text: 6 fact that these time-varying currents are bound to pro- (hence the classical equivalence) but would acquire dif- duce electromagnetic ﬁeld outside the solenoid. Its con- ferent phase shifts resulting in observable discrepancies. tribution to the phase shift appears to cancel exactly the Thus we conclude that in the dynamic case the main sig- time-dependent part resulting from the magnetic ﬂux in- nature of the AB eﬀect will still be present. side the torus. This again highlights the peculiarity of VI. SUMMARY AND DISCUSSION NR sources where the time-dependent phase shift arises naturally. We have shown that in the Weyl gauge the vector po- One might raise a natural objection to calling the de- tential of an arbitrary NR source is spatially localized. scribed thought experiment as time-dependent AB ef- We have also proven that this apparently local condi- fect. After all, the local impact of the electric ﬁeld is tion for an electromagnetic source to be non-radiating still present. It is easy to see though, that this impact is equivalent (although in a non-trivial way) to the non- cannot account for the phase shift (28). To make the local criterion formulated in [1] by Devaney and Wolf. arguments precise let us consider another non-radiating Using the obtained local non-radiating condition we con- − source, which is characterized by zero scalar potential ﬁrmed that the relation D = ikT , which holds for point and the following vector poten(cid:90)t(cid:90)ial NR sources, is also valid for spatially extended but phys- ically small NR sources, where D and T are correspond- − − ingly the total electric and toroidal dipole moments of Acap(t, r) = iωtσn d2s δ(r rs) (29) the system. D We have shown that any NR source can be viewed as a distribution of elementary dynamic anapoles – NR Notation here is the same as in formula (23). The elec- point sources of the most fundamental type. Such an tric ﬁeld derived from (29) is time-independent and co- approach allows one to build concrete examples of spa- incides with the electric ﬁeld of the disk-shaped dynamic tially extended NR sources and study their properties. anapole (24) taken at the moment t = 0, E (r) = cap As an illustration, we considered a simple scenario for E(r, 0). Since electric ﬁeld is static and homogeneous, the dynamic version of the Aharonov-Bohm eﬀect in the such a non-radiating source resembles an ordinary elec- context of non-radiating sources. We came up with an tric capacitor. However, by insisting that the ﬁelds out- explicit example of an NR source for which the phase side the capacitor are strictly zero, we have also incorpo- shift in a dynamic experiment would arise exactly as in rated magnetic ﬁeld at the bou(cid:90)ndary the static case, while retaining its dependence on time. Apart from the electrodynamics in general, the for- − − malism developed here will be of particular importance H (r, t) = ωtσ dl δ(r r ) (30) cap l for the ﬁelds of metamaterials and nanophotonics, which C currently witness a surge of interest in the properties of the dynamic anapole and non-radiating systems (see The phase shift for the contour penetrating the capac- [13] and references therein). Indeed, a number of recent itor is given by magnetic ﬂux, which linearly grows with works have already conﬁrmed the key role of anapole time excitations in controlling scattering properties of very simple electromagnetic systems, such as nanodisks and − eσ nanowires [15–23]. Correspondingly, one may want to re- δφcap(t) = ωt (cid:126)c (31) visit the analysis of the electromagnetic response of struc- turally more complex metamaterials, where the dynamic Hereby, the oscillating anapole and non-radiating ca- anapoles could underpin, for example, the microscopic pacitor have the same electric ﬁeld at t = 0, but diﬀerent mechanisms of electromagnetic transparency [14] and magnetic ﬂuxes leading to diﬀerent phase shifts (28) and high-Q eﬀects. Our approach to non-radiating sources (31). could be also useful in the analysis of non-radiating Taking a slightly shifted perspective one can say that modes of antennas and scattering suppression in stealth the essence of the AB eﬀect is that the experimental se- applications. In particular, it might aid in designing tups equivalent in the classical sense do not in general stealth antennas and minimizing the radar cross-section exhibit the same behavior at the quantum level. In- of other elements that protrude from the airframe (such deed, the standard AB experimental conﬁguration with as meteorological sensors, guns, landing gear etc). static magnetic ﬁeld conﬁned to an excluded region is ACKNOWLEDGMENTS classically equivalent to the complete absence of electro- magnetic ﬁelds, yet it causes a measurable phase shift in The authors gratefully acknowledge the ﬁnancial sup- probability amplitudes of a charged particle. Compari- port of the Ministry of Education and Science of the son of the oscillating anapole and non-radiating capacitor Russian Federation in the framework of Increase Com- № shows that a similar discrepancy is expected in the time- petitiveness Program of NUST ”MISiS” ( K4-2015-031) dependent experiment. Charged particles traveling fast the Russian Foundation for Basic Research (Grant Agree- enough would feel the same electric ﬁeld in both cases ment No. 16-02-00789) ##LTLine##

text: fact

text: that

text: these

text: time-varying

text: currents

text: are

text: bound

text: to

text: pro-

text: (hence

text: the

text: classical

text: equivalence)

text: but

text: would

text: acquire

text: dif-

text: duce

text: electromagnetic

text: ﬁeld

text: outside

text: the

text: solenoid.

text: Its

text: con-

text: ferent

text: phase

text: shifts

text: resulting

text: in

text: observable

text: discrepancies.

text: tribution

text: to

text: the

text: phase

text: shift

text: appears

text: to

text: cancel

text: exactly

text: the

text: Thus

text: we

text: conclude

text: that

text: in

text: the

text: dynamic

text: case

text: the

text: main

text: sig-

text: time-dependent

text: part

text: resulting

text: from

text: the

text: magnetic

text: ﬂux

text: in-

text: nature

text: of

text: the

text: AB

text: eﬀect

text: will

text: still

text: be

text: present.

text: side

text: the

text: torus.

text: This

text: again

text: highlights

text: the

text: peculiarity

text: of

text: VI.

text: SUMMARY

text: AND

text: DISCUSSION

text: NR

text: sources

text: where

text: the

text: time-dependent

text: phase

text: shift

text: arises

text: naturally.

text: We

text: have

text: shown

text: that

text: in

text: the

text: Weyl

text: gauge

text: the

text: vector

text: po-

text: One

text: might

text: raise

text: a

text: natural

text: objection

text: to

text: calling

text: the

text: de-

text: tential

text: of

text: an

text: arbitrary

text: NR

text: source

text: is

text: spatially

text: localized.

text: scribed

text: thought

text: experiment

text: as

text: time-dependent

text: AB

text: ef-

text: We

text: have

text: also

text: proven

text: that

text: this

text: apparently

text: local

text: condi-

text: fect.

text: After

text: all,

text: the

text: local

text: impact

text: of

text: the

text: electric

text: ﬁeld

text: is

text: tion

text: for

text: an

text: electromagnetic

text: source

text: to

text: be

text: non-radiating

text: still

text: present.

text: It

text: is

text: easy

text: to

text: see

text: though,

text: that

text: this

text: impact

text: is

text: equivalent

text: (although

text: in

text: a

text: non-trivial

text: way)

text: to

text: the

text: non-

text: cannot

text: account

text: for

text: the

text: phase

text: shift

text: To

text: make

text: the

text: local

text: criterion

text: formulated

text: in

text: by

text: Devaney

text: and

text: Wolf.

text: arguments

text: precise

text: let

text: us

text: consider

text: another

text: non-radiating

text: Using

text: the

text: obtained

text: local

text: non-radiating

text: condition

text: we

text: con-

text: source,

text: which

text: is

text: characterized

text: by

text: zero

text: scalar

text: potential

text: ﬁrmed

text: that

text: the

text: relation

text: D

text: ikT

text: which

text: holds

text: for

text: point

text: and

text: the

text: following

text: vector

text: NR

text: sources,

text: is

text: also

text: valid

text: for

text: spatially

text: extended

text: but

text: phys-

text: ically

text: small

text: NR

text: sources,

text: where

text: D

text: and

text: T

text: are

text: correspond-

text: ingly

text: the

text: total

text: electric

text: and

text: toroidal

text: dipole

text: moments

text: of

text: Acap(t,

text: r)

text: iωtσn

text: δ(r

text: rs)

text: the

text: system.

text: D

text: We

text: have

text: shown

text: that

text: any

text: NR

text: source

text: can

text: be

text: viewed

text: as

text: a

text: distribution

text: of

text: elementary

text: dynamic

text: anapoles

text: NR

text: Notation

text: here

text: is

text: the

text: same

text: as

text: in

text: formula

text: The

text: elec-

text: point

text: sources

text: of

text: the

text: most

text: fundamental

text: type.

text: Such

text: an

text: tric

text: ﬁeld

text: derived

text: from

text: is

text: time-independent

text: and

text: co-

text: approach

text: allows

text: one

text: to

text: build

text: concrete

text: examples

text: of

text: spa-

text: incides

text: with

text: the

text: electric

text: ﬁeld

text: of

text: the

text: disk-shaped

text: dynamic

text: tially

text: extended

text: NR

text: sources

text: and

text: study

text: their

text: properties.

text: anapole

text: taken

text: at

text: the

text: moment

text: t

text: E

text: (r)

text: cap

text: As

text: an

text: illustration,

text: we

text: considered

text: a

text: simple

text: scenario

text: for

text: E(r,

text: Since

text: electric

text: ﬁeld

text: is

text: static

text: and

text: homogeneous,

text: the

text: dynamic

text: version

text: of

text: the

text: Aharonov-Bohm

text: eﬀect

text: in

text: the

text: such

text: a

text: non-radiating

text: source

text: resembles

text: an

text: ordinary

text: elec-

text: context

text: of

text: non-radiating

text: sources.

text: We

text: came

text: up

text: with

text: an

text: tric

text: capacitor.

text: However,

text: by

text: insisting

text: that

text: the

text: ﬁelds

text: out-

text: explicit

text: example

text: of

text: an

text: NR

text: source

text: for

text: which

text: the

text: phase

text: side

text: the

text: capacitor

text: are

text: strictly

text: zero,

text: we

text: have

text: also

text: incorpo-

text: shift

text: in

text: a

text: dynamic

text: experiment

text: would

text: arise

text: exactly

text: as

text: in

text: rated

text: magnetic

text: ﬁeld

text: at

text: the

text: the

text: static

text: case,

text: while

text: retaining

text: its

text: dependence

text: on

text: time.

text: Apart

text: from

text: the

text: electrodynamics

text: in

text: general,

text: the

text: for-

text: malism

text: developed

text: here

text: will

text: be

text: of

text: particular

text: importance

text: H

text: (r,

text: t)

text: ωtσ

text: dl

text: δ(r

text: r

text: cap

text: l

text: for

text: the

text: ﬁelds

text: of

text: metamaterials

text: and

text: nanophotonics,

text: which

text: C

text: currently

text: witness

text: a

text: surge

text: of

text: interest

text: in

text: the

text: properties

text: of

text: the

text: dynamic

text: anapole

text: and

text: non-radiating

text: systems

text: (see

text: The

text: phase

text: shift

text: for

text: the

text: contour

text: penetrating

text: the

text: capac-

text: and

text: references

text: therein).

text: Indeed,

text: a

text: number

text: of

text: recent

text: itor

text: is

text: given

text: by

text: magnetic

text: ﬂux,

text: which

text: linearly

text: grows

text: with

text: works

text: have

text: already

text: conﬁrmed

text: the

text: key

text: role

text: of

text: anapole

text: time

text: excitations

text: in

text: controlling

text: scattering

text: properties

text: of

text: very

text: simple

text: electromagnetic

text: systems,

text: such

text: as

text: nanodisks

text: and

text: eσ

text: nanowires

text: Correspondingly,

text: one

text: may

text: want

text: to

text: re-

text: δφcap(t)

text: ωt

text: visit

text: the

text: analysis

text: of

text: the

text: electromagnetic

text: response

text: of

text: struc-

text: turally

text: more

text: complex

text: metamaterials,

text: where

text: the

text: dynamic

text: Hereby,

text: the

text: oscillating

text: anapole

text: and

text: non-radiating

text: ca-

text: anapoles

text: could

text: underpin,

text: for

text: example,

text: the

text: microscopic

text: pacitor

text: have

text: the

text: same

text: electric

text: ﬁeld

text: at

text: t

text: but

text: diﬀerent

text: mechanisms

text: of

text: electromagnetic

text: transparency

text: and

text: magnetic

text: ﬂuxes

text: leading

text: to

text: diﬀerent

text: phase

text: shifts

text: and

text: high-Q

text: eﬀects.

text: Our

text: approach

text: to

text: non-radiating

text: sources

text: could

text: be

text: also

text: useful

text: in

text: the

text: analysis

text: of

text: non-radiating

text: Taking

text: a

text: slightly

text: shifted

text: perspective

text: one

text: can

text: say

text: that

text: modes

text: of

text: antennas

text: and

text: scattering

text: suppression

text: in

text: stealth

text: the

text: essence

text: of

text: the

text: AB

text: eﬀect

text: is

text: that

text: the

text: experimental

text: se-

text: applications.

text: In

text: particular,

text: it

text: might

text: aid

text: in

text: designing

text: tups

text: equivalent

text: in

text: the

text: classical

text: sense

text: do

text: not

text: in

text: general

text: stealth

text: antennas

text: and

text: minimizing

text: the

text: radar

text: cross-section

text: exhibit

text: the

text: same

text: behavior

text: at

text: the

text: quantum

text: level.

text: In-

text: of

text: other

text: elements

text: that

text: protrude

text: from

text: the

text: airframe

text: (such

text: deed,

text: the

text: standard

text: AB

text: experimental

text: conﬁguration

text: with

text: as

text: meteorological

text: sensors,

text: guns,

text: landing

text: gear

text: etc).

text: static

text: magnetic

text: ﬁeld

text: conﬁned

text: to

text: an

text: excluded

text: region

text: is

text: ACKNOWLEDGMENTS

text: classically

text: equivalent

text: to

text: the

text: complete

text: absence

text: of

text: electro-

text: magnetic

text: ﬁelds,

text: yet

text: it

text: causes

text: a

text: measurable

text: phase

text: shift

text: in

text: The

text: authors

text: gratefully

text: acknowledge

text: the

text: ﬁnancial

text: sup-

text: probability

text: amplitudes

text: of

text: a

text: charged

text: particle.

text: Compari-

text: port

text: of

text: the

text: Ministry

text: of

text: Education

text: and

text: Science

text: of

text: the

text: son

text: of

text: the

text: oscillating

text: anapole

text: and

text: non-radiating

text: capacitor

text: Russian

text: Federation

text: in

text: the

text: framework

text: of

text: Increase

text: Com-

text: shows

text: that

text: a

text: similar

text: discrepancy

text: is

text: expected

text: in

text: the

text: time-

text: petitiveness

text: Program

text: of

text: NUST

text: ”MISiS”

text: dependent

text: experiment.

text: Charged

text: particles

text: traveling

text: fast

text: the

text: Russian

text: Foundation

text: for

text: Basic

text: Research

text: (Grant

text: Agree-

text: enough

text: would

text: feel

text: the

text: same

text: electric

text: ﬁeld

text: in

text: both

text: cases

text: ment

text: No.

text: ##LTLine##


text: SUIS: An Online Graphical Signature-Based User Identiﬁcation System Shahid Alam Department of Computer Science and Engineering Qatar University, Doha, Qatar Email: salam@qu.edu.qa Abstract— Humans possess a large amount of, and almost systems are based on reproducing an already pre-drawn image limitless, visual memory, that assists them to remember pictures (outline drawing). far better than words. This phenomenon has recently motivated Elizabeth et al. [17] presents a study about the relationship the computer security researchers’ in academia and industry to design and develop graphical user identiﬁcation systems (GUISs). between memory and graphical passwords. The results indicate Cognometric GUISs are more memorable than drawmetric GUISs, that the recognition-based (cognometric) graphical passwords but takes more time to authenticate. None of the previously are more memorable than recall-based (drawmetric) graphi- proposed GUISs combines the advantages of both cognometric cal passwords, but takes more time to login. None of the and drawmetric systems. A signature personify a person and previously proposed GUISs [4, 5, 6, 7, 8, 9, 10, 12, 15, a graphical signature is easier to recall than other drawings. This paper proposes a new graphical Signature-based User 16, 19, 20, 21, 22, 23] combines the advantages of both Identiﬁcation System named SUIS. It is based on a 2D grid cognometric and drawmetric systems. The systems proposed technology, that is used to draw, digitize and store the signature in [19] and [8] use signature-based schemes but do not use for user identiﬁcation. SUIS is categorized as both a cognometric a 2D grid technology (draw, digitize and store the signature), and drawmetric system. Unlike other systems that use 2D grid: and hence are difﬁcult to recall. The system proposed in [12] We take one cell in a grid as one pixel in the drawing; for signature matching, the signature drawn has to follow the same uses a 2D grid technology but is not a signature-based scheme. grid cells as the signature stored, independent of the sequence; Therefore we do not categorize these three [8, 12, 19] as both and that the system is not based on any machine learning cognometric and drawmetric systems. model. Increasing the number of grid cells increases the password Handwritten signatures have long been used as a proof of space, and decreasing the size of the grid cell increases the precision of the signature. These characteristics makes SUIS: authorship. In general, signatures are authentic, unforegeable, (1) Rigorous enough to be a password system, but easy enough not reusable, unalterable and unrepudiatable [19]. Signatures to be usable. (2) Independent of the language and device used personify a person and are easier to recall than other drawings to draw the signature. (3) Efﬁcient and practical to be used for when drawn on a 2D grid. Therefore we can categorize online authentication systems. such signatures as recall-based graphical passwords that have the same or close enough memorability as recognition-based I. INTRODUCTION AND MOTIVATION graphical passwords. The traditional methods for user identiﬁcation systems rely In this paper we propose a new graphical Signature-based on entering a username and a text password. One of the User Identiﬁcation System named SUIS. It is based on a 2D major vulnerabilities of this technique is the difﬁculty of grid technology, that is used to draw, digitize and store the remembering passwords. Therefore the users tend to pick short signature for user identiﬁcation. SUIS is categorized as both passwords or passwords that are easy to remember. These a cognometric and drawmetric system. The 2D grid was ﬁrst passwords can be easily broken. Passwords that are hard to used in Draw a Secret [12] for user identiﬁcation. Our model break are often hard to remember. is different than proposed in [12]: The ability of humans to remember pictures far better • We take one cell in a grid, as one pixel in the drawing. than words [13, 14] has recently motivated computer security This makes it much simpler to implement the model and researchers’ in academia [1, 6, 7, 8, 9, 10, 12, 18, 19, 21, 23] compare the signature in practice. and industry [4, 5, 15, 16, 20, 22] to study, design and develop • For signature matching, the signature drawn has to follow graphical user identiﬁcation systems (GUISs). More than 10 the same grid cells as the signature stored, independent of US patents on GUISs have been issued, the ﬁrst in 1996 the sequence. This increases the usability of the system, to Greg E. Blonder [2] and the last in 2014 to Microsoft but decreases the password space. Increasing the usability Corporation [11], but yet GUISs are not widely used. here means, since the users do not have to follow the same There are three basic categories of GUISs [6]: Cognomet- sequence they can draw and remember more complex rics, Locimetrics, and Drawmetrics. Cognometric systems are signatures. The password space can be increased by based on the human cognitive abilities, such as the ability to increasing the number of grid cells. remember and recall images. Locimetric systems are based on locating or identifying a point in an image. Drawmetric Some of the characteristics of SUIS are as follows: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: SUIS:

label: title

text: An

text: Online

text: Graphical

text: Signature-Based

text: User

text: Identiﬁcation

text: System

text: Shahid

label: author

text: Alam

text: Department

label: paragraph

text: of

text: Computer

text: Science

text: and

text: Engineering

text: Qatar

text: University,

text: Doha,

text: Qatar

text: Email:

text: salam@qu.edu.qa

text: Abstract—

text: Humans

label: abstract

text: possess

text: a

text: large

text: amount

text: of,

text: and

text: almost

text: systems

text: are

text: based

text: on

text: reproducing

text: an

text: already

text: pre-drawn

text: image

text: limitless,

text: visual

text: memory,

text: that

text: assists

text: them

text: to

text: remember

text: pictures

text: (outline

text: drawing).

text: far

text: better

text: than

text: words.

text: This

text: phenomenon

text: has

text: recently

text: motivated

text: Elizabeth

text: et

text: al.

text: presents

text: a

text: study

text: about

text: the

text: relationship

text: the

text: computer

text: security

text: researchers’

text: in

text: academia

text: and

text: industry

text: to

text: design

text: and

text: develop

text: graphical

text: user

text: identiﬁcation

text: systems

text: (GUISs).

text: between

text: memory

text: and

text: graphical

text: passwords.

text: The

text: results

text: indicate

text: Cognometric

text: GUISs

text: are

text: more

text: memorable

text: than

text: drawmetric

text: GUISs,

text: that

text: the

text: recognition-based

text: (cognometric)

text: graphical

text: passwords

text: but

text: takes

text: more

text: time

text: to

text: authenticate.

text: of

text: the

text: previously

text: are

text: more

text: memorable

text: than

text: recall-based

text: (drawmetric)

text: graphi-

text: proposed

text: GUISs

text: combines

text: the

text: advantages

text: of

text: both

text: cognometric

text: cal

text: passwords,

text: but

text: takes

text: more

text: time

text: to

text: login.

text: of

text: the

text: and

text: drawmetric

text: systems.

text: A

text: signature

text: personify

text: a

text: person

text: and

text: previously

text: proposed

text: GUISs

text: a

text: graphical

text: signature

text: is

text: easier

text: to

text: recall

text: than

text: other

text: drawings.

text: This

text: paper

text: proposes

text: a

text: new

text: graphical

text: Signature-based

text: User

text: combines

text: the

text: advantages

text: of

text: both

text: Identiﬁcation

text: System

text: named

text: SUIS.

text: It

text: is

text: based

text: on

text: a

text: grid

text: cognometric

text: and

text: drawmetric

text: systems.

text: The

text: systems

text: proposed

text: technology,

text: that

text: is

text: used

text: to

text: draw,

text: digitize

text: and

text: store

text: the

text: signature

text: in

text: and

text: use

text: signature-based

text: schemes

text: but

text: do

text: not

text: use

text: for

text: user

text: identiﬁcation.

text: SUIS

text: is

text: categorized

text: as

text: both

text: a

text: cognometric

text: a

text: grid

text: technology

text: (draw,

text: digitize

text: and

text: store

text: the

text: signature),

text: and

text: drawmetric

text: system.

text: Unlike

text: other

text: systems

text: that

text: use

text: grid:

text: and

text: hence

text: are

text: difﬁcult

text: to

text: recall.

text: The

text: system

text: proposed

text: in

text: We

text: take

text: one

text: cell

text: in

text: a

text: grid

text: as

text: one

text: pixel

text: in

text: the

text: drawing;

text: for

text: signature

text: matching,

text: the

text: signature

text: drawn

text: has

text: to

text: follow

text: the

text: same

text: uses

text: a

text: grid

text: technology

text: but

text: is

text: not

text: a

text: signature-based

text: scheme.

text: grid

text: cells

text: as

text: the

text: signature

text: stored,

text: independent

text: of

text: the

text: sequence;

text: Therefore

text: we

text: do

text: not

text: categorize

text: these

text: three

text: as

text: both

text: and

text: that

text: the

text: system

text: is

text: not

text: based

text: on

text: any

text: machine

text: learning

text: cognometric

text: and

text: drawmetric

text: systems.

text: model.

text: Increasing

text: the

text: number

text: of

text: grid

text: cells

text: increases

text: the

text: password

text: Handwritten

text: signatures

text: have

text: long

text: been

text: used

text: as

text: a

text: proof

text: of

text: space,

text: and

text: decreasing

text: the

text: size

text: of

text: the

text: grid

text: cell

text: increases

text: the

text: precision

text: of

text: the

text: signature.

text: These

text: characteristics

text: makes

text: SUIS:

text: authorship.

text: In

text: general,

text: signatures

text: are

text: authentic,

text: unforegeable,

text: Rigorous

text: enough

text: to

text: be

text: a

text: password

text: system,

text: but

text: easy

text: enough

text: not

text: reusable,

text: unalterable

text: and

text: unrepudiatable

text: Signatures

text: to

text: be

text: usable.

text: Independent

text: of

text: the

text: language

text: and

text: device

text: used

text: personify

text: a

text: person

text: and

text: are

text: easier

text: to

text: recall

text: than

text: other

text: drawings

text: to

text: draw

text: the

text: signature.

text: Efﬁcient

text: and

text: practical

text: to

text: be

text: used

text: for

text: when

text: drawn

text: on

text: a

text: grid.

text: Therefore

text: we

text: can

text: categorize

text: online

text: authentication

text: systems.

text: such

text: signatures

text: as

text: recall-based

text: graphical

text: passwords

text: that

text: have

text: the

text: same

text: or

text: close

text: enough

text: memorability

text: as

text: recognition-based

text: I.

text: INTRODUCTION

text: AND

text: MOTIVATION

text: graphical

text: passwords.

text: The

text: traditional

text: methods

text: for

text: user

text: identiﬁcation

text: systems

text: rely

text: In

text: this

text: paper

text: we

text: propose

text: a

text: new

text: graphical

text: Signature-based

text: on

text: entering

text: a

text: username

text: and

text: a

text: text

text: password.

text: One

text: of

text: the

text: User

text: Identiﬁcation

text: System

text: named

text: SUIS.

text: It

text: is

text: based

text: on

text: a

text: major

text: vulnerabilities

text: of

text: this

text: technique

text: is

text: the

text: difﬁculty

text: of

text: grid

text: technology,

text: that

text: is

text: used

text: to

text: draw,

text: digitize

text: and

text: store

text: the

text: remembering

text: passwords.

text: Therefore

text: the

text: users

text: tend

text: to

text: pick

text: short

text: signature

text: for

text: user

text: identiﬁcation.

text: SUIS

text: is

text: categorized

text: as

text: both

text: passwords

text: or

text: passwords

text: that

text: are

text: easy

text: to

text: remember.

text: These

text: a

text: cognometric

text: and

text: drawmetric

text: system.

text: The

text: grid

text: was

text: ﬁrst

text: passwords

text: can

text: be

text: easily

text: broken.

text: Passwords

text: that

text: are

text: hard

text: to

text: used

text: in

text: Draw

text: a

text: Secret

text: for

text: user

text: identiﬁcation.

text: Our

text: model

text: break

text: are

text: often

text: hard

text: to

text: remember.

text: is

text: different

text: than

text: proposed

text: in

text: The

text: ability

text: of

text: humans

text: to

text: remember

text: pictures

text: far

text: better

text: We

text: take

text: one

text: cell

text: in

text: a

text: grid,

text: as

text: one

text: pixel

text: in

text: the

text: drawing.

text: than

text: words

text: has

text: recently

text: motivated

text: computer

text: security

text: This

text: makes

text: it

text: much

text: simpler

text: to

text: implement

text: the

text: model

text: and

text: researchers’

text: in

text: academia

text: compare

text: the

text: signature

text: in

text: practice.

text: and

text: industry

text: to

text: study,

text: design

text: and

text: develop

text: For

text: signature

text: matching,

text: the

text: signature

text: drawn

text: has

text: to

text: follow

text: graphical

text: user

text: identiﬁcation

text: systems

text: (GUISs).

text: More

text: than

text: the

text: same

text: grid

text: cells

text: as

text: the

text: signature

text: stored,

text: independent

text: of

text: US

text: patents

text: on

text: GUISs

text: have

text: been

text: issued,

text: the

text: ﬁrst

text: in

text: the

text: sequence.

text: This

text: increases

text: the

text: usability

text: of

text: the

text: system,

text: to

text: Greg

text: E.

text: Blonder

text: and

text: the

text: last

text: in

text: to

text: Microsoft

text: but

text: decreases

text: the

text: password

text: space.

text: Increasing

text: the

text: usability

text: Corporation

text: but

text: yet

text: GUISs

text: are

text: not

text: widely

text: used.

text: here

text: means,

text: since

text: the

text: users

text: do

text: not

text: have

text: to

text: follow

text: the

text: same

text: There

text: are

text: three

text: basic

text: categories

text: of

text: GUISs

text: Cognomet-

text: sequence

text: they

text: can

text: draw

text: and

text: remember

text: more

text: complex

text: rics,

text: Locimetrics,

text: and

text: Drawmetrics.

text: Cognometric

text: systems

text: are

text: signatures.

text: The

text: password

text: space

text: can

text: be

text: increased

text: by

text: based

text: on

text: the

text: human

text: cognitive

text: abilities,

text: such

text: as

text: the

text: ability

text: to

text: increasing

text: the

text: number

text: of

text: grid

text: cells.

text: remember

text: and

text: recall

text: images.

text: Locimetric

text: systems

text: are

text: based

text: on

text: locating

text: or

text: identifying

text: a

text: point

text: in

text: an

text: image.

text: Drawmetric

text: Some

text: of

text: the

text: characteristics

text: of

text: SUIS

text: are

text: as

text: follows:

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text: 1) SUIS is easier and faster to compare for signature Experiments were carried out with 21 users. The successful matching. rate achieved was 93%. To calculate the acceleration they used 2) SUIS takes into consideration the angular changes at a the classical physics formula a = F , where F is the user’s m coarse level. force to push the mouse and m is the mass of the mouse. Al- 3) SUIS, by using a digitization technique, provides extra though, the number of participants in the experiments carried security and protection on top of encryption. out in [19] were small to make any deﬁnite conclusion, the 4) The signature window i.e; the grid area (number of grid successful rate reported is very encouraging, and indicates that cells) in SUIS can be increased to increase the password some of the parameters can be used for a successful signature space (2n where n = number of grid cells) of the system. matching. The size of an average grid (extended) used in our SUIS indirectly uses the signature points as part of the grid × empirical study is 7 7 for which the password space cells. We do not use the last three parameters selected by × is 249 > 10 1014. The password space for a text-based [19], as we believe these are not the true representation of a user identiﬁcation system for 8 characters (average size) user drawing a signature. In practice the last three parameters × password is 9510 > 6 1015. There are 95 possible are more dependent than the ﬁrst two parameters, on different character sets including space. environments, such as the mood of the user (e.g; in sickness, 5) SUIS is rigorous enough to be a password system, but sadness and excitement etc), times of the day or night, etc, easy enough to be usable. and hence can produce more true negatives. 6) Precision of the signature in SUIS can be changed by Jermyn et al. [12] proposed a system called Draw a Secret decreasing the size of the grid cell of the system. For (DAS), that allows the user to draw a unique password. example, a signature drawn with mouse/ﬁnger needs less The password is drawn on a 2D grid. If the stored and the precision, whereas a signature drawn with pen or other drawn password touches the same grid cells in the same such pointing devices needs more precision. sequence, then the user is authenticated. It becomes difﬁcult to 7) SUIS is independent of the language and the device authenticate if the strokes of the user are too close to the grid used to write/draw the signature, but is more suitable lines. In this case either, the user is presented with the internal for touch-based systems (the empirical study presented representation to conﬁrm if the cells were actually touched by in this paper was performed on a touch-based system), the drawing, or, the system does not accept a drawing that is such as, mobile devices and laptops. too close to a grid line. 8) SUIS is efﬁcient and suitable to be used for online As passwords depend on users, the proposed system in [12] (veriﬁcation is performed immediately after a password lacks an empirical study for its usability. It has only been is submitted) authentication systems. Our system is not tested through paper prototypes. Because of lack of a suitable based on any machine learning model as used in [8]. A user study we cannot comment on its effectiveness as a GUIS. machine learning model is not suitable and practical to Everitt et al. [8] proposed a neural network-based system be used for identifying a user for online authentication using graphical signatures. The system use a hybrid approach, systems. Moreover, for successful signature matching using both text and graphical passwords. The input devices such a model needs a large set of training data, i.e; the used are mouse for graphical-based password and keyboard for forged samples of the signatures. text-based password. The authenticity of a user is conﬁrmed by The remainder of the paper is organized as follows. Section the typing style and the signature match. For the typing style describes and compares research efforts that are similar to they measure two times, one is the time between the two key SUIS. Section describes SUIS in detail. We conclude in presses, and the other is the time a key is held down. This idea Section . for the key metrics is similar in concept to the one used in [19] for the mouse metrics. For matching two graphical signatures II. RELATED WORKS they use the signature traces, and measure the change in angles This section discusses some of the previous research efforts and euclidean distances in the two signatures. on GUISs that are similar to the system proposed in this paper. The experiments were carried out with 41 participants A thorough survey of these and other systems referenced in between the ages of 20 and 30. The results show that they Section can be found in [1, 18]. achieved a false accept rate (FAR) of 4.4% and a range of Syukri et al. [19] proposed a system that uses signatures false reject rate (FRR) from 0.2% – 38.6%. FAR is the rate drawn with mouse for identifying a user. Through exper- at which forged samples are accepted as genuine and FRR is imental evaluation they selected parameters for signature the rate at which genuine samples are rejected as forgeries. veriﬁcation and matching. User veriﬁcation threshold was The system [8] is based on a machine learning model and set to 70%. The size of the signature window was set to hence needs training in addition to registration and veriﬁcation. × 1024 512 pixels. The distance threshold was set to 50 pixels. The training data is usually created by asking users to provide The parameters selected for signature veriﬁcation were: (1) a set of forged signature samples for other users or these forge Number of signature points. (2) Coordinate of signature points. signature samples are generated automatically. We think this is (3) Signature writing time. (4) Signature writing velocity. (5) one of the major problems of using a machine learning model Signature writing acceleration. in such systems. ##LTLine##

label: list

text: SUIS

text: is

text: easier

text: and

text: faster

text: to

text: compare

text: for

text: signature

text: Experiments

text: were

text: carried

text: out

text: with

text: users.

text: The

text: successful

text: matching.

text: rate

text: achieved

text: was

text: To

text: calculate

text: the

text: acceleration

text: they

text: used

text: SUIS

text: takes

text: into

text: consideration

text: the

text: angular

text: changes

text: at

text: a

text: the

text: classical

text: physics

text: formula

text: a

text: F

text: where

text: F

text: is

text: the

text: user’s

text: m

text: coarse

text: level.

text: force

text: to

text: push

text: the

text: mouse

text: and

text: m

text: is

text: the

text: mass

text: of

text: the

text: mouse.

text: Al-

text: SUIS,

text: by

text: using

text: a

text: digitization

text: technique,

text: provides

text: extra

text: though,

text: the

text: number

text: of

text: participants

text: in

text: the

text: experiments

text: carried

text: security

text: and

text: protection

text: on

text: top

text: of

text: encryption.

text: out

text: in

text: were

text: small

text: to

text: make

text: any

text: deﬁnite

text: conclusion,

text: the

text: The

text: signature

text: window

text: i.e;

text: the

text: grid

text: area

text: (number

text: of

text: grid

text: successful

text: rate

text: reported

text: is

text: very

text: encouraging,

text: and

text: indicates

text: that

text: cells)

text: in

text: SUIS

text: can

text: be

text: increased

text: to

text: increase

text: the

text: password

text: some

text: of

text: the

text: parameters

text: can

text: be

text: used

text: for

text: a

text: successful

text: signature

text: space

text: where

text: n

text: number

text: of

text: grid

text: cells)

text: of

text: the

text: system.

text: matching.

text: The

text: size

text: of

text: an

text: average

text: grid

text: (extended)

text: used

text: in

text: our

text: SUIS

text: indirectly

text: uses

text: the

text: signature

text: points

text: as

text: part

text: of

text: the

text: grid

text: empirical

text: study

text: is

text: for

text: which

text: the

text: password

text: space

text: cells.

text: We

text: do

text: not

text: use

text: the

text: last

text: three

text: parameters

text: selected

text: by

text: is

text: The

text: password

text: space

text: for

text: a

text: text-based

text: as

text: we

text: believe

text: these

text: are

text: not

text: the

text: true

text: representation

text: of

text: a

text: user

text: identiﬁcation

text: system

text: for

text: characters

text: (average

text: size)

text: user

text: drawing

text: a

text: signature.

text: In

text: practice

text: the

text: last

text: three

text: parameters

text: password

text: is

text: There

text: are

text: possible

text: are

text: more

text: dependent

text: than

text: the

text: ﬁrst

text: two

text: parameters,

text: on

text: different

text: character

text: sets

text: including

text: space.

text: environments,

text: such

text: as

text: the

text: mood

text: of

text: the

text: user

text: (e.g;

text: in

text: sickness,

text: SUIS

text: is

text: rigorous

text: enough

text: to

text: be

text: a

text: password

text: system,

text: but

text: sadness

text: and

text: excitement

text: etc),

text: times

text: of

text: the

text: day

text: or

text: night,

text: etc,

text: easy

text: enough

text: to

text: be

text: usable.

text: and

text: hence

text: can

text: produce

text: more

text: true

text: negatives.

text: Precision

text: of

text: the

text: signature

text: in

text: SUIS

text: can

text: be

text: changed

text: by

text: Jermyn

text: et

text: al.

text: proposed

text: a

text: system

text: called

text: Draw

text: a

text: Secret

text: decreasing

text: the

text: size

text: of

text: the

text: grid

text: cell

text: of

text: the

text: system.

text: For

text: (DAS),

text: that

text: allows

text: the

text: user

text: to

text: draw

text: a

text: unique

text: password.

text: example,

text: a

text: signature

text: drawn

text: with

text: mouse/ﬁnger

text: needs

text: less

text: The

text: password

text: is

text: drawn

text: on

text: a

text: grid.

text: If

text: the

text: stored

text: and

text: the

text: precision,

text: whereas

text: a

text: signature

text: drawn

text: with

text: pen

text: or

text: other

text: drawn

text: password

text: touches

text: the

text: same

text: grid

text: cells

text: in

text: the

text: same

text: such

text: pointing

text: devices

text: needs

text: more

text: precision.

text: sequence,

text: then

text: the

text: user

text: is

text: authenticated.

text: It

text: becomes

text: difﬁcult

text: to

text: SUIS

text: is

text: independent

text: of

text: the

text: language

text: and

text: the

text: device

text: authenticate

text: if

text: the

text: strokes

text: of

text: the

text: user

text: are

text: too

text: close

text: to

text: the

text: grid

text: used

text: to

text: write/draw

text: the

text: signature,

text: but

text: is

text: more

text: suitable

text: lines.

text: In

text: this

text: case

text: either,

text: the

text: user

text: is

text: presented

text: with

text: the

text: internal

text: for

text: touch-based

text: systems

text: (the

text: empirical

text: study

text: presented

text: representation

text: to

text: conﬁrm

text: if

text: the

text: cells

text: were

text: actually

text: touched

text: by

text: in

text: this

text: paper

text: was

text: performed

text: on

text: a

text: touch-based

text: system),

text: the

text: drawing,

text: or,

text: the

text: system

text: does

text: not

text: accept

text: a

text: drawing

text: that

text: is

text: such

text: as,

text: mobile

text: devices

text: and

text: laptops.

text: too

text: close

text: to

text: a

text: grid

text: line.

text: SUIS

text: is

text: efﬁcient

text: and

text: suitable

text: to

text: be

text: used

text: for

text: online

text: As

text: passwords

text: depend

text: on

text: users,

text: the

text: proposed

text: system

text: in

text: (veriﬁcation

text: is

text: performed

text: immediately

text: after

text: a

text: password

text: lacks

text: an

text: empirical

text: study

text: for

text: its

text: usability.

text: It

text: has

text: only

text: been

text: is

text: submitted)

text: authentication

text: systems.

text: Our

text: system

text: is

text: not

text: tested

text: through

text: paper

text: prototypes.

text: Because

text: of

text: lack

text: of

text: a

text: suitable

text: based

text: on

text: any

text: machine

text: learning

text: model

text: as

text: used

text: in

text: A

text: user

text: study

text: we

text: cannot

text: comment

text: on

text: its

text: effectiveness

text: as

text: a

text: GUIS.

text: machine

text: learning

text: model

text: is

text: not

text: suitable

text: and

text: practical

text: to

text: Everitt

text: et

text: al.

text: proposed

text: a

text: neural

text: network-based

text: system

text: be

text: used

text: for

text: identifying

text: a

text: user

text: for

text: online

text: authentication

text: using

text: graphical

text: signatures.

text: The

text: system

text: use

text: a

text: hybrid

text: approach,

text: systems.

text: Moreover,

text: for

text: successful

text: signature

text: matching

text: using

text: both

text: text

text: and

text: graphical

text: passwords.

text: The

text: input

text: devices

text: such

text: a

text: model

text: needs

text: a

text: large

text: set

text: of

text: training

text: data,

text: i.e;

text: the

text: used

text: are

text: mouse

text: for

text: graphical-based

text: password

text: and

text: keyboard

text: for

text: forged

text: samples

text: of

text: the

text: signatures.

text: text-based

text: password.

text: The

text: authenticity

text: of

text: a

text: user

text: is

text: conﬁrmed

text: by

text: The

text: remainder

text: of

text: the

text: paper

text: is

text: organized

text: as

text: follows.

text: Section

text: the

text: typing

text: style

text: and

text: the

text: signature

text: match.

text: For

text: the

text: typing

text: style

text: describes

text: and

text: compares

text: research

text: efforts

text: that

text: are

text: similar

text: to

text: they

text: measure

text: two

text: times,

text: one

text: is

text: the

text: time

text: between

text: the

text: two

text: key

text: SUIS.

text: Section

text: describes

text: SUIS

text: in

text: detail.

text: We

text: conclude

text: in

text: presses,

text: and

text: the

text: other

text: is

text: the

text: time

text: a

text: key

text: is

text: held

text: down.

text: This

text: idea

text: Section

text: for

text: the

text: key

text: metrics

text: is

text: similar

text: in

text: concept

text: to

text: the

text: one

text: used

text: in

text: for

text: the

text: mouse

text: metrics.

text: For

text: matching

text: two

text: graphical

text: signatures

text: II.

text: RELATED

text: WORKS

text: they

text: use

text: the

text: signature

text: traces,

text: and

text: measure

text: the

text: change

text: in

text: angles

text: This

text: section

text: discusses

text: some

text: of

text: the

text: previous

text: research

text: efforts

text: and

text: euclidean

text: distances

text: in

text: the

text: two

text: signatures.

text: on

text: GUISs

text: that

text: are

text: similar

text: to

text: the

text: system

text: proposed

text: in

text: this

text: paper.

text: The

text: experiments

text: were

text: carried

text: out

text: with

text: participants

text: A

text: thorough

text: survey

text: of

text: these

text: and

text: other

text: systems

text: referenced

text: in

text: between

text: the

text: ages

text: of

text: and

text: The

text: results

text: show

text: that

text: they

text: Section

text: can

text: be

text: found

text: in

text: achieved

text: a

text: false

text: accept

text: rate

text: (FAR)

text: of

text: and

text: a

text: range

text: of

text: Syukri

text: et

text: al.

text: proposed

text: a

text: system

text: that

text: uses

text: signatures

text: false

text: reject

text: rate

text: (FRR)

text: from

text: FAR

text: is

text: the

text: rate

text: drawn

text: with

text: mouse

text: for

text: identifying

text: a

text: user.

text: Through

text: exper-

text: at

text: which

text: forged

text: samples

text: are

text: accepted

text: as

text: genuine

text: and

text: FRR

text: is

text: imental

text: evaluation

text: they

text: selected

text: parameters

text: for

text: signature

text: the

text: rate

text: at

text: which

text: genuine

text: samples

text: are

text: rejected

text: as

text: forgeries.

text: veriﬁcation

text: and

text: matching.

text: User

text: veriﬁcation

text: threshold

text: was

text: The

text: system

text: is

text: based

text: on

text: a

text: machine

text: learning

text: model

text: and

text: set

text: to

text: The

text: size

text: of

text: the

text: signature

text: window

text: was

text: set

text: to

text: hence

text: needs

text: training

text: in

text: addition

text: to

text: registration

text: and

text: veriﬁcation.

text: pixels.

text: The

text: distance

text: threshold

text: was

text: set

text: to

text: pixels.

text: The

text: training

text: data

text: is

text: usually

text: created

text: by

text: asking

text: users

text: to

text: provide

text: The

text: parameters

text: selected

text: for

text: signature

text: veriﬁcation

text: were:

text: a

text: set

text: of

text: forged

text: signature

text: samples

text: for

text: other

text: users

text: or

text: these

text: forge

text: Number

text: of

text: signature

text: points.

text: Coordinate

text: of

text: signature

text: points.

text: signature

text: samples

text: are

text: generated

text: automatically.

text: We

text: think

text: this

text: is

text: Signature

text: writing

text: time.

text: Signature

text: writing

text: velocity.

text: one

text: of

text: the

text: major

text: problems

text: of

text: using

text: a

text: machine

text: learning

text: model

text: Signature

text: writing

text: acceleration.

text: in

text: such

text: systems.

text: ##LTLine##


text: 30 K. BURNS, H. MASUR, C. MATHEUS AND A. WILKINSON To see that µ(T 1S) < ∞, we use the expression for d vol from (28) and integrate: (cid:90) (cid:18)(cid:90) (cid:19) µ(T 1S) = δ−1d vol = O δ0 xr−1 dx < ∞. T1S 0 (cid:5) The ﬂow ψ is a time change of ϕ ; that is, it has the same orbits, but they are t t traversed at a diﬀerent speed, depending on the distance to the singular locus. Indeed, R R deﬁning the cocycle τ : T 1S × → by the implicit formula (cid:90) τ(v,t) dx (37) = t, δ(ϕ (v)) 0 x R we have that ψ (v) = ϕ (v), for all v ∈ T 1S, t ∈ . This gives an alternate way to t τ(v,t) see the completeness of the ﬂow ψ: the function δ clearly remains positive along orbits of ψ for all time. Theorem 4.12. The ﬂow ψ is an Anosov ﬂow in the (cid:63)-metric. That is, there exists a t Dψ -invariant, continuous splitting of the tangent bundle: t (cid:0) (cid:1) R T T 1S = Eu ⊕ ψ˙ ⊕ Es ψ ψ and constants C > 0, λ > 1 such that for every v ∈ T 1S, and every t > 0: − • ξ ∈ Eu(v) =⇒ (cid:107)Dψ− (ξ)(cid:107) ≤ Cλ t(cid:107)ξ(cid:107) , and ψ t (cid:63) − (cid:63) • ξ ∈ Es (v) =⇒ (cid:107)Dψ (ξ)(cid:107) ≤ Cλ t(cid:107)ξ(cid:107) . ψ t (cid:63) (cid:63) From Theorem 4.12 we obtain several important properties of both ψ and ϕ . The t t ﬁrst is ergodicity. Since volume preserving Anosov ﬂows are ergodic, the ﬂows ϕ and t ψ have the same orbits, and the (cid:63) volume is equivalent to (i.e. has the same zero sets t as) the original volume on T 1S, we obtain: Corollary 4.13. The ﬂow ψ is ergodic with respect to the invariant volume µ. Conse- t quently, ϕ is ergodic with respect to volume. t In the next corollary we obtain a splitting of T T 1S, invariant under Dϕ . t Corollary 4.14. Dϕ has an invariant singular hyperbolic splitting t (cid:0) (cid:1) R T T 1S = Eu ⊕ ϕ˙ ⊕ Es, ϕ ϕ R R with Eu and Es given by intersecting Eu ⊕ ψ˙ and Es ⊕ ψ˙ with the smooth, Dϕ - ϕ ϕ ⊥ ψ ψ t invariant bundle ϕ˙ . Since the weak stable and unstable distributions of a C3 Anosov ﬂow in dimension 3 are C1+α, for some α > 0, we also obtain: R R Corollary 4.15. The distributions Eu ⊕ ψ˙ and Es ⊕ ψ˙ are C1+α, for some α > 0. ψ ψ The distributions Eu and Es are also C1+α, when measured in the (cid:63) metric. Thus in ϕ ϕ the compact part δ ≥ δ , the distributions Eu and Es are uniformly C1+α. 0 ϕ ϕ ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: K.

text: BURNS,

text: H.

text: MASUR,

text: C.

text: MATHEUS

text: AND

text: A.

text: WILKINSON

text: To

text: see

text: that

text: µ(T

text: we

text: use

text: the

text: expression

text: for

text: d

text: vol

text: from

text: and

text: integrate:

text: µ(T

text: δ−1d

text: vol

text: O

text: δ0

text: dx

text: The

text: ﬂow

text: ψ

text: is

text: a

text: time

text: change

text: of

text: ϕ

text: that

text: is,

text: it

text: has

text: the

text: same

text: orbits,

text: but

text: they

text: are

text: t

text: t

text: traversed

text: at

text: a

text: diﬀerent

text: speed,

text: depending

text: on

text: the

text: distance

text: to

text: the

text: singular

text: locus.

text: Indeed,

text: R

text: R

text: deﬁning

text: the

text: cocycle

text: τ

text: T

text: by

text: the

text: implicit

text: formula

label: equation

text: τ(v,t)

text: dx

text: t,

text: δ(ϕ

text: (v))

text: x

text: R

text: we

text: have

text: that

text: ψ

text: (v)

text: ϕ

text: (v),

text: for

text: all

text: v

text: T

text: t

text: This

text: gives

text: an

text: alternate

text: way

text: to

text: t

text: τ(v,t)

text: see

text: the

text: completeness

text: of

text: the

text: ﬂow

text: ψ:

text: the

text: function

text: δ

text: clearly

text: remains

text: positive

text: along

text: orbits

text: of

text: ψ

text: for

text: all

text: time.

text: Theorem

text: The

text: ﬂow

text: ψ

text: is

text: an

text: Anosov

text: ﬂow

text: in

text: the

text: That

text: is,

text: there

text: exists

text: a

text: t

text: Dψ

text: -invariant,

text: continuous

text: splitting

text: of

text: the

text: tangent

text: bundle:

text: t

text: R

text: T

text: T

text: Eu

text: ψ˙

text: Es

text: ψ

text: ψ

text: and

text: constants

text: C

text: λ

text: such

text: that

text: for

text: every

text: v

text: T

text: and

text: every

text: t

text: ξ

text: Eu(v)

text: Cλ

text: and

text: ψ

text: t

text: ξ

text: Es

text: (v)

text: Cλ

text: ψ

text: t

text: From

text: Theorem

text: we

text: obtain

text: several

text: important

text: properties

text: of

text: both

text: ψ

text: and

text: ϕ

text: The

text: t

text: t

text: ﬁrst

text: is

text: ergodicity.

text: Since

text: volume

text: preserving

text: Anosov

text: ﬂows

text: are

text: ergodic,

text: the

text: ﬂows

text: ϕ

text: and

text: t

text: ψ

text: have

text: the

text: same

text: orbits,

text: and

text: the

text: volume

text: is

text: equivalent

text: to

text: (i.e.

text: has

text: the

text: same

text: zero

text: sets

text: t

text: as)

text: the

text: original

text: volume

text: on

text: T

text: we

text: obtain:

text: Corollary

text: The

text: ﬂow

text: ψ

text: is

text: ergodic

text: with

text: respect

text: to

text: the

text: invariant

text: volume

text: µ.

text: Conse-

text: t

text: quently,

text: ϕ

text: is

text: ergodic

text: with

text: respect

text: to

text: volume.

text: t

text: In

text: the

text: next

text: corollary

text: we

text: obtain

text: a

text: splitting

text: of

text: T

text: T

text: invariant

text: under

text: Dϕ

text: t

text: Corollary

text: Dϕ

text: has

text: an

text: invariant

text: singular

text: hyperbolic

text: splitting

text: t

text: R

text: T

text: T

text: Eu

text: ϕ˙

text: Es,

text: ϕ

text: ϕ

text: R

text: R

text: with

text: Eu

text: and

text: Es

text: given

text: by

text: intersecting

text: Eu

text: ψ˙

text: and

text: Es

text: ψ˙

text: with

text: the

text: smooth,

text: Dϕ

text: ϕ

text: ϕ

text: ψ

text: ψ

text: t

text: invariant

text: bundle

text: ϕ˙

text: Since

text: the

text: weak

text: stable

text: and

text: unstable

text: distributions

text: of

text: a

text: Anosov

text: ﬂow

text: in

text: dimension

text: are

text: for

text: some

text: α

text: we

text: also

text: obtain:

text: R

text: R

text: Corollary

text: The

text: distributions

text: Eu

text: ψ˙

text: and

text: Es

text: ψ˙

text: are

text: for

text: some

text: α

text: ψ

text: ψ

text: The

text: distributions

text: Eu

text: and

text: Es

text: are

text: also

text: when

text: measured

text: in

text: the

text: metric.

text: Thus

text: in

text: ϕ

text: ϕ

text: the

text: compact

text: part

text: δ

text: δ

text: the

text: distributions

text: Eu

text: and

text: Es

text: are

text: uniformly

text: ϕ

text: ϕ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 36 K. BURNS, H. MASUR, C. MATHEUS AND A. WILKINSON We will work with δ ≤ δ /2 so that for δ(p) ≤ δ , we have δ(p) = δ(p) and χ(p) = 2 1 2 r − 1 − (cid:15), where χ is the function appearing in Proposition 4.1. Let T (δ ) ⊂ T 1S be the torus consisting of all unit tangent vectors to S with footpoint 2 in Z(δ ): 2 T (δ ) = T 1 S. 2 Z(δ2) This torus is transverse to the vector ﬁeld ϕ˙, except at the two circles − C+ := {J∇δ(p) : p ∈ Z}, and C := {−J∇δ(p) : p ∈ Z}. Let W(cid:99)u and W(cid:99)s be the laminations of T (δ ) obtained by intersecting leaves of the 2 − weak foliations Wcu and Wcs with T (δ ). On T (δ )\(C+∪C ), the laminations W(cid:99)u and 2 2 W(cid:99)s are transverse foliations with 1-dimensional leaves. Each lamination W(cid:99)u and W(cid:99)s has exactly one closed leaf, the curves Zu and Zs respectively, which are also unstable and stable manifolds for ϕ . t Wcs ✓ r Z Zu C+ Wcu Zs ✓ C r ± Figure 1. The laminations W(cid:99)u and W(cid:99)s. The singular loci C are the labeled circles where the leaves of W(cid:99)u and W(cid:99)s become tangent. For η > 0 we deﬁne two open subsets T (δ , η ) and T (δ , η ) of T (δ ) as follows: 0 in 2 0 out 2 0 2 T (δ , η ) := {v ∈ T (δ ) : a(v) < 0 & |b(v)| ≤ η }, in 2 0 2 0 and T (δ , η ) := {v ∈ T (δ ) : a(v) > 0 & |b(v)| ≤ η }, out 2 0 2 0 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: K.

text: BURNS,

text: H.

text: MASUR,

text: C.

text: MATHEUS

text: AND

text: A.

text: WILKINSON

text: We

text: will

text: work

text: with

text: δ

text: δ

text: so

text: that

text: for

text: δ(p)

text: δ

text: we

text: have

text: δ(p)

text: δ(p)

text: and

text: χ(p)

text: r

text: where

text: χ

text: is

text: the

text: function

text: appearing

text: in

text: Proposition

text: Let

text: T

text: (δ

text: T

text: be

text: the

text: torus

text: consisting

text: of

text: all

text: unit

text: tangent

text: vectors

text: to

text: S

text: with

text: footpoint

text: in

text: Z(δ

text: T

text: (δ

text: T

text: S.

text: This

text: torus

text: is

text: transverse

text: to

text: the

text: vector

text: ﬁeld

text: ϕ˙,

text: except

text: at

text: the

text: two

text: circles

text: C+

text: {J∇δ(p)

text: p

text: Z},

text: and

text: C

text: {−J∇δ(p)

text: p

text: Z}.

text: Let

text: and

text: be

text: the

text: laminations

text: of

text: T

text: (δ

text: obtained

text: by

text: intersecting

text: leaves

text: of

text: the

text: weak

text: foliations

text: Wcu

text: and

text: Wcs

text: with

text: T

text: (δ

text: On

text: T

text: (δ

text: )\(C+∪C

text: the

text: laminations

text: and

text: are

text: transverse

text: foliations

text: with

text: leaves.

text: Each

text: lamination

text: and

text: has

text: exactly

text: one

text: closed

text: leaf,

text: the

text: curves

text: Zu

text: and

text: Zs

text: respectively,

text: which

text: are

text: also

text: unstable

text: and

text: stable

text: manifolds

text: for

text: ϕ

text: t

text: Wcs

text: r

text: Z

text: Zu

text: C+

text: Wcu

text: Zs

text: C

text: r

text: Figure

text: The

text: laminations

text: and

text: The

text: singular

text: loci

text: C

text: are

text: the

text: labeled

text: circles

text: where

text: the

text: leaves

text: of

text: and

text: become

text: tangent.

text: For

text: η

text: we

text: deﬁne

text: two

text: open

text: subsets

text: T

text: (δ

text: η

text: and

text: T

text: (δ

text: η

text: of

text: T

text: (δ

text: as

text: follows:

text: in

text: out

text: T

text: (δ

text: η

text: {v

text: T

text: (δ

text: a(v)

text: |b(v)|

text: η

text: in

text: and

text: T

text: (δ

text: η

text: {v

text: T

text: (δ

text: a(v)

text: |b(v)|

text: η

text: out

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: RATES OF MIXING 41 Fix Θ ⊂ T 1S the thick part deﬁned by Θ = {v ∈ T 1S : δ(v) ≥ δ }. 2 Note that Σ ∪ Σ ⊂ Θ. Lemma 5.6 implies that there exists U > 0 such that for in out 0 any v ∈ Θ, there exists w ∈ Wu(v, η ) and t ∈ (0, U ] such that ϕ (w) ∈ T (δ , η ). 0 0 t in 2 0 Let s be the maximum time needed for a piece of unstable interval to double in length 0 under ϕ . Let U = U + s + 2δ . Denote by Zcs the singular set consisting of all vectors t 0 0 2 v ∈ T 1S with δ(v) ≤ δ and such that ϕ (v) hits the cusp in time t ≤ δ ; that is: 2 t 2 Zcs := ϕ (Zs). [0,δ2] We begin by chopping Zu into a collection G of intervals of length in [η , 2η ). We 0 0 0 say that an open interval G ⊂ Zu is active gap interval at time t ≥ 0 if ϕ (G)∩Zcs = ∅ [0,t] and |ϕ (G)| ∈ [η , 2η ). Thus G consists of active gap intervals at time 0. t 0 0 0 Recall from Corollary 3.4 that for all v ∈ Σ \ Zs, ϕ (v) ∈ Θ. This implies that if in 2δ2 G ⊂ Zu is any piece of unstable manifold of length less than η such that: 0 • ϕ (G) ∩ Σ (cid:54)= ∅, and t0 in • πcs(ϕ (G)) contains a fundamental interval, t0 then there exists t ∈ (0, 2δ ) such that G is an active gap interval at time t + t . 1 2 0 1 We now describe an algorithm for evolving an active gap interval to produce new active gap intervals and other intervals called border intervals. Let G ⊂ Zu be an active gap interval at some time t ≥ 0. We then ﬂow G forward 0 until the ﬁrst t > t when one of two things happens: 0 (a) |ϕ (G)| = 2η , or t 0 (b) ϕ (G) ∩ Zs (cid:54)= ∅. t Either (a) or (b) will occur within time s . 0 If (a) happens ﬁrst, we chop G into two new gap intervals, G and G , so that 1 2 |ϕ (G )| = |ϕ (G )| = η . We say that G and G are born and become active at time t 1 t 2 0 1 2 t and write t (G ) = t (G ) = t (G ) = t (G ) = t. Note that G = G ∪ {v } ∪ G , b 1 a 1 a 2 b 2 1 0 2 where v is the point where the interval G is cut. Since distortion is bounded by 4/3 on 0 intervals of length ≤ 2η , we have: 0 (cid:20) (cid:21) (cid:20) (cid:21) |G | 3 4 |G | 3 2 1 i (42) ∈ , , and ∈ , , i = 1, 2. |G | 4 3 |G| 8 3 2 We say that G is inactive in the time interval [t, ∞), and that G , G are inactive in 1 2 the time interval [0, t). If (b) happens ﬁrst, then G gives birth to two border intervals B and B and two gap 1 2 intervals G , G as follows. Let v ∈ G be the unique point satisfying ϕ (v) ∈ Zs. Let 1 2 t X and X be the components of G \ {v} that lie to the left and right of v, respectively. 1 2 For i = 1, 2, let B ⊂ X be the largest interval satisfying: i i • {v} ∪ B is a closed interval, i • πcsϕ (B ) is a countable union of fundamental intervals, and t i • πcsϕ (G ) contains exactly one fundamental interval, where G = X \ B . t i i i i For i = 1, 2, we deﬁne the birthday of B and G to be t (B ) = t (G ) = t. Let t (G ) i i b i b i a i to be the ﬁrst time such that ϕ is an active interval. Note that since ϕ (G ) meets ta(Gi) t i ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: RATES

text: OF

text: MIXING

text: Fix

text: Θ

text: T

text: the

text: thick

text: part

text: deﬁned

text: by

text: Θ

text: {v

text: T

text: δ(v)

text: δ

text: Note

text: that

text: Σ

text: Σ

text: Θ.

text: Lemma

text: implies

text: that

text: there

text: exists

text: U

text: such

text: that

text: for

text: in

text: out

text: any

text: v

text: Θ,

text: there

text: exists

text: w

text: Wu(v,

text: η

text: and

text: t

text: U

text: such

text: that

text: ϕ

text: (w)

text: T

text: (δ

text: η

text: t

text: in

text: Let

text: s

text: be

text: the

text: maximum

text: time

text: needed

text: for

text: a

text: piece

text: of

text: unstable

text: interval

text: to

text: double

text: in

text: length

text: under

text: ϕ

text: Let

text: U

text: U

text: s

text: 2δ

text: Denote

text: by

text: Zcs

text: the

text: singular

text: set

text: consisting

text: of

text: all

text: vectors

text: t

text: v

text: T

text: with

text: δ(v)

text: δ

text: and

text: such

text: that

text: ϕ

text: (v)

text: hits

text: the

text: cusp

text: in

text: time

text: t

text: δ

text: that

text: is:

text: t

text: Zcs

text: ϕ

text: (Zs).

text: We

text: begin

text: by

text: chopping

text: Zu

text: into

text: a

text: collection

text: G

text: of

text: intervals

text: of

text: length

text: in

text: [η

text: 2η

text: We

text: say

text: that

text: an

text: open

text: interval

text: G

text: Zu

text: is

text: active

text: gap

text: interval

text: at

text: time

text: t

text: if

text: ϕ

text: (G)∩Zcs

text: and

text: |ϕ

text: (G)|

text: [η

text: 2η

text: Thus

text: G

text: consists

text: of

text: active

text: gap

text: intervals

text: at

text: time

text: t

text: Recall

text: from

text: Corollary

text: that

text: for

text: all

text: v

text: Σ

text: Zs,

text: ϕ

text: (v)

text: Θ.

text: This

text: implies

text: that

text: if

text: in

text: G

text: Zu

text: is

text: any

text: piece

text: of

text: unstable

text: manifold

text: of

text: length

text: less

text: than

text: η

text: such

text: that:

text: ϕ

text: (G)

text: Σ

text: and

text: in

text: πcs(ϕ

text: (G))

text: contains

text: a

text: fundamental

text: interval,

text: then

text: there

text: exists

text: t

text: 2δ

text: such

text: that

text: G

text: is

text: an

text: active

text: gap

text: interval

text: at

text: time

text: t

text: t

text: We

text: now

text: describe

text: an

text: algorithm

text: for

text: evolving

text: an

text: active

text: gap

text: interval

text: to

text: produce

text: new

text: active

text: gap

text: intervals

text: and

text: other

text: intervals

text: called

text: border

text: intervals.

text: Let

text: G

text: Zu

text: be

text: an

text: active

text: gap

text: interval

text: at

text: some

text: time

text: t

text: We

text: then

text: ﬂow

text: G

text: forward

text: until

text: the

text: ﬁrst

text: t

text: t

text: when

text: one

text: of

text: two

text: things

text: happens:

text: (a)

text: |ϕ

text: (G)|

text: 2η

text: or

text: t

text: (b)

text: ϕ

text: (G)

text: Zs

text: t

text: Either

text: (a)

text: or

text: (b)

text: will

text: occur

text: within

text: time

text: s

text: If

text: (a)

text: happens

text: ﬁrst,

text: we

text: chop

text: G

text: into

text: two

text: new

text: gap

text: intervals,

text: G

text: and

text: G

text: so

text: that

text: |ϕ

text: (G

text: |ϕ

text: (G

text: η

text: We

text: say

text: that

text: G

text: and

text: G

text: are

text: born

text: and

text: become

text: active

text: at

text: time

text: t

text: t

text: t

text: and

text: write

text: t

text: (G

text: t

text: (G

text: t

text: (G

text: t

text: (G

text: t.

text: Note

text: that

text: G

text: G

text: {v

text: G

text: b

text: a

text: a

text: b

text: where

text: v

text: is

text: the

text: point

text: where

text: the

text: interval

text: G

text: is

text: cut.

text: Since

text: distortion

text: is

text: bounded

text: by

text: on

text: intervals

text: of

text: length

text: 2η

text: we

text: have:

text: |G

text: |G

text: i

text: and

text: i

text: |G

text: |G|

text: We

text: say

text: that

text: G

text: is

text: inactive

text: in

text: the

text: time

text: interval

text: [t,

text: and

text: that

text: G

text: G

text: are

text: inactive

text: in

text: the

text: time

text: interval

text: t).

text: If

text: (b)

text: happens

text: ﬁrst,

text: then

text: G

text: gives

text: birth

text: to

text: two

text: border

text: intervals

text: B

text: and

text: B

text: and

text: two

text: gap

text: intervals

text: G

text: G

text: as

text: follows.

text: Let

text: v

text: G

text: be

text: the

text: unique

text: point

text: satisfying

text: ϕ

text: (v)

text: Zs.

text: Let

text: t

text: X

text: and

text: X

text: be

text: the

text: components

text: of

text: G

text: {v}

text: that

text: lie

text: to

text: the

text: left

text: and

text: right

text: of

text: v,

text: respectively.

text: For

text: i

text: let

text: B

text: X

text: be

text: the

text: largest

text: interval

text: satisfying:

text: i

text: i

text: {v}

text: B

text: is

text: a

text: closed

text: interval,

text: i

text: πcsϕ

text: (B

text: is

text: a

text: countable

text: union

text: of

text: fundamental

text: intervals,

text: and

text: t

text: i

text: πcsϕ

text: (G

text: contains

text: exactly

text: one

text: fundamental

text: interval,

text: where

text: G

text: X

text: B

text: t

text: i

text: i

text: i

text: i

text: For

text: i

text: we

text: deﬁne

text: the

text: birthday

text: of

text: B

text: and

text: G

text: to

text: be

text: t

text: (B

text: t

text: (G

text: t.

text: Let

text: t

text: (G

text: i

text: i

text: b

text: i

text: b

text: i

text: a

text: i

text: to

text: be

text: the

text: ﬁrst

text: time

text: such

text: that

text: ϕ

text: is

text: an

text: active

text: interval.

text: Note

text: that

text: since

text: ϕ

text: (G

text: meets

text: ta(Gi)

text: t

text: i

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 44 K. BURNS, H. MASUR, C. MATHEUS AND A. WILKINSON Proof. Since tR is bounded it suﬃces to ﬁnd λ ∈ (0, 1) such that 1 (cid:12)(cid:91) (cid:12) (cid:12) (cid:12) (cid:12) {B ∈ B∞ : t (B) ≥ k}(cid:12) ≤ Cλk. b 1 But this follows immediately from the construction with λ = λ appearing in Lemma 5.7. 1 0 (cid:5) (cid:83) R We now deﬁne the return time function R: I → . Recall the projection >0 πs : Σ → Zu along the leaves of W(cid:99)s. The ﬁbers of πs are local W(cid:99)s manifolds. out Over each point v ∈ Σ there lies a unique point w(v) ∈ Ws (πs(v)) such that out loc w(v) = ϕ (v), for some small value of r(v). Let Σ = w(Σ ). r(v) out out R Lemma 5.9. The function r : Σ → is uniformly C1+α, and Σ is a C1+α mani- out out fold. The map w : Σ → Σ is a C1+α diﬀeomorphism. The manifold Σ is C1+α out out out foliated by local Ws-leaves: (cid:91) (cid:48) Σ ⊂ Ws (v ), out loc (cid:48)∈ v ∆0 and the projection πs : Σ → Zu along these local leaves is a C1+α submersion. 0 Proof. This follows from the fact that the foliation Ws is uniformly C1+α. (cid:5) (cid:83) R We deﬁne R: I → by R(v) = R (v) + r(ϕ (v)); it has the property that 0 R(cid:83)0(v) (cid:48) ϕ (v) ∈ Σ . Lemma 5.9 implies that for each v ∈ I, there exists a unique v ∈ ∆ R(v) (cid:48) out (cid:48) 0 – namely, v = πsϕ (v) – such that ϕ (v) ∈ Ws (v ), giving conclusion (2). R(v) R(v) loc For ∆ ∈ I, we deﬁne h : ∆ → ∆ to be the inverse of the map F = πcs ◦ ϕ · = j j 0 j j R( ) πs ◦ ϕ · : ∆ → ∆ . This is well-deﬁned, because R( ) j 0 πs(ϕ · (∆ )) = πs(R(πcs ◦ ϕ (∆ ))) = ∆ , R( ) j tb(∆j) j 0 since πcsϕ (∆ ) is a fundamental interval. Since πs is a submersion, and the map tb(∆j) j v (cid:55)→ R(πcsϕ )(v) is a diﬀeomorphism from ∆ onto its image, the composition is a t (B) j b diﬀeomorphism from ∆ to ∆ . Thus its inverse h : ∆ → ∆ is a diﬀeomorphism. This j 0 j 0 j establishes conclusion (3). (cid:48) − Note that |h (v)| ∼ (cid:107)Du ϕ (cid:107) 1, and so conclusion (4) holds. Conclusion (5) j hj(v) R0(hj(v)) follows from the facts that ϕ has bounded distortion and the map πcs is uniformly C1+α. t − Indeed note that the map F = h 1 can also be expressed in the following way. We ﬁx j j some point vˆ ∈ ∆ and consider the image ϕ (∆ ) under the constant time ﬂow ϕ . j R(vˆ) j − R(vˆ) This is a piece of unstable manifold that meets Σ . The map F = h 1 is just the out j j composition F = πcs ◦ ϕ of this ﬂow with the center-stable projection πcs : U → Zu, j R(vˆ) deﬁned in the beginning of the section. This latter projection is a uniformly C1+α submersion and a local diﬀeomorphism when restricted to local unstable manifolds, since the foliation Wcs is uniformly C1+α. Thus h is uniformly C1+α. j Let’s examine the map R ◦ h . Again ﬁx a point vˆ ∈ ∆ , and consider the image j j ϕ (∆ ), which is a piece of unstable manifold meeting Σ at the point ϕ (vˆ). R0(vˆ) j out R R0(vˆ) It follows that there is a uniformly bounded C2 function rˆ: ϕ (∆ ) → such that R0(vˆ) j ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: K.

text: BURNS,

text: H.

text: MASUR,

text: C.

text: MATHEUS

text: AND

text: A.

text: WILKINSON

text: Proof.

text: Since

text: tR

text: is

text: bounded

text: it

text: suﬃces

text: to

text: ﬁnd

text: λ

text: such

text: that

text: {B

text: B∞

text: t

text: (B)

text: Cλk.

text: b

text: But

text: this

text: follows

text: immediately

text: from

text: the

text: construction

text: with

text: λ

text: λ

text: appearing

text: in

text: Lemma

text: R

text: We

text: now

text: deﬁne

text: the

text: return

text: time

text: function

text: R:

text: I

text: Recall

text: the

text: projection

text: πs

text: Σ

text: Zu

text: along

text: the

text: leaves

text: of

text: The

text: ﬁbers

text: of

text: πs

text: are

text: local

text: manifolds.

text: out

text: Over

text: each

text: point

text: v

text: Σ

text: there

text: lies

text: a

text: unique

text: point

text: w(v)

text: Ws

text: (πs(v))

text: such

text: that

text: out

text: loc

text: w(v)

text: ϕ

text: (v),

text: for

text: some

text: small

text: value

text: of

text: r(v).

text: Let

text: Σ

text: w(Σ

text: r(v)

text: out

text: out

text: R

text: Lemma

text: The

text: function

text: r

text: Σ

text: is

text: uniformly

text: and

text: Σ

text: is

text: a

text: mani-

text: out

text: out

text: fold.

text: The

text: map

text: w

text: Σ

text: Σ

text: is

text: a

text: diﬀeomorphism.

text: The

text: manifold

text: Σ

text: is

text: out

text: out

text: out

text: foliated

text: by

text: local

text: Ws-leaves:

text: Σ

text: Ws

text: (v

text: out

text: loc

text: v

text: ∆0

text: and

text: the

text: projection

text: πs

text: Σ

text: Zu

text: along

text: these

text: local

text: leaves

text: is

text: a

text: submersion.

text: Proof.

text: This

text: follows

text: from

text: the

text: fact

text: that

text: the

text: foliation

text: Ws

text: is

text: uniformly

text: R

text: We

text: deﬁne

text: R:

text: I

text: by

text: R(v)

text: R

text: (v)

text: r(ϕ

text: (v));

text: it

text: has

text: the

text: property

text: that

text: ϕ

text: (v)

text: Σ

text: Lemma

text: implies

text: that

text: for

text: each

text: v

text: I,

text: there

text: exists

text: a

text: unique

text: v

text: R(v)

text: out

text: namely,

text: v

text: πsϕ

text: (v)

text: such

text: that

text: ϕ

text: (v)

text: Ws

text: (v

text: giving

text: conclusion

text: R(v)

text: R(v)

text: loc

text: For

text: I,

text: we

text: deﬁne

text: h

text: to

text: be

text: the

text: inverse

text: of

text: the

text: map

text: F

text: πcs

text: ϕ

text: j

text: j

text: j

text: j

text: R(

text: πs

text: ϕ

text: This

text: is

text: well-deﬁned,

text: because

text: R(

text: j

text: πs(ϕ

text: πs(R(πcs

text: ϕ

text: R(

text: j

text: tb(∆j)

text: j

text: since

text: πcsϕ

text: is

text: a

text: fundamental

text: interval.

text: Since

text: πs

text: is

text: a

text: submersion,

text: and

text: the

text: map

text: tb(∆j)

text: j

text: v

text: R(πcsϕ

text: )(v)

text: is

text: a

text: diﬀeomorphism

text: from

text: onto

text: its

text: image,

text: the

text: composition

text: is

text: a

text: t

text: (B)

text: j

text: b

text: diﬀeomorphism

text: from

text: to

text: Thus

text: its

text: inverse

text: h

text: is

text: a

text: diﬀeomorphism.

text: This

text: j

text: j

text: j

text: establishes

text: conclusion

text: Note

text: that

text: |h

text: (v)|

text: ϕ

text: and

text: so

text: conclusion

text: holds.

text: Conclusion

text: j

text: hj(v)

text: follows

text: from

text: the

text: facts

text: that

text: ϕ

text: has

text: bounded

text: distortion

text: and

text: the

text: map

text: πcs

text: is

text: uniformly

text: t

text: Indeed

text: note

text: that

text: the

text: map

text: F

text: h

text: can

text: also

text: be

text: expressed

text: in

text: the

text: following

text: way.

text: We

text: ﬁx

text: j

text: j

text: some

text: point

text: vˆ

text: and

text: consider

text: the

text: image

text: ϕ

text: under

text: the

text: constant

text: time

text: ﬂow

text: ϕ

text: j

text: R(vˆ)

text: j

text: R(vˆ)

text: This

text: is

text: a

text: piece

text: of

text: unstable

text: manifold

text: that

text: meets

text: Σ

text: The

text: map

text: F

text: h

text: is

text: just

text: the

text: out

text: j

text: j

text: composition

text: F

text: πcs

text: ϕ

text: of

text: this

text: ﬂow

text: with

text: the

text: center-stable

text: projection

text: πcs

text: U

text: Zu,

text: j

text: R(vˆ)

text: deﬁned

text: in

text: the

text: beginning

text: of

text: the

text: section.

text: This

text: latter

text: projection

text: is

text: a

text: uniformly

text: submersion

text: and

text: a

text: local

text: diﬀeomorphism

text: when

text: restricted

text: to

text: local

text: unstable

text: manifolds,

text: since

text: the

text: foliation

text: Wcs

text: is

text: uniformly

text: Thus

text: h

text: is

text: uniformly

text: j

text: Let’s

text: examine

text: the

text: map

text: R

text: h

text: Again

text: ﬁx

text: a

text: point

text: vˆ

text: and

text: consider

text: the

text: image

text: j

text: j

text: ϕ

text: which

text: is

text: a

text: piece

text: of

text: unstable

text: manifold

text: meeting

text: Σ

text: at

text: the

text: point

text: ϕ

text: (vˆ).

text: j

text: out

text: R

text: It

text: follows

text: that

text: there

text: is

text: a

text: uniformly

text: bounded

text: function

text: rˆ:

text: ϕ

text: such

text: that

text: j

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: RATES OF MIXING 45 ϕ · sends the piece of unstable manifold ϕ (∆ ) to W(cid:99)u (ϕ (vˆ)) ⊂ Σ . Then rˆ( ) R0(vˆ) j loc R0(vˆ) out R (v) = R (vˆ) + rˆ(ϕ (v)), and so R(v) = R (vˆ) + rˆ(ϕ (v)) + r(ϕ (v)). Thus 0 0 R0(vˆ) 0 R0(vˆ) R0(v) (cid:48) (cid:48) |R (v)| ≤ |rˆ (ϕ (v))|(cid:107)Duϕ (cid:107) R0(vˆ) v R0(vˆ) (cid:48) (cid:0) (cid:48) (cid:1) +|r (ϕ (v))| (cid:107)Duϕ (cid:107)| + (cid:107)ϕ˙(ϕ (v)(cid:107)|rˆ (ϕ (v))|(cid:107)Duϕ (cid:107) (cid:48) R0(v) v R0(v) (cid:48) R0(v) R0(cid:48) (vˆ) v R0(vˆ) = |rˆ (ϕ (v))|(cid:107)Duϕ (cid:107)(1 + |r (ϕ (v))|) + |r (ϕ (v))|(cid:107)Duϕ (cid:107)|. R0(vˆ) (cid:48) (cid:48)v R0(vˆ) R0(v) (cid:48) R0(v) v R0(v) − The derivatives r and rˆ are uniformly bounded. Since |h (v)| (cid:16) (cid:107)Du ϕ (cid:107) 1, j hj(v)(cid:48) R0(hj(v)) we obtain that there exists a uniform constant C ≥ 1 such that |(R ◦ h ) | ≤ C, for all j j. This gives conclusion (6). Since the function r is bounded, Lemma 5.8 implies that for each k > 0, we have (cid:12)(cid:110) (cid:91) (cid:111)(cid:12) (cid:12) (cid:12) (cid:12) v ∈ I : R(v) ≥ k (cid:12) ≤ Cλk; (cid:48) since |h | (cid:16) |∆ |, this gives conclusion (7) of Theorem 5.1, where (cid:15) > 0 is chosen so that j j λ exp(C(cid:15)) < 1. Finally we verify that the UNI Condition in conclusion (8) holds. This is a direct consequence of the fact that ϕ preserves a contact 1-form ω, which implies that the t foliations Ws and Wu are not jointly integrable. The details are carried out in Lemma 12 of [1] (in the Axiom A context) and Lemma 4.2 and Corollary 4.3 of [4] (close to the current context). (cid:5) References [1] V. Arau´jo, O. Butterley and P. Varandas, Open sets of Axiom A ﬂows with exponentially mixing attractors, Proc. Amer. Math. Soc. 144 (2016), no. 7, 2971–2984. [2] A. Avila and S. Gou¨ezel, Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichmu¨ller ﬂow, Ann. of Math. 178 (2013), 385–442. [3] A. Avila, S. Gou¨ezel and J.-C. Yoccoz, Exponential mixing for the Teichmu¨ller ﬂow. Publ. Math. Inst. Hautes E´tudes Sci. 104 (2006), 143–211. [4] V. Arau´jo and I. Melbourne, Exponential decay of correlations for nonuniformly hyperbolic ﬂows with a C1+α stable foliation, including the classical Lorenz attractor, preprint. arXiv:1504.04316 (2015). [5] V. Baladi and B. Vall´ee, Exponential decay of correlations for surface semi-ﬂows without ﬁnite Markov partitions. Proc. Amer. Math. Soc. 133 (2005), no. 3, 865–874. [6] V. Baladi, M. Demers and C. Liverani, Exponential Decay of Correlations for Finite Horizon Sinai Billiard Flows, preprint. arXiv:1506.02836 (2015). [7] K. Burns, H. Masur, C. Matheus, and A. Wilkinson, Rates of mixing for the Weil-Petersson geodesic ﬂow I: no rapid mixing in non-exceptional moduli spaces, preprint. arXiv:1312.6012 (2015). [8] K. Burns, H. Masur, C. Matheus, and A. Wilkinson, Calculating the (cid:63) connection, preprint, http: //math.uchicago.edu/~wilkinso/papers/connection.pdf. [9] K. Burns, H. Masur and A. Wilkinson, The Weil-Petersson geodesic ﬂow is ergodic. Ann. of Math. (2) 175 (2012), no. 2, 835–908. [10] D. Dolgopyat, On decay of correlations in Anosov ﬂows. Ann. of Math. (2) 147 (1998), no. 2, 357–390. [11] U. Hamensta¨dt, Teichmueller ﬂow and Weil-Petersson ﬂow, preprint. arXiv:1505.01113 (2015). [12] E. Heintze and C.-H. Im Hof, Geometry of horospheres. J. Diﬀerential Geom. 12 (1977) 481–491. [13] O. Kowalski, Curvature of the Induced Riemannian Metric on the Tangent Bundle of a Riemannian Manifold, J. Reine Angew. Math. 250 (1971), 124–129.

text: RATES

text: OF

text: MIXING

text: ϕ

text: sends

text: the

text: piece

text: of

text: unstable

text: manifold

text: ϕ

text: to

text: (ϕ

text: (vˆ))

text: Σ

text: Then

text: rˆ(

text: j

text: loc

text: out

text: R

text: (v)

text: R

text: (vˆ)

text: rˆ(ϕ

text: (v)),

text: and

text: so

text: R(v)

text: R

text: (vˆ)

text: rˆ(ϕ

text: (v))

text: r(ϕ

text: (v)).

text: Thus

text: |R

text: (v)|

text: |rˆ

text: (ϕ

text: v

text: +|r

text: (ϕ

text: (v))|

text: (ϕ

text: v

text: (vˆ)

text: v

text: |rˆ

text: (ϕ

text: |r

text: (ϕ

text: (v))|)

text: |r

text: (ϕ

text: v

text: The

text: derivatives

text: r

text: and

text: rˆ

text: are

text: uniformly

text: bounded.

text: Since

text: |h

text: (v)|

text: ϕ

text: j

text: we

text: obtain

text: that

text: there

text: exists

text: a

text: uniform

text: constant

text: C

text: such

text: that

text: |(R

text: h

text: C,

text: for

text: all

text: j

text: j.

text: This

text: gives

text: conclusion

text: Since

text: the

text: function

text: r

text: is

text: bounded,

text: Lemma

text: implies

text: that

text: for

text: each

text: k

text: we

text: have

text: v

text: I

text: R(v)

text: k

text: Cλk;

text: since

text: |h

text: this

text: gives

text: conclusion

text: of

text: Theorem

text: where

text: is

text: chosen

text: so

text: that

text: j

text: j

text: λ

text: Finally

text: we

text: verify

text: that

text: the

text: UNI

text: Condition

text: in

text: conclusion

text: holds.

text: This

text: is

text: a

text: direct

text: consequence

text: of

text: the

text: fact

text: that

text: ϕ

text: preserves

text: a

text: contact

text: ω,

text: which

text: implies

text: that

text: the

text: t

text: foliations

text: Ws

text: and

text: Wu

text: are

text: not

text: jointly

text: integrable.

text: The

text: details

text: are

text: carried

text: out

text: in

text: Lemma

text: of

text: (in

text: the

text: Axiom

text: A

text: context)

text: and

text: Lemma

text: and

text: Corollary

text: of

text: (close

text: to

text: the

text: current

text: context).

text: References

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