Generating Functions for Special Flows over the 1-Step Countable Topological Markov Chains

0.23 MB·English

by  Dawoud Ahmadi Dastjerdi

Generating Functions for Special Flows over the
1-Step Countable Topological Markov Chains
1
1
0
2 D. Ahmadi Dastjerdi

n S. Lamei
a
DepartmentofMathematics, TheUniversityofGuilan,P.O.1914,Rasht,Iran.
J
3 e-mail: [email protected]
2 DepartmentofMathematics, TheUniversityofGuilan,P.O.1914,Rasht,Iran.

e-mail:[email protected]
]
S
D
.
h Abstract
t
a Let Y be a topological Markov chain with finite leading and follower
m
sets. SpecialflowoverY whoseheight functiondependson thetimezero
[ ofelementsofY isconstructed. Thenaformulaforcomputingtheentropy

of thisflow will be given. Asan application, we give alower estimate for
1
the entropy of a class of geodesic flows on the modular surface. We also
v
4 give sufficient conditions to guarantee the existence of a measure with
7 maximal entropy.
3
4
.
1
1 Introduction
0
1
1 Therearetwomainroutinestocomputetheentropyofnon-compactdynamical
: systems. The first is to use (T,ǫ)-spanning sets introduced by Bowenonmetric
v
i of spaces [2, 7, 10], and the second is to use the topological pressure from the
X
thermodynamic formalism [1, 5, 6, 9]. Our concern is the latter and in partic-
r ular, we consider special flows over countable Markov chains. These flows are
a
mainly associated with geodesic flows on non-compact manifolds with negative
curvature. For instance in [3], it has been shown that the geodesic flows on
themodularsurfacecanberepresentedbyspecialflowovercountablealphabet.
However, even in this case, depending on the properties, several definitions for
entropies are given [1, 5, 6, 9].
In this paper we construct a special flow constructed from a certain class
of topological Markov chains. Namely, we let the base Y of the flow be taken
from a 1-step topological Bernoulli scheme TBS with countable states so that
the follower and leading sets are finite. This means that we partition the set of
alphabet to {P1,...,Pm} such that if y = {yi}i∈Z, y′ = {yi′}i∈Z are in Y with
2 NOTATIONS AND MAIN DEFINITIONS 2
y , y′ ∈P then y ,y′ ∈P and y ,y′ ∈P for some 1≤i,j,k ≤m. We call
0 0 k 1 1 i −1 −1 j
any Y satisfying this condition reducible to finite type or briefly RFT.
We let Y to be a 1-step topological Markov chain where this will able us to
define the height function depending only on the zero coordinate of an element
of y where y = {yi}i∈Z. Our main objective is to find the generating function
φ:R→Rdependingontheheightfunction. Thenbyapplyingsomeconditions,
necessary for the results in [9], the topological entropy of the flow is −ln(x )
0
where x is the unique point where φ(x ) = 1. In section 4, we show how this
0 0
is applied in application, by giving some examples which arises in the study of
geodesic flows in the modular surface. Then in Section 5, we show that results
in [8] can be deduced from our method. In Section 6, based on results in [8]
and[9],sufficientconditionsforhavingameasurewithmaximalentropyforthe
flow has been given.
Acknowledgments. We thank S. Savchenko for bringing to our attention a
series which helped us to give the example 3 in section 3. Also we thank M.
Kessebohmer for his useful comments.
2 Notations and main definitions
Now we recall some notations and definitions many adopted from [8]. Let G
be a connected directed graph with a countable vertex set V(G) and edge set
E(G)⊆V(G)×V(G). IfE(G)=V(G)×V(G), thenwedenoteGbyG which
0
is called a complete graph. A path γ with length ℓ(γ)=n in G from v to v is
0 n
the sequence γ =(v ...,v ), n≥1 of vertices of G such that (v ,v )∈E(G)
0 n k k+1
for 0≤k ≤n−1. A path γ =(v ,...,v ), n≥1 in G is called a simple v-cycle
0 n
if v = v = v and v 6= v for 1 ≤ i ≤ n−1. Denote by C(G;v) the set of all
0 n i
simple v-cycles in the graph G.
Let Y(G) = {(...,y ,y ,y ,...) : y ∈ V(G), (y ,y ) ∈ E(G),i ∈ Z}
i−1 i i+1 i i i+1
be the set of two-sided infinite paths in G and the shift transformation T :
Y(G)−→Y(G)isdefinedas(Ty) =y ,fory ∈Y(G)andi∈Z. Thesystem
i i+1
(Y(G),T) is called a countable Topological Markov Chain TMC. In the case of
completegraph,thedynamicalsystem(Y(G ),T)iscalledcountableTopological
0
Bernoulli Scheme TBS.ATMC will be alocal perturbation of a countable TBS
if D =E(G )−E(G) is finite.
0
Let (Y(G),T) be a given TMC. Consider on Y(G) a continuous positive
function f : Y(G) −→ (0,∞) such that ∞ f(Tky) = ∞ f(T−ky) = ∞,
k=1 k=1
y ∈ Y(G). The set of such functions which depend only on zero coordinate y
0
of the sequence y is denoted by Fo(Y(GP)). A good candPidate for such f is a
specialflow. Tobemoreprecise,letY (G)={(y,u): y ∈Y(G), 0≤u≤f(y)}
f
with the points (y,f(y)) and (Ty,0) identified. For 0 ≤ u,u+t ≤ f(y) we let
Tt(y,t)=(y,u+t). The family T ={Tt}, t∈R is a special flow constructed
f f f
over its base Y(G).
Consider the following series
F (x)= xf(v), (1)
f,V
v∈V
X
3 COMPUTING GENERATING FUNCTION 3
which is defined for x ≥ 0. This series is convergent at zero and we will call
r(F ) = sup{x ≥ 0 : xf(v) is convergent} the radius of convergent of
f,V v∈V
F .
f,V
Let f ∈Fo(Y(G)). TPhe generating function of simple v-cycles with respect
to the special flow T constructed over a TMC (Y(G),T) is defined to be the
f
series
φ (x)= xf∗(γ), x≥0 (2)
G,f,w
γ∈CX(G;w)
wheref∗(γ)= n−1f(v ),γ =(v ,...,v ). Theradiusofconvergencer(φ )
i=0 i 0 n G,f,w
∈[0,1) is defined as was defined for (1).
P
3 Computing Generating Function
LetV(G) andE(G) be asabove. Considera “weighted”adjacentmatrixA =
G
[aij], that is, a matrix where aij =xf(vi) if (vi,vj)∈E(G) and zero otherwise.
For each v ∈ V(G), let V+ = {v′ ∈ V(G) : (v,v′) ∈ E(G)} and V− = {v′ ∈
v v
V(G) : (v′,v) ∈ E(G)} be the follower and leading set for v respectively. Set
V = {v ∈ V(G) : ∃v′ ∈ V(G) ∋ (v,v′) 6∈ E(G) or (v′,v) 6∈ E(G)}. Note that
V may be infinite or even equal to V(G). Let ρ be an equivalence relation on
Ve(G) defined by v ∼ρ v′ ⇔ (V+ = V+,V− = V−) and P be the associated
v v′ v v′
peartition. We are interested in cases where |P|<∞.
We alter a bit the above notations and will produce a quotient set for G
which is again a connected directed graph. To achieve that fix w ∈ V(G) and
let W = P ∨{{w},V(G)−{w}} be the set of all non-empty intersections
{w}
of P with the partition {{w},V(G)−{w}}. Then |W | < ∞ and let V =
{w} 0
{w},V ,...,V be the elements of W .
1 m {w}
For v ∈ V define the follower and leading sets for V as V+ = V+ and
i i i v
V− =V− respectively. NotethatanyV ,V+ orV− canbewrittenastheunion
i v i i i
of some of elements of W . Therefore, a directed graph H arises with vertex
{w}
setW andthe edgesetE(H)={(V ,V ):(V ,V )∈W ×W ,(v ,v )∈
{w} i j i j {w} {w} i j
E(G),v ∈ V ,v ∈ V }. We call H the quotient graph for G. Graph H is
i i j j
connected because G is connected.
By reindexing the elements of W , we may assume {V ,...,V } = V+ .
{w} 1 k {w}
Let H be a tree with root {w} and V , ..., V in its second level. We wish
{w} 1 k
to extend H to a tree T whose first two levels are exactly H and any
{w} {w} {w}
path starting from {w} ends at {w}. By this we mean the third level of T
{w}
consists of the follower sets of V ’s, V 6= {w} and 1 ≤ j ≤ k. Again the next
j j
levelconsistsofthe followersetsofverticesofthirdlevelwhicharenot{w}and
so on.
Theorem 1. Let k be the number of vertices at the second level of T . Then
{w}
all the elements of W appear at the vertices of T at most up to level
{w} {w}
m−k+2.
3 COMPUTING GENERATING FUNCTION 4
Proof. Ifk =mwearedone. Soassumek <m. Thenthethirdlevelmusthave
a vertex which is not in {V ,...,V }. Otherwise, that vertex will not appear
1 m
in any level which is in contradiction with the fact that H is connected. By
the samereasoning,anyhigherlevelmusthaveonenew vertexuntilallofthem
have appeared.
This theoremjustifies thatsuchT exists. Because by replacing{w} with
{w}
any other vertex V in W and using the same proof as the above theorem,
{w}
{w} will appear at least once as vertex in a tree with root at V.
The next lemma states that how in our case the computation of generating
functioncanbesimplified. Letα =α (x)= xf(v) andsetα =α (x)=
i i v∈Vi ij ij
α (x) if (V ,V )∈E(H) and zero otherwise.
i i j
P
Lemma 1. Suppose (Y(G),T) is an RFT and f ∈F0(Y(G )) with r(F )>
0 f,V
0. Then there exist series A (x) which are the solution of the follower set of
i
equations
A (x)=α (x)+α (x)A (x)+α (x)A (x)+...+α (x)A (x), (3)
i i0 i1 1 i2 2 im m
for 1≤i≤m so that the generating function for the flow T is
f
φ (x)=α (x)+α (x)A (x)+α (x)A (x)+...+α (x)A (x). (4)
G,f,w 00 01 1 02 2 0m m
Here r(φ )=min{r(A ),...,r(A )}≤r(F ).
G,f,w 1 m f,V
Proof. Let
A (x)= xf∗(γ)
i
vX∈Viγ=(Xv,...,w)
be a series on all paths in G starting at a vertex v ∈V and ending at w. Then
i
A (x) = xf(v) xf∗(γ′)
i
!
vX∈Vi VjX∈Vi+vX′∈Vjγ′=(Xv′,...,w)
= α (x)+α (x)A (x)+α (x)A (x)+...+α (x)A (x).
i0 i1 1 i2 2 im m
m
Since V ∩V =∅ for i6=j, F (x)= α (x). Hence
i j f,V i=1 i
r(Ff,V)=min{Pr(αi),...,r(αm)}
and since each A (x) is a rational map in variables α (x),...,α (x), therefore
i 1 m
min{r(A ),...,r(A )}≤r(F ). Also
1 m f,V
φ (x) = xf∗(γ)
G,f,w
γ∈CX(G;w)
= xf(w) xf∗(γ)
ViX∈Vw+vX∈Viγ=(Xv,...,w)
= α (x)+α (x)A (x)+α (x)A (x)+...+α (x)A (x).
00 01 1 02 2 0m m
By the way A (x) is defined above, r(φ ) = min{r(A (x)),...,r(A (x))}.
i G,f,w 1 m
3 COMPUTING GENERATING FUNCTION 5
Set A(x) = (A (x),...,A (x)) and α(x) = (α (x),...,α (x)) and consider
1 m 1 m
them as column vectors when it applies and let
α (x)−1 α (x) ... α (x)
11 12 1m
α (x) α (x)−1 ... α (x)
21 22 2m
M(x)= . . (5)
.
.
 
 αm1(x) α21(x) ... αmm(x)−1
 
Then statement (3) in the conclusion of Lemma (1) implies
M(x)A(x)=−α(x). (6)
Consider(6) asasetofequationswithunknownA(x). Inthe nexttheorem,we
will find x such that A(x) satisfies (6) and A(x) is a solution of (3), that is, we
will find r(φ ). In fact for x>0, a solution of (6) is a solution of (3) if and
G,f,w
only if A (x)>0.
i
Set x˜ = r(F ) if M(x) is invertible for 0 ≤ x < r(F ), otherwise set
0 f,V f,V
x˜ = inf{x : 0 ≤ x < r(F ), detM(x) = 0}. Since M(0) is invertible and
0 f,V
M(x) is continuous then clearly x˜ >0 if and only if r(F )>0.
0 f,V
Theorem 2. Suppose the hypothesis of Lemma (1) is satisfied and A(x), M(x)
andx˜ areas above. Thenr(φ )=x˜ andifx˜ <r(F ), lim A (x)=
0 G,f,w 0 0 f,V x→x˜0 i
lim φ (x)=∞.
x→x˜0 G,f,w
Proof. Let e = (0,0,...,1,0,...,0) be the unit vector whose ith entry is 1. Let
i
m m
P = { t e : t ≥ 0} and N = { t e : t ≤ 0}. The boundary of
i=1 i i i i=1 i i i
m
P consists of m sets P = { t e : t ≥ 0}, 1 ≤ j ≤ m. These are
P j i=1,i6=j Pi i i
m−1 dimensional manifolds with boundaries. Let M : v(x) 7→ M(x)v(x),
x
S+(x) = M (P) and S (x) P= M (P ). Then ∂P = ∪m P and if M(x) is
x j x j j=1 j
invertible, ∂S+ =∪m S (x).
j=1 j
RecallthatifW is asubspaceofRm ofcodimension1,thenRm\W consists
oftwounboundedcomponents. ButwhenM(x)isinvertible,∂S+(x)isahome-
omorphicimage of sucha W andhence Rm\∂S+(x) consistsof two unbounded
components as well. Also for any x and j we have S (x)∩No ={0} where No
j
is the interior of N. That is because the jth entry of a nonzero vector in S (x)
j
whichisequaltot α (x)+...+t α (x)+t α (x)+...+t α (x)
1 1j j−1 (j−1)j j+1 (j+1)j m mj
for nonzero t ’s is never negative. Hence as far as M(x) is invertible, N lies in
i
one side of Rm\∂S+(x).
Note that M(0) = −Id and hence it is invertible and P = M−1(N). Also
0
by continuity, for small positive x, M(x) remains invertible and in fact P ⊂
M−1(N). It may happen that we have x such that M(x) is not invertible and
x
set as above x˜ to be the smallest positive real such that detM(x˜)=0. Since
0 0
the entries ofM(x)does notdecreaseasx increases,soS+(x )⊆S+(x ) when
1 2
x < x < x˜. Hence M−1S+(x ) ⊆ M−1S+(x ) and then M−1(S+(0)) =
1 2 0 x2 1 x2 2 x
M−1(N)⊆P for 0<x<x˜.
x 0
Inparticular,A(x)=M−1(−α)∈Po wherebyuniquenessofsolutionsA(x)
x
will be the same as (3). We are done if we show that for each i
lim A (x)=∞. (7)
i
x→x˜0
3 COMPUTING GENERATING FUNCTION 6
First note that since A (x) is increasing, the limit exists and M (−α) ∈
i x˜0
T := M (Rm) . The dimension of T is at most m − 1 and we prove (7)
x˜0
by claiming that T dose not intersect Po. Because if (7) is not satisfied, by our
claim, the only possibility is that T is a subspace intersecting ∂P and contain-
ing lim A(x) = (lim A (x),...,lim A (x)). But then for some i,
x→x˜0 x→x˜0 1 x→x˜0 m
A (x˜)=0 which contradicts the fact that A (x˜)>A(x)>0 for x˜ >x>0.
i 0 i 0 0
Nowweprovetheclaim. LetS−(x)=∪m { m t M(x)(−e ):t ≥0}.
j=1 i=1,i6=j i i i
We have S−(0) = P and by similar argument as we did for S+(x), P will be
P
on one side of Rm\S−(x) for small x. Our claim is established if we show that
S+(x)∩Po = ∅ for 0 < x < x˜. If it was not the case, then by continuity
0
there is x , 0 < x < x˜ such that S+(x ) intersects S−(x ) in at least one
1 1 0 1 1
nonzero vector. That means there is v 6= 0 such that v = t M(x )e =
ti≥0 i 1 i
t′M(x )(−e ) or equivalently (t +t′)M(x )e =0 which implies
ti≥0 i 1 i ti,t′i≥0 i i 1 Pi
that M(x) is nonsingular for x <x˜ which is absurd.
P 1 0P
Corollary1. Suppose(Y(G),T)isanRFTandf ∈Fo(Y(G)). Thenφ (x)
G,f,w
is C1.
Proof. This is clear if r(φ )=0. Otherwise the proof follows from the fact
G,f,w
that F (x) is C1 (see proof of Theorem 2 in [8]). Because then α (x) is C1
f,V ij
and since M(x) is invertible, A (x) is a rationalmap onα (x)’s and hence C1.
i ij
Now since φ (x) is a polynomial on α (x)’s and A (x)’s, must be C1.
G,f,w ij i
Stillsomeotherresultsmaybeinteresting. Forinstancefor1≤i≤mwrite
(3) as
−α (x)=α (x)A (x)+...+(α (x)−1)A (x)+...+α (x)A (x),
i0 i1 1 ii i im m
and note that α (x) and A (x) are non-negative. This implies at least one
ij i
coefficientofA (x)onright,inourcase(α (x)−1),mustbenegativeorα (x)<
i ii ii
m
1. So if α (x) > 0 for all i, then F = α (x) is uniformly bounded on
ii f,V i=1 i
the domainofφ (x). Thoughthislastresultholds anytimeifx˜ <r(F ).
G,f,w 0 f,V
P
Remark 1. Let (Y(G),T) be a topological Markov chain and f ∈ Fo(Y(G))
with generating function φ (x). Also suppose h(T ) the topological entropy
G,f,w f
of the flow T on Y(G) is not infinity. Then by a result in [9], for an arbitrary
f
w∈V(G)
h(T )=inf h≥0: e−hf∗(γ) ≤1 .
f
 
 γ∈CX(G;w) 
Since γ∈C(G;w)e−hf∗(γ) =φG,f,w(e−h), wehaveh(Tf)=−ln(xˆf)wherexˆf =
sup{x≥0:φ (x)≤1}. Therefore, the problem of computing h(T ) reduces
P G,f,w f
to find xˆ . By the fact that φ (x) is increasing, then xˆ is either the unique
f G,f,w f
solution of φ (x)=1 or xˆ =r(φ ).
G,f,w f G,f,w
4 APPLICATIONS 7
4 Applications
In this section we give three examples. The first two arises in the study of the
geodesicflowsinthemodularsurface. ThefirstisalocalperturbationofaTBS
andourgoalis to comparethe algorithmgivenin [8] andthe one deducedfrom
Theorem 2. For this reason, we choose exactly the same example appearing as
Example 1 in [8]. We only produce φ (x) and refer the reader to [8] for the
G,f,w
facts behind this example andalsoseeinghowthis functionis appliedto obtain
an approximation for entropy of the respective dynamical system.
The second example with some more details is not a local perturbation of a
countable TBS. For that example φ (x) is obtained and an estimate of the
G,f,w
entropy will be given. The last example illustrates a case where r(φ ) =
G,f,w
r(F ).
f,V
Example 1. Let V(G) = {3,4,5,6,...}, w = 3 and take the set of forbid-
den edges to be
D ={(3,3),(3,4),(3,5),(4,3),(5,3)}.
Thisexampleoccursinthecodingofgeodesicflowsonthemodularsurface. See
Example 1 in [8] for a brief explanation. There the following formula is defined
xf(3)(F (x)−xf(3)−xf(4)−xf(5))(1−xf(4)−xf(5))
f,V
φ (x)= (8)
G,f,3 1+xf(3)−F (x)
f,V
for the case when denominator is positive.
By the above method the relation ρ on V(G), is
W ={V ={3},V ={4,5},V =V(G)\{3,4,5}}.
{3} 0 1 2
Hence m =2 and φ (x) =xf(3)A (x). Statement (3) in the above theorem
G,f,3 2
implies
A (x)=α (x)+α A (x)+α A (x)=(xf(4)+xf(5))(A (x)+A (x)),
1 10 11 1 12 2 1 2
A (x)=α (x)+α A (x)+α A (x)=( xf(v))(A (x)+A (x)+1),
(cid:26) 2 20 21 1 22 2 v∈V2 1 2
where all α =α (x) are real functions and inPfact α =α =xf(4)+xf(5),
ij ij 11 12
α −1 α
α = α = ∞ xf(i) and α = xf(3). So, M = 11 12 and
21 22 i=6 20 α α −1
21 22
(cid:18) (cid:19)
if detM 6= 0,Pthen A1(x) = M−1 0 . Therefore, A (x) = α12α20 ,
A2(x) −α20 1 1−α11−α22
(cid:18) (cid:19) (cid:18) (cid:19)
A (x) = α20(1−α11). But 1−α −α >0, because A ’s and α ’s are positive
2 1−α11−α22 11 21 i ij
bydefinitionand1−α >0bytheresultsfollowingCorollary(1). Thatmeans
11
φ (x) = xf(3)α20(x)(1−α11(x)). By evaluating α (x), the formula (8) will be
G,f,3 1−α11(x)−α22(x) ij
established. NotethatdetM =1−α (x)−α (x)=1+xf(3)−F (x)which
11 22 f,V
is the denominator in (8).
4 APPLICATIONS 8
Example 2. Recall from [4] that any bi-infinite sequence of non-zero inte-
gers{...,v ,v ,v ,...},|v |6=1suchthat|1+ 1|≤ 1 isrealizedasageometric
−1 0 1 i v v′ 2
code of anorientedgeodesic onthe modular surface. These codes areproduced
by choosing a suitable cross section, that is, a set which is hit infinitely many
times in past and future by geodesic. So let V(G)={v ∈Z:|v|≥2} and
D = {(−3,−3),(−3,−4),(−3,−5),(−4,−3),(−5,−3),
1
(3,3),(3,4),(3,5),(4,3),(5,3)},
D = {(−v,−2),(−2,−v),(v,2),(2,v):v ≥2}.
2
Set D = D ∪ D , that is, (v,v′) ∈ E(G) if and only if |1 + 1| ≤ 1. Let
1 2 v v′ 2
X = {{...,v ,v ,v ,...} : |v | ≥ 2,|1 + 1| ≤ 1}, σ(v ) = v and (X,σ) the
−1 0 1 i v v′ 2 i i+1
associated system. In fact, (X ∪{...,1,−1,1,−1,...},σ) is the maximal 1-step
countable topological Markov chain in the set of all admissible codes known as
geometric codes [3, Theorem 2.3].
By putting w=2, W ={V ={2}, V ={3}, V ={4,5}, V ={6,7,...},
{2} 0 1 2 3
V ={−2},V ={−3},V ={−4,−5},V ={...,−7,−6}},oneseesthat(X,σ)
4 5 6 7
is an RFT. Hence we apply our technique to give an estimation for the entropy
of (X,σ).
Define f({...,v ,v ,v ,...}) = 2ln|cv |, c = 1.25 and let σ be the special
−1 0 1 0 f
flow over X with the ceiling function f. (To keep the continuity of argument,
we later give some explanation to justify choosing such an f.) We may assume
f is defined on V(G) and f(v)=2ln|cv|.
Note that φ (x) = xf(2)(A (x)+A (x)+A (x)+A (x)). Also if α =
G,f,2 4 5 6 7 i
α (x)andA =A (x),thenA ’sarethesolutionofthefollowersetofequations.
i i i i
A =α (A +A +A +A +A )
1 1 3 4 5 6 7
A =α (A +A +A +A +A +A )
 2 2 2 3 4 5 6 7
A =α (A +A +A +A +A +A +A )
 AA543 ==αα543((11++1 AA11++2 AA22++3 AA33)+4 A7)5 6 7
A =α (1+A +A +A +A +A )
6 6 1 2 3 6 7
Here α = α =Ax7f(=2),αα7(1=+Aα1+=Ax2f(+3),Aα3+=A5α+A=6x+f(A4)7)+. xf(5) and α =
0 4 1 5 2 6 3
α = ∞ xf(v). Therefore, the entropy will be −lnxˆ = 0.8665 where xˆ is
7 v=6 f f
the unique solutionofφ (x)=1. (We used the computer softwareMaple to
G,f,2
P
perform the computations.)
Nowweexplainwhysuchanf waschosenabove. Letx={...,v ,v ,v ,...}
−1 0 1
be a geometric code for an oriented geodesic γ. Then w(x) =v (x)− 1
0 v1(x)−.1..
called minus continued fraction, represents the attractive end point of γ. Let
h, the ceiling function, be the first return time function of oriented geodesic.
This h records the time between two hits of cross sections by geodesic and is
cohomologous to g(x) = 2ln|w(x)|. Since two cohomologous ceiling functions
give the same entropy for special flows on the same base space, take g to be
the ceiling function over X. Note that if |v | = 2 and |1 + 1 | ≤ 1, then
i vi vi+1 2
5 AN EQUIVALENT FORMULA FOR GENERATING FUNCTION 9
v v <0 and |v | can be arbitrary large. Therefore,
i i+1 i+1
1 1 |v (x)|
0
|w(x)|≤|v (x)|+| |≤|v (x)|+ ≤|v (x)|+ .
0 v (x)− 1 0 2 0 4
1 v2(x)−.1..
Thisinturnshowsthat|w(x)|≤1.25|v (x)|=c|v (x)|.But,thisimpliesg(x)≤
0 0
f(x) and hence h(Tg)≥h(Tf)∼=0.8665.
Weliketomentionthatourestimateimprovesslightlytheestimateobtained
in [4]. There they consider X′ = {(...,v ,v ,v ,...) : |v | ≥ 3,v ∈ Z ⊂ X}.
−1 0 1 i i
Then σ is invariant on X′ and let σ′ = σ . Let V′(G) = {v ∈ V : |v | ≥
|X i
3}⊆V(G) and D′ =D . It is proved in [4] that the special flow associated to
1
(X′,σ′) is a local perturbation of a countable TBS and based on the results in
[8], they estimate the entropy to be greater than 0.84171.
Recall that entropy of the geodesic flows in modular surface is 1 [3], if we
roughly agree that bigger entropies of subsystems are due to richer dynamics,
hence (X ∪{...,1,−1,1,−1,...},σ) with entropy greater than 0.8665 is a fairly
rich subsystem of the geodesic flows in modular surface.
Example 3. LetG be a graphwith vertexsetV(G)={v =1,v =2,...}and
0 1
edge set E(G) = {(v ,v ) : v 6= v and either v or v is v }. Let w = {1}.
i j i j i j 0
ThenW ={V ={1},V =V(G)−{1}}. SoM(x)=[−1]whichisinvertible
{1} 0 1
for 0 ≤ x ≤ r(F ). Therefore, by Theorem 2, r(φ ) = r(F ). Also
f,V G,f,1 f,V
φ (x)=xf(1)( xf(v)).
G,f,1 v∈V(G)−{1}
Nowletf beanincreasingfunctionsuchthattakesthevalue1onv =1and
0
value k ∈N, k ≥2Pexactly ⌊2k⌋ times, where ⌊r⌋ denotes the integer part of r.
k2
Thenr(F )= 1 andφ (x)=x1( xf(v))=x( ∞ ⌊2k⌋xk)≤
f,V 2 G,f,1 v∈V(G)−{1} k=2 k2
φ (1)<0.85. Hence xˆ = 1. See Remark (1).
G,f,1 2 f 2 P P
5 An equivalent formula for generating function
Inthissectionbyarathernewapproachwegiveanexplicitformulaforφ (x)
G,f,w
whereinthespecialcaseoflocalperturbationofaTBS,theformulawillexactly
be the one given in [8]. (See corollary 2). Let W = {V = {w},V ,...,V }
{w} 0 1 m
be the partition of V(G) as before. Let V = {v ∈ V : ∃v′ ∈ V ∋ (v,v′) 6∈
G
E(G)}\{w}. We reindex V ’s so that for some 1≤ℓ≤m,
i
e
V = V ∪...∪V ;
G 1 ℓ
∪m−1 V = {v ∈V(G) :(v,v′)∈E(G), v′ ∈V(G),
 i=ℓ+1 i
 Vem = {v ∈V(G) :((vv,′′v,v′))∈6∈EE((GG)),,(∃vv′,′′v∈)∈VE(G(G)});, (9)
v′ ∈V(G)}.

5 AN EQUIVALENT FORMULA FOR GENERATING FUNCTION 10
Let M =M(x) be the matrix in (5) which is obtained from W \{w}. Let ℓ
{w}
be as in (9) and
α −1 α ... α
11 12 1ℓ
α α −1 ... α
21 22 2ℓ
C = .  (10)
.
.
 
 α α ... α −1
 ℓ1 ℓ2 ℓℓ 
 
an ℓ×ℓ sub-matrix of M on the upper left corner. Note that B = C +Id
represents a weighted adjacent matrix for the vertices of V . The next lemma
G
shows when φ (x) is defined, then C is invertible.
G,f,w
e
Lemma 2. Suppose x is the smallest real number such that detC =0. Then
C
x ≥r(φ ).
C G,f,w
Proof. We have
∞ [B(x)](n) = (−1)e(Vi)+e(Vj)det((Id −B(x))e(Vj)e(Vi)) (11)
ViVj det(Id−B(x))
n=0
X
where (Id −B(x)) is the sub-matrix of Id−B(x) = −C obtained by
e(Vj)e(Vi)
deletingV throwandV thcolumn. Infact(11)isthesameidentityas(2.18)in
j i
[8]. ItholdsbecauseifweletA=[a ]beap×pmatrixoverCandifan bethe
ij ij
ijthentryofmatrixA(n) then a(n)zn = (−1)i+jdet((Id−zA)ji),1≤i,j ≤p.
n≥0 ij det(Id−zA)
Note that the right hand of (11) is by definition [C−1] , the V V th entry
P ViVj i j
of C−1. This also shows that the radius of convergence of the series on the left
is x . But ∞ [B(x)](n) = xf∗(γ) where the sum on the right is over the
C n=0 ViVj
paths γ from all v in V to all v in V with all possible lengths. If there is not
i i j j
P P
any path from V to V in V then ∞ [B(x)](n) =0. Let H be the reduced
i j G n=0 ViVj
graph of G. Since the graph H with vertices W is connected so we can fix
{w}
P
onepathγ1 fromw tovi ∈eVi andonepathγ2 fromvj ∈Vj tow. Thenγ1γγ2 ∈
C(G;w)and Vi,Vj∈VeG γ∈[B(x)]nViVj xf∗(γ1γγ2) ≤φG,f,w(x)whichimpliesxC ≥
r(φ ).
G,f,w P P
Recall that α = α (x) = xf(v), 1 ≤ i ≤ m. Then A ’s satisfy (3) or
i i v∈Vi i
equivalently
P
A =α +α A +...+α A
1 10 11 1 1m m
A =α +α A +...+α A
2 20 21 1 2m m

.
 A..Aℓℓ+=1 =αℓ0α+ℓ+1α(ℓ11A+1A+1.+..+A2α+ℓm..A.m+Am) (12)
.
.
 .Am =αm(1+A1+A2+...+Am).