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An introduction to ergodic theory - Walters P.

Walters P. An introduction to ergodic theory - London, 1982. - 251 p. Previous << 1 .. 78 79 80 81 82 83 < 84 > 85 86 87 88 89 90 .. 99 >> Next By Remark 14, for any <5 > 0 we have lim¬£_0 P(T,f,e) ^ 3 + P(T,f) so limf_0 P(T,f,c) ^ P(T,f). ‚ñ°
To obtain definitions of pressure involving open covers we generalise Theorem 7.7. We need the following definitions.
Definition 9.7. If / e C(A\ R), n ^ 1 and a is an open cover of X put
‚ñ°
Definition 9.6. For / e C(X, R) and e > 0 put
–Ý(–ì,/,–µ) = lim sup - log P‚Äû{T,f,e).
Remarks
is a finite subcover of \Jl= –æ T ‚Äò¬´} and
is a finite subcover of \/7=o T *a}-Clearly qJJJ,a) <i p‚Äû(T,J,<x).
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9 Topological Pressure and Its Relationship with Invariant Measures
Theorem 9.2. Lei T.X -* X be continuous anil f e C(X, R).
(i) If —Ç. is an open cover of X with Lebesgue number 6 then qn( T. f.y.) < Qn(Tj,6.2)<Pn(TJ,6l2).
(ii) If e > 0 and —É is an open cot er with di.im(;‚Äô) ‚Ä¢¬£ e then QJT. f.r.) ‚ñÝ\$_ PJLTM * PJLT,f,r).
Proof. We know from Remark 13 lhat QJT.f.r.) < PjT. f,i.) for all r. > 0.
(i) If F is an (n,S/2) spanning set then .V = 1J*,f –ü" –æ –¢~‚Äò–ì–¶–¢'.\: 6/2). Since each B(T‚Äòx:3,2) is a subset of a member of a v,e have q‚Äû{T,f,x) ^ Y^‚ÇF f<sn/)u> ancj hence <y‚Äû(7\/,c0 < Qn(TJ,8,2).
(ii) Let ¬£ be a (/!,¬£) sepaiated subset of X. Since no member of SJ* 7, J T~'y contains two elements of E we have <- pn(T,f,y). Therefore PATJ.c) ¬£ pJT.f.y). ‚ñ°
Remarks
(16) If a. ‚Ä¢/ are open covers of .Y and at < —É then q¬£T.f.u) < qn(T,f,y).
(17) If d{\, j') < diam(a) implies |/(x) ‚Äî /(v)| ^ 6 then pK(T,f,z)^
Lemma 9.3. If f ‚Ç C(X,R) and i is an open cover of X then
lim - Iogp‚Äû(T,/,a)
–ª-*or –ò
exists arid equals inl‚Äû(l jn) log p‚Äû( T, f. a).
Proof. By Theorem 4.9 it suffices to show pn+l(T,/, x)<%^T,f, a) ‚Ä¢ pt(T,/, a). If/? is a finite subcover of V?=o T~‚Äò-x and —É is a finite subcover of Vi*o T~‚Äòa then fiw T~‚Äùy is a finite subcover of V*=o-1 T~‚Äòa, and we have
¬£ sup <; ( ¬£ sup ¬£ sup t?(S|</>u>Y
DcfivT~ny xeD \Bc fl xe –í /\–°–µ —É *cC /
Therefore p‚Äû+k( T, f, a) < p‚Äû( T, f, a) ‚Ä¢ pk( T, f, a). ‚ñ°
The following gives definitions of pressure using open covers.
Theorem 9.4. If T :X -* X is continuous and f 6 C(X,R) then each of the following equals P{T,f).
(i) lima_0[supa{lim‚Äû‚ÄûOD(l/n)log/>II(7\/,a)| a is an open cover of X with diam(a) < 5}].
(ii) lim*-., [–ù—Ç–ø_^ (1 jn) log p‚Äû(T,/, a*)] if {at} is a sequence of opencovers with diamfa*) -¬ª0.
(iii) Iim^0[supa{Iiminfn^a,(l/n)log<yn(T,/,a)|oc is an open cover of X with diam(a) ^ <)}].
¬ß9.1 Topological Pressure
211
fiv) Iima-.0[su^ limsup‚Äû_x (1 n) log q‚Äû(T,f,x)\a is an open cover of X with diam(a) < <>}].
(v) [lim sup‚Äû_a (1'n)qn[T,f,xk)] if {a*} is a sequence of open covers with diam(a*) -¬ª0.
(vi) sup, {lim sup‚Äû_¬ª(l/n)logf/n(T,/,a)|a is an open cover of X}.
(vii) lim^o liminf‚Äû_T (1 n)Q‚Äû{T,f,c).
(viii) Km^o lim inf,,-..^ (1 vi) log P‚Äû(T,/, –≥).
(i) If d > 0 and —É is an open cover with diam(y) < <5 then P‚Äû(T,f,S) ^ p‚Äû(T,f,f) (Theorem 9.2(ii)). Therefore
is an open cover of A' with diam(y) < <>}, using Lemma 9.3. Therefore P(T,f) is no larger than the expression in (i).
If a is a cover and S is a Lebesgue number for ot then </‚Äû(T,/, a) < P‚Äû(T,/, S/2) by Theorem 9.2(i). Also if = sup{|/(.v) ‚Äî /(y)|:</(.\\>) ^ diam(a)} then P‚Äû{T,f,a) <. em¬∞q‚Äû{T,f,x), by Remark 17. Hence
Proof
pn(T,f,x)<.e'"'Pn{TJ,5l2)
so
lim - log pn(T,f, —É) < ra + P(T,f),
n-*x –ò
and
lim I sup< lim - logp‚Äû(7\/, a) diamfa) < ¬ª/
sup< lim
Therefore (i) is proved. The same reasoning proves (ii).
(iii) We know qn(T,f, a) ^ p‚Äû(T,f, a) for all a. Also if
Ta = sup{|/(x) - /(y)\:d(x, y) ^ diam(a)} then p‚Äû(T,f, a) ^ em‚Äúq‚Äû(T,f, a)
(Remark 17). Therefore
e nt*P‚Äû(T,f, a) ^ q‚Äû(T,f, a) ¬£ p‚Äû(T,f, a)
so
-–≥, + lim - log p‚Äû{T,f, a) ^ lim inf- log q‚Äû(7\/, a)
^ lim sup logqn(TJ, a)
^ lim -\ogp‚Äû{T,f,a).
–Ø-¬ª a –ò
The formulae in (iii), (iv), and (v) follow from (i) and (ii).
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9 Topological Pressure and Its Relationship with Invariant Measures
(vi) Let a be an open cover of X and let 2c be a Lebesgue number for a. By Theorem 9.2 q‚Äû( T.f, a) < Q‚Äû( T.f. <;) .,o that lim sup‚Äû_, (I n)q^ T.f, a) <; Q\T,f, e) < P(T,f). Therefore the expression in (v) is majorised by P(T,f). The opposite inequality follows from (iv).
(vii) and (viii) Let at denote the cover of X be all open balls of radius 2c and -/¬£ denote any cover by balls of radius (e/2). Then. by. Theorem 9.2 and Remark 17,
e~nTAcpST, f, 3tt) ^ q‚Äû(T,f. a,) < Q‚Äû(T. f. e) < P‚Äû[T.f, r.) < p‚Äû(T.f. yj
where t4c = sup‚Äô|/(.v) ‚Äî f(\)|:</(.v, i) <L 4e}. Then (vii) and (viii) follow by taking lim infs in this expression and using (ii). ‚ñ°
Remarks
(18) For some examples sup [limn_ , (1 n) logp‚Äû(T,/, —è) | a is an open cover of X} is strictly larger than P( T,f).
(19) From (vi) of Theorem 9.4 we see that P(T,f) does not depend on the metric on X.
As one may expect, from our .nowledge of topological entropy, the definition of pressure car be simplified for expansive homeomorphisms. We shall need the following lemma. Previous << 1 .. 78 79 80 81 82 83 < 84 > 85 86 87 88 89 90 .. 99 >> Next 