Thermodynamic formalism for some systems with countable Markov structures

0.14 MB·

by  Michael Jakobson

Thermodynamic formalism for some systems
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1 with countable Markov structures
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n
a Michael Jakobson
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January 25, 2016
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To thememoryofDmitryViktorovichAnosov
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t Abstract
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Westudyergodicpropertiesofcertainpiecewisesmoothtwo-dimensional
[
systemsbyconstructingcountableMarkovpartitions. Usingthermodynamic

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formalismweproveexponentialdecayofcorreleations. Thatextendsthere-
v
3 sultsof[8],[9]. OurapproachismotivatedbytheoriginalmethodofAnosov
1 andSinaifrom[5].
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6
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1 Motivation: Folklore Theorem in dimension 1
.
1
0
6 Awell-knownFolkloreTheoreminone-dimensionaldynamicscanbeformulated
1
: as follows.
v
i
X FolkloreTheorem. LetI =[0,1]betheunitinterval,andsuppose{I ,I ,...}
1 2
r is a countable collection of disjoint open subintervals of I such that I has the
a i i
full Lebesgue measure in I. Suppose there are constants K >1 andSK >0 and
0 1
mappings f :I →I satisfyingthefollowingconditions.
i i
1. f extends to a C2 diffeomorphism from the closure of I onto [0,1], and
i i
inf |Df (z)|>K forall i.
z∈Ii i 0
|D2f (z)|
2. sup i |I |<K for alli.
z∈Ii |Dfi(z)| i 1
1
Then, the mappingF(z) defined by F(z)= f (z) for z∈I, has a uniqueinvariant
i i
ergodicprobabilitymeasure m equivalentto LebesguemeasureonI.
For the proof of the Folklore theorem , the ergodic properties of m and the
historyofthequestionseeforexample[3]and [15].
In [8] , [9] the Folklore Theorem was generalized to two-dimensional maps F
which piecewise coincide with certain hyperbolic diffeomorphisms f . As in the
i
one-dimensional situation there is an essential difference between a finite and an
infinitenumberof f . Inthecaseofaninfinitenumberof f ,theirderivativesgrow
i i
withiand relationsbetween first and secondderivativesbecomecrucial.
Models with infinitely many f appear when we study non-hyperbolic systems,
i
suchasquadratic-likemapsindimension1,andHenon-likemapsindimension2.
2 Model under consideration. Geometric and hy-
perbolicity conditions
1. As in [8] , [9] we consider the following 2-d model. Let Q be the unit
square. Let x = {E ,E ,...,} be a countable collection of closed curvi-
1 2
linear rectangles in Q. Assume that each E lies inside a domain of def-
i
inition of a C2 diffeomorphism f which maps E onto its image S ⊂ Q.
i i i
We assume each E connects the top and the bottom of Q. Thus each E is
i i
bounded from above and from below by two subintervals of the line seg-
ments{(x,y):y=1, 0≤x≤1}and{(x,y):y=0, 0≤x≤1}. Hyperbol-
icityconditionsthatweformulatebelowimplythattheleftandrightbound-
aries of E are graphs of smooth functions x(i)(y) with dx(i) ≤a where a
i dy
(cid:12) (cid:12)
isareal numbersatisfying0<a <1. (cid:12) (cid:12)
(cid:12) (cid:12)
The images f (E)=S are narrow strips connecting the left and right sides
i i i
ofQ and thattheyare boundedon theleftand right bythetwo subintervals
ofthelinesegments{(x,y):x=0, 0≤y≤1}and{(x,y):x=1, 0≤y≤1}
andaboveandbelowbythegraphsofsmoothfunctionsYi(X),| dY(i) |≤a .
dX
Weare sayingthatE′sarefullheight inQ whiletheS′sare fullwidthinQ.
i i
2. For z ∈ Q, let ℓ be the horizontal line through z. We define d (E) =
z z i
diam(ℓ E ), d =max d (E ), d =min d (E). We assume
z i i,max z∈Q z i i,min z∈Q z i
thefolloTwing
2
Geometricconditions.
G1. Fori6= j holdsint E ∩int E =0/ int S ∩int S =0/ .
i j i j
G2. mes(Q\∪ intE )=0 wheremes standsforLebesguemeasure.
i i
G3. −(cid:229) d logd <¥ .
i i,max i,min
3. In the standard coordinate system for a map F :(x,y)→(F (x,y),F (x,y))
1 2
weuse DF(x,y) to denotethedifferential of F at somepoint (x,y) and F ,
jx
F , F , F ,etc., forpartial derivativesofF , j =1,2.
jy jxx jxy j
LetJ (z)=|F (z)F (z)−F (z)F (z)| betheabsolutevalueoftheJaco-
F 1x 2y 1y 2x
biandeterminantofF at z.
Hyperbolicityconditions.
Thereexistconstants0<a <1and K >1 suchthat foreach i themap
0
F(z)= f (z)forz∈E
i i
satisfies
H1. |F (z)|+a |F (z)|+a 2|F (z)|≤a |F (z)|
2x 2y 1y 1x
H2. |F (z)|−a |F (z)|≥K .
1x 1y 0
H3. |F (z)|+a |F (z)|+a 2|F (z)|≤a |F (z)|
1y 2y 2x 1x
H4. |F (z)|−a |F (z)|≥J (z)K .
1x 2x F 0
Fora real number0<a <1, wedefine thecones
Kau ={(v ,v ):|v |≤a |v |}
1 2 2 1
Kas ={(v ,v ):|v |≤a |v |}
1 2 1 2
and the corresponding cone fields Kau (z),Kas (z) in the tangent spaces at
pointsz∈R2.
The following proposition proved in [9] relates conditions H1-H4 above
with the usual definition of hyperbolicity in terms of cone conditions. It
shows that conditions H1 and H2 imply that the Kau cone is mapped into
3
itselfby DF and expandedby afactor nosmallerthanK whileH3 and H4
0
imply that the Kas cone is mapped into itself by DF−1 and expanded by a
factornosmallerthanK .
0
Unlessotherwisestated,weusethemaxnormonR2,|(v ,v )|=max(|v |,|v |).
1 2 1 2
Proposition2.1 UnderconditionsH1-H4 above, wehave
DF(Kau )⊆Kau (1)
v∈Kau ⇒|DFv|≥K |v| (2)
0
DF−1(Kas )⊆Kas (3)
v∈Kas ⇒|DF−1v|≥K |v| (4)
0
Remark2.2 Thefirstversionofhyperbolicityconditionsappearedin[14].
It was developed in particular in [4] and [7] . Here we use hyperbolicity
conditionsfrom [9]. In [8] we used hyperolicity conditionsfrom [4] which
implied the invariance of cones and uniform expansion with respect to the
sumnorm|v|=|v |+|v |.
1 2
4. Themap
F(z)= f (z)forz∈int E
i i
isdefinedalmosteverywhereonQ. LetQ˜ = intE ,and,defineQ˜ ,n>0,
0 i i n
inductively by Q˜ =Q˜ F−1Q˜ . Let Q˜ =S Q˜ be the set of points
n 0 n−1 n≥0 n
whose forward orbits alwTays stay in int E .TThen, Q˜ has full Lebesgue
i i
measureinQ, andF mapsQ˜ intoitselSf.
ThehyperbolicityconditionsH1–H4implytheestimateson thederivatives
of the boundary curves of E and S which we described earlier. They also
i i
imply that any intersection f E E is full width in E . Further, E =
i i j j ij
T
4
E f−1E is a full height subrectangle of E and S = f f E is a full
i i j i ij j i ij
wiTdthsubstripinQ.
Givenafinitestringi ...i , wedefineinductively
0 n−1
E =E f−1E .
i0...in−1 i0 i0 i1i2...in−1
\
Then,each setE is afullheightsubrectangleofE .
i0...in−1 i0
Analogously,forastringi ...i wedefine
−m −1
S = f (S E )
i−m...i−1 i−1 i−m...i−2 i−1
\
andgetthatS isafullwidthstripinQ. ItiseasytoseethatS =
i−m...i−1 i−m...i−1
f ◦f ◦...◦f (E )andthat f−1(S )isafull-widthsubstrip
i−1 i−2 i−m i−m...i−1 i0 i−m...i−1
ofE .
i
0
Wealsodefine curvilinearrectangles R by
i−m...i−1,i0...in−1
R =S E
i−m...i−1,i0...in−1 i−m...i−1 i0...in−1
\
Ifthere are no negativeindices then respectiverectangleis full heightin Q.
Forinfinitestrings,wehavethefollowingProposition.
Proposition2.3 Any C1 map F satisfying the above geometric conditions
G1–G3andhyperbolicityconditionsH1–H4hasa ”topologicalattractor”
L = S
i−k...i−1
[ \
...i−n...i−1k≥1
The infinite intersections ¥ S defineC1 curves y(x), |dy/dx|≤a
k=1 i−k...i−1
whicharetheunstablemaTnifoldsforthepointsoftheattractor. Theinfinite
intersections ¥ E define C1 curves x(y), |dx/dy| ≤ a which are
k=1 i0...ik−1
thestablemanTifoldsforthepointsoftheattractor. Theinfiniteintersections
¥ ¥
R
i−m...i−1,i0...in−1
\ \
m=1n=1
definepointsoftheattractor.
5
Proposition 2.3 is a well known fact in hyperbolic theory. For example it
follows from Theorem 1 in [4]. See also [10]. The union of the stable
manifolds has full measure in Q. The trajectories of all points in this set
convergeto L . Thatis thereason to call L atopologicalattractor.
5. An F−invariant Borel probablility measure m on Q is called a Sinai−
Ruelle−Bowen measure (or SRB-measure) for F if m is ergodic and there
is a set A ⊂ Q of positive Lebesgue measure such that for x ∈ A and any
continuousreal-valuedfunctionf :Q→R, wehave
lim 1n(cid:229)−1f (Fkx)= f dm . (5)
n→¥ n Z
k=0
Existence of an SRB measure is a much stronger result, than 2.3. It allows
to describe statistical properties of trajectories in a set of positive phase
volume. It requires someadditionalassumptions.
3 Distortion conditions
As we have a countable number of domains the derivatives of f grow. We for-
i
mulate certain assumptions on the second derivatives. We use the distance func-
tion d((x,y),(x ,y ))=max(|x−x |,|y−y |) associated with the norm |v|=
1 1 1 1
max(|v |,|v |) onvectorsv=(v ,v ).
1 2 1 2
As above, for a point z ∈ Q, let l denote the horizontal line through z, and
z
if E ⊆Q, let d (E) denote the diameter of the horizontal section l E. We call
z z
d (E) thez−widthofE. T
z
In given coordinate systems we write f (x,y) = (f (x,y), f (x,y)). We use
i i1 i2
f , f , f , f , etc. forpartial derivativesof f , j=1,2.
ijx ijy ijxx ijxy ij
Wedefine
|D2f (z)|= max | f (z)|.
i
ijkl
j=1,2,(k,l)=(x,x),(x,y),(y,y)
Nextweformulatedistortionconditionswhichareusedtocontrolthefluctua-
tion of the derivatives of iterates of F along unstable manifolds, and to construct
Sinai localmeasures.
SupposethereisaconstantC >0suchthatthefollowingdistortionconditions
0
hold
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|D2f (z)|
D1. sup i d (E )<C .
z∈Ei,i≥1 | fi1x(z)| z i 0
Ourconditionsimplythefollowingtheoremprovedin[8], [9].
Theorem 3.1 Let F be a piecewise smooth mappingas above satisfyingthe geo-
metricconditions G1–G3, the hyperbolicityconditionsH1–H4 and the distortion
conditionD1.
Then,F hasanSRBmeasurem supportedonL whosebasinhasfullLebesgue
measureinQ. Dynamicalsystem(F,m ) satisfiesthefollowingproperties.
1. (F,m )is measure-theoreticallyisomorphictoa Bernoullishift.
2. F hasfiniteentropywith respecttothemeasure m , andtheentropyformula
holds
hm (F)= log|DuF|dm (6)
Z
whereDuF(z)isthenormofthederivativeofF intheunstabledirectionat
z.
3.
1
hm (F)= lim log|DFn(z)| (7)
n→¥ n
where thelatter limitexists for Lebesguealmost allz and is independentof
suchz.
4 Additional hyperbolicityand distortion conditions
and statement of the main theorem
When applying thermodynamic formalism to hyperbolic attractors one considers
the function f (z) = −log(DuF(z)). Thermodynamic formalism is based on the
fact that the pullback of f (z) into a symbolic space determined by some Markov
partitionis alocallyHo¨lderfunction.
WeproveHo¨lderpropertyoff (z)assuminganextrahyperbolicitycondition,and
adistortionconditionD2 strongerthan D1.
Hyperbolicity conditionH5.
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H5. 1 +a 2 <1.
K2
0
DistortionconditionD2.
|D2f (z)|
D2. sup i <C .
z∈Ei,i≥1 | fi1x(z)| 0
Remark4.1 Condition D2 is too strong to be useful for quadratic-like systems.
In dimension 1 it reads as | Fixx | < c instead of | Fixx | < c. However instead of
Fix Fi2x
D2 one can assume additional hyperbolicity conditions, which can be vaguely
formulated as ”contraction of f grows faster than expansion” . That approach
i
willbediscussedina forthcomingpaper.
AssumingadditionallyH5 and D2 we provethat Ho¨lderfunctionshaveexponen-
tialdecay ofcorrelations.
Let Hg be the space of functions on Q satisfying Ho¨lder property with exponent
g
|f (x)−f (y)|≤c|x−y|g
Thenthefollowingtheoremholds.
Theorem 4.2 Let F be a piecewise smooth mappingas above satisfyingthe geo-
metricconditions G1–G3, the hyperbolicityconditionsH1–H5 and the distortion
condition D2. Then (F,m ) has exponential decay of correlations for f ,y ∈Hg .
Namelythereexist h (g )<1 andC=C(f ,y )suchthat
| f (y ◦Fn)dm − f dm y dm |<Ch n (8)
R R R
5 Ho¨lder properties of log(DuF(z))
1. AlthoughMarkovpartitionsarepartitionsoftheattractor,weneedtocheck
Ho¨lderpropertyonactualtwo-dimensionalcurvilinearrectanglesR .
i−m...i−1,i0...in−1
Wecall respectivepartitionMarkovas well. InourmodelMarkovpartition
consistsofinitialfull heightrectanglesE .
i
We consider rectangles R with m≥0,n≥1. We use notation
i−m...i−1,i0...in−1
8
m=0 if there are no negativecoordinates, which means R =
i−m...i−1,i0...in−1
R is a full height rectangle. The assumption n≥1, means that vari-
i0,...,in−1
ations are measured between points which belong to the same full height
rectangle.
BydefinitionthefunctionlogDuF islocallyHo¨lderifform≥0,n≥1the
variationoflogDuF on R satisfies
i−m...i−1,i0...in−1
var(logDuF)|R <Cq min(m,n) (9)
i−m...i−1,i0...in−1 0
forsomeC>0, q <1.
0
Proposition5.1 logDuF isa locallyHo¨lderfunction.
We proveProposition 5.1 with some q andC determined by hyperbolicity
0
anddistortionconditions.
(a) The sets R are bounded from above and below by some
i−m...i−1,i0...in−1
arcsoftwounstablecurvesG u ,whichareimagesofsomepieces
i−m...i−1
of the top and bottom of Q˜, and from left and right by some arcs of
twostablecurvesG s ,whicharepreimagesofsomepiecestheleft
i0...in−1
and rightboundariesofQ˜ .
LetZ ,Z ∈R betwopointsontheattractor. Weconnect
1 2 i−m...i−1,i0...in−1
Z ,Z by two pieces of their unstable manifolds to two points Z ,Z
1 2 3 4
which belongtothesamestablemanifold. Let
g =g (Z ,Z )⊂Wu(Z ), g =g (Z ,Z )⊂Wu(Z ), g =g (Z ,Z )⊂
1 1 3 1 2 2 4 2 3 3 4
Ws(Z ) berespectivecurvesall locatedinsideR .
3 i−m...i−1,i0...in−1
Weestimate
|logDuF(Z )−logDuF(Z )|≤|logDuF(Z )−logDuF(Z )|+
1 2 1 3
|logDuF(Z )−logDuF(Z )|+|logDuF(Z )−logDuF(Z )|
3 4 4 2
(b) First we estimate |logDuF(Z )−logDuF(Z )|. We connect Z and
1 3 1
Z by a chain of small rectangles R ⊂ R covering g .
3 i−m...i−1,i0...in−1 1
Then |logDuF(Z )−logDuF(Z )| is majorated by the sum of sim-
1 3
ilardifferences forpointsz ,z ∈Wu(Z )⊂R.
1 2 1
BecauseofconeconditionswecanchooserectanglesR=D x×D ysat-
isfying|D y|<a |D x|. Let R beoneofsuch rectangles.
Hyperbolicityconditionsimplythefollowingproperties,see[9].
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i. Any unit vector in Kau at a point z ∈ E , in particular a tangent
i
vectortoWu(z),has coordinates(1,a ) with|a |<a .
z z
ii.
|DuF(z)|=|F (z)+a F (z)| (10)
1x z 1y
iii.
|F |
1y <a (11)
|F |
1x
iv.
|F |
2x <a (12)
|F |
1x
v.
|F | 1
2y < +a 2 (13)
|F | K2
1x 0
Assuming without loss of generality F >0 for all x∈E we get that
1x i
variationoflog|DuF |betweentwopointsz ,z ∈Wu(Z )⊂Requals
1 2 1
F F
1y 1y
log F (z ) 1+a (z ) −log F (z ) 1+a (z ) (14)
1x 1 z1F 1 1x 2 z2F 2
(cid:2) (cid:0) 1x (cid:1)(cid:3) (cid:2) (cid:0) 1x (cid:1)(cid:3)
Wesplititintotwo expressionsand estimateseparately
logF (z )−logF (z ) (15)
1x 1 1x 2
and
F F
1y 1y
log 1+a (z ) −log 1+a (z ) (16)
z 1 z 2
1F 2F
(cid:0) 1x (cid:1) (cid:0) 1x (cid:1)
Werewrite16 as
F F
a 1y(z )−a 1y(z )
log 1+ z1F1x 1 z2F1x 2 (17)
F
(cid:0) 1+az2F11yx(z2) (cid:1)
Asdenominatorofthefractionin17isuniformlyboundedawayfrom
0,westimatethenumeratorandrewriteitasasumoftwoexpressions
|F (z )F (z )−F (z )F (z )|
1y 1 1x 2 1y 2 1x 1
|a | (18)
z
1 F (z )F (z )
1x 1 1x 2
and
|a −a || F1y(z2) | (19)
z1 z2 F1x(z2)
10