7095, Institut d’Astrophysique de Paris, F- ing, Germany 75014, Paris, France 20 Faculty of Physics, Ludwig-Maximilians University, Schein- 9 Department of Physics & Astronomy, University College erstr. 1, 81679 Munich, Germany London, Gower Street, London, WC1E 6BT, UK 21 Department of Astronomy, University of California, Berke- 10 Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7095, ley, 501 Campbell Hall, Berkeley, CA 94720, USA Institut d’Astrophysique de Paris, F-75014, Paris, France 22 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, 11 Institute of Cosmology & Gravitation, University of Berkeley, CA 94720, USA Portsmouth, Portsmouth, PO1 3FX, UK 23 Max Planck Institute for Extraterrestrial Physics, Giessen- 12 Laborat´orio Interinstitucional de e-Astronomia - LIneA, bachstrasse, 85748 Garching, Germany

text: Draft

text: version

text: May

text: Preprint

text: typeset

text: using

text: LATEX

text: style

text: emulateapj

text: v.

text: ASSESSMENT

text: OF

text: SYSTEMATIC

text: CHROMATIC

text: ERRORS

text: THAT

text: IMPACT

text: PHOTOMETRIC

text: PRECISION

text: IN

text: LARGE-AREA

text: SKY

text: SURVEYS

text: T.

label: author

text: S.

text: D.

text: L.

text: J.

text: L.

text: D.

text: R.

text: J.

text: G.

text: M.

text: S.

text: D.

text: L.

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text: J.

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text: C.

text: B.

text: R.

text: A.

text: A.

text: A.

text: K.

text: A.

text: M.

text: E.

text: V.

text: M.

text: I.

text: R.

text: C.

text: M.

text: F.

text: E.

text: G.

text: D.

text: V.

text: (The

text: DES

text: Collaboration)

text: Draft

text: version

text: May

text: ABSTRACT

text: Meeting

text: the

text: science

text: goals

text: for

text: many

text: current

text: and

text: future

text: ground-based

text: optical

text: large-area

text: sky

text: surveys

text: requires

text: that

text: the

text: calibrated

text: broadband

text: photometry

text: is

text: both

text: stable

text: in

text: time

text: and

text: uniform

text: over

text: the

text: sky

text: to

text: precision

text: or

text: better.

text: Past

text: and

text: current

text: surveys

text: have

text: achieved

text: photometric

text: precision

text: of

text: by

text: calibrating

text: the

text: survey’s

text: stellar

text: photometry

text: with

text: repeated

text: measurements

text: of

text: a

text: large

text: number

text: of

text: stars

text: observed

text: in

text: multiple

text: epochs.

text: The

text: calibration

text: techniques

text: employed

text: by

text: these

text: surveys

text: only

text: consider

text: the

text: relative

text: frame-by-frame

text: photometric

text: zeropoint

text: oﬀset

text: and

text: the

text: focal

text: plane

text: position-

text: dependent

text: illumination

text: corrections,

text: which

text: are

text: independent

text: of

text: the

text: source

text: color.

text: However,

text: variations

text: in

text: the

text: wavelength

text: dependence

text: of

text: the

text: atmospheric

text: transmission

text: and

text: the

text: instrumental

text: throughput

text: induce

text: source

text: color-dependent

text: systematic

text: errors.

text: These

text: systematic

text: errors

text: must

text: also

text: be

text: considered

text: to

text: achieve

text: the

text: most

text: precise

text: photometric

text: measurements.

text: In

text: this

text: paper,

text: we

text: examine

text: such

text: systematic

text: chromatic

text: errors

text: using

text: photometry

text: from

text: the

text: Dark

text: Energy

text: Survey

text: (DES)

text: as

text: an

text: example.

text: We

text: ﬁrst

text: deﬁne

text: a

text: natural

text: magnitude

text: system

text: for

text: DES

text: and

text: calculate

text: the

text: systematic

text: errors

text: on

text: stellar

text: magnitudes,

text: when

text: the

text: atmospheric

text: transmission

text: and

text: instrumental

text: throughput

text: deviate

text: from

text: the

text: natural

text: system.

text: We

text: conclude

text: that

text: the

text: systematic

text: chromatic

text: errors

text: caused

text: by

text: the

text: change

text: of

text: airmass

text: in

text: each

text: exposure,

text: the

text: change

text: of

text: the

text: precipitable

text: water

text: vapor

text: and

text: aerosol

text: in

text: the

text: atmosphere

text: over

text: time,

text: and

text: the

text: non-uniformity

text: of

text: instrumental

text: throughput

text: over

text: the

text: focal

text: plane,

text: can

text: be

text: up

text: to

text: in

text: some

text: bandpasses.

text: We

text: then

text: compare

text: the

text: calculated

text: systematic

text: chromatic

text: errors

text: with

text: the

text: observed

text: DES

text: data.

text: For

text: the

text: test

text: sample

text: data,

text: we

text: correct

text: these

text: errors

text: using

text: measurements

text: of

text: the

text: atmospheric

text: transmission

text: and

text: instrumental

text: throughput

text: from

text: auxiliary

text: calibration

text: systems.

text: The

text: residual

text: after

text: correction

text: is

text: less

text: than

text: Moreover,

text: we

text: calculate

text: such

text: systematic

text: chromatic

text: errors

text: for

text: Type

text: Ia

text: supernovae

text: and

text: elliptical

text: galaxies

text: and

text: ﬁnd

text: that

text: the

text: chromatic

text: errors

text: for

text: non-stellar

text: objects

text: are

text: redshift-dependent

text: and

text: can

text: be

text: larger

text: than

text: those

text: for

text: stars

text: at

text: certain

text: redshifts.

text: Keywords:

text: George

text: P.

text: and

text: Cynthia

text: Woods

text: Mitchell

text: Institute

text: for

text: Funda-

text: Rua

text: Gal.

text: Jos´e

text: Cristino

text: Rio

text: de

text: Janeiro,

text: RJ

text: mental

text: Physics

text: and

text: Astronomy,

text: and

text: Department

text: of

text: Physics

text: and

text: Brazil

text: Astronomy,

text: Texas

text: A&M

text: University,

text: College

text: Station,

text: TX

text: Observat´orio

text: Nacional,

text: Rua

text: Gal.

text: Jos´e

text: Cristino

text: Rio

text: de

text: USA

text: Janeiro,

text: RJ

text: Brazil

text: Fermi

text: National

text: Accelerator

text: Laboratory,

text: P.

text: O.

text: Box

text: Department

text: of

text: Astronomy,

text: University

text: of

text: Illinois,

text: W.

text: Batavia,

text: IL

text: USA

text: Green

text: Street,

text: Urbana,

text: IL

text: USA

text: Kavli

text: Institute

text: for

text: Cosmological

text: Physics,

text: University

text: of

text: National

text: Center

text: for

text: Supercomputing

text: Applications,

text: Chicago,

text: Chicago,

text: IL

text: USA

text: West

text: Clark

text: St.,

text: Urbana,

text: IL

text: USA

text: Department

text: of

text: Physics

text: and

text: Astronomy,

text: University

text: of

text: Penn-

text: Institut

text: de

text: Ci`encies

text: de

text: l’Espai,

text: IEEC-CSIC,

text: Campus

text: UAB,

text: sylvania,

text: Philadelphia,

text: PA

text: USA

text: Carrer

text: de

text: Can

text: Magrans,

text: s/n,

text: Bellaterra,

text: Barcelona,

text: Spain

text: SLAC

text: National

text: Accelerator

text: Laboratory,

text: Menlo

text: Park,

text: CA

text: Institut

text: de

text: F´ısica

text: d’Altes

text: Energies

text: (IFAE),

text: The

text: Barcelona

text: USA

text: Institute

text: of

text: Science

text: and

text: Technology,

text: Campus

text: UAB,

text: Bel-

text: Kavli

text: Institute

text: for

text: Particle

text: Astrophysics

text: Cosmology,

text: P.

text: O.

text: laterra

text: (Barcelona)

text: Spain

text: Box

text: Stanford

text: University,

text: Stanford,

text: CA

text: USA

text: School

text: of

text: Physics

text: and

text: Astronomy,

text: University

text: of

text: Southamp-

text: Cerro

text: Tololo

text: Inter-American

text: Observatory,

text: National

text: Optical

text: ton,

text: Southampton,

text: UK

text: Astronomy

text: Observatory,

text: Casilla

text: La

text: Serena,

text: Chile

text: Excellence

text: Cluster

text: Universe,

text: Boltzmannstr.

text: Garch-

text: CNRS,

text: UMR

text: Institut

text: d’Astrophysique

text: de

text: Paris,

text: F-

text: ing,

text: Germany

text: Paris,

text: France

text: Faculty

text: of

text: Physics,

text: Ludwig-Maximilians

text: University,

text: Schein-

text: Department

text: of

text: Physics

text: Astronomy,

text: University

text: College

text: erstr.

text: Munich,

text: Germany

text: London,

text: Gower

text: Street,

text: London,

text: UK

text: Department

text: of

text: Astronomy,

text: University

text: of

text: California,

text: Berke-

text: Sorbonne

text: Universit´es,

text: UPMC

text: Univ

text: Paris

text: UMR

text: ley,

text: Campbell

text: Hall,

text: Berkeley,

text: CA

text: USA

text: Institut

text: d’Astrophysique

text: de

text: Paris,

text: Paris,

text: France

text: Lawrence

text: Berkeley

text: National

text: Laboratory,

text: Cyclotron

text: Road,

text: Institute

text: of

text: Cosmology

text: Gravitation,

text: University

text: of

text: Berkeley,

text: CA

text: USA

text: Portsmouth,

text: Portsmouth,

text: UK

text: Max

text: Planck

text: Institute

text: for

text: Extraterrestrial

text: Physics,

text: Giessen-

text: Laborat´orio

text: Interinstitucional

text: de

text: e-Astronomia

text: LIneA,

text: bachstrasse,

text: Garching,

text: Germany


text: 12 Li et al. 1.0 0.8 0.6 0.4 0.2 0.0 400 500 600 700 800 900 1000 1100 wavelength[nm] 50 15 0 0 −50 −15 10 10 0 0 −10 −10 10 10 0 0 −10 −10 −1 0 1 2 3 4 5 −1 0 1 2 3 4 5 g−i g−i Figure 4. Top panel: Solid lines are the ﬁducial instrumental throughput φinst(λ) for ugrizY -bands. Dashed lines are the instrumental ref throughput with a 2 nm shift towards the longer waveleng−th. Bottom panels: Synthetic SCE in th−e ugriz−Y -bands for O5V-M6V stars introduced by this 2 nm bandpass shift, as a function of g i color. Note that the scales for the u and g band are diﬀerent from the other four bands. P. Melchior28, C. J. Miller29,30, R. Miquel31,17, J. J. Mohr19,20,23, E. Neilsen2, R. C. Nichol11, B. Nord2, R. Ogando12,13, A. A. Plazas32, A. K. Romer33, A. Roodman6,5, M. Sako4, E. Sanchez34, V. Scarpine2, M. Schubnell30, I. Sevilla-Noarbe34,14, R. C. Smith7, M. Soares-Santos2, F. Sobreira2,12, E. Suchyta4, G. Tarle30, D. Thomas11, V. Vikram35 (The DES Collaboration) Keywords: 1 George P. and Cynthia Woods Mitchell Institute for Funda- 94025, USA mental Physics and Astronomy, and Department of Physics and 6 Kavli Institute for Particle Astrophysics & Cosmology, P. O. Astronomy, Texas A&M University, College Station, TX 77843, Box 2450, Stanford University, Stanford, CA 94305, USA USA 7 Cerro Tololo Inter-American Observatory, National Optical 2 Fermi National Accelerator Laboratory, P. O. Box 500, Astronomy Observatory, Casilla 603, La Serena, Chile Batavia, IL 60510, USA 8 CNRS, UMR 7095, Institut d’Astrophysique de Paris, F- 3 Kavli Institute for Cosmological Physics, University of 75014, Paris, France Chicago, Chicago, IL 60637, USA 9 Department of Physics & Astronomy, University College 4 Department of Physics and Astronomy, University of Penn- London, Gower Street, London, WC1E 6BT, UK sylvania, Philadelphia, PA 19104, USA 10 Sorbonne Universit´es, UPMC Univ Paris 06, UMR 7095, 5 SLAC National Accelerator Laboratory, Menlo Park, CA Institut d’Astrophysique de Paris, F-75014, Paris, France tsniφ )gamm(u∆ )gamm(r∆ )gamm(z∆ )gamm(g∆ )gamm(i∆ )gamm(Y∆ ##LTFigure## ##LTFigure## ##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## 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