Specification and thermodynamical properties for semigroups actions - - PowerPoint PPT Presentation
Specification and thermodynamical properties for semigroups actions Paulo Varandas Federal University of Bahia (joint with F. Rodrigues - UFRGS) AMS/EMS/SPM Meeting - Porto, July 2015 Topological pressure for individual dynamics Classical
Paulo Varandas Federal University of Bahia (joint with F. Rodrigues - UFRGS) AMS/EMS/SPM Meeting - Porto, July 2015
Classical results
Classical results
M compact metric space f : M ! M continuous ' : M ! R continuous potential The topological pressure Ptop(f , ') = lim
"→0 lim sup n→∞
1 n log{sup
En
X
x∈En
eSn'(x)}
Ptop(f , ') = sup
µ∈M1(f )
Z ' dµ
periodic points Ptop(f , ') = lim sup
n→∞
1 n log
f n(x)=x
eSn'(x)
Classical results
[Adler, Konheim, McAndrew 65’] Definition of topological entropy [Ruelle 68’] The pressure function 7! Ptop(f , ' + ) is analytic [Bowen 71’] Specification ) Positive entropy [Ruelle 73’ Walters 75’] Variational principle for continuous maps [Parry 64’ Bowen 71’, 74’] Specification & expansiveness ) 9! equilibrium state µ' for every H¨
as weak∗-limit of 1 Zn X
x∈Pern(f )
eSn'(x)x with Zn = X
x∈Pern(f )
eSn'(x) and htop(f ) = lim
n→∞
1 n log ]Pern(f )
A continuous map f : X ! X satisfies the specification property if for any > 0 there exists an integer p() 1 such that the following holds: for every k 1, any points x1, . . . , xk, and any sequence of positive integers n1, . . . , nk and p1, . . . , pk with pi p() there exists a point x in X such that d ⇣ f j(x), f j(x1) ⌘ , 8 0 j n1 and d ⇣ f j+n1+p1+···+ni−1+pi−1(x) , f j(xi) ⌘ for every 2 i k and 0 j ni.
Motivated by the number of different applications the following classes of dynamical systems have been intensively studied:
Non-autonomous (or sequential) dynamical systems F = (fk)k≥1 Fn = fn · · · f2 f1 for n 1 Some difficulties include:
’Topological & probabilistic complexity’
(G, ) finitely generated (semi)group G1 = {id, g1, g2, . . . , gm} generators & G = S
n∈N0 Gn
g 2 Gn if and only if g = gin . . . gi2gi1 with gij 2 G1
(concatenations of at most n elements of G1) (G, ) is a group (G, ) is a semigroup generators G1 = {id, g ±
1 , g ± 2 , . . . , g ± m }
generators G1 = {id, g1, g2, . . . , gm} (Gn)n∈N increasing family in G (Gn)n∈N may be non-increasing dG(h, g) := |h−1g| distance no natural distance
Notation: G ∗
1 = G1 \ {id} and
G ∗
n = {g = gin . . . gi2gi1 with gij 2 G ∗ 1 }
We say that T : G ⇥ X ! X is a continuous semigroup action on a topological space X if:
The orbit of x 2 X is the set OT(x) = {gx : g 2 G}. x 2 X is ’periodic point’ (period n) if gn(x) = x for some gn 2 Gn Per(G) = S
n≥1 Per(Gn) set of periodic orbits.
We say that T : G ⇥ X ! X is a continuous semigroup action on a topological space X if:
The orbit of x 2 X is the set OT(x) = {gx : g 2 G}. x 2 X is ’periodic point’ (period n) if gn(x) = x for some gn 2 Gn Per(G) = S
n≥1 Per(Gn) set of periodic orbits.
Motivational example: geodesics and moving billiards table f : S1 ! S1 be smooth expanding map (Bowen-Series map) R↵ : S1 ! S1 rotation angle ↵ G semigroup generated by G1 = {id, f , R↵}
Bijection Z+ ⇥ Z4 7! hg1(x) = R π
4 (x), g2(x) = 4x(mod1)i
Non-injective Z2
+ 7! hg1(x) = 2x(mod1), g2(x) = 4x(mod1)i
Bijection F2 (free group) 7! hg1, g2i Anosov diffeos g2 / 2 Z(g1)
Some (different) notions and contributions: [Ruelle 73’] [Ghys, Langevin, Walczak 88’] [Friedland 95’] [Bufetov 99’] [Lind, Schmidt 02’] [Bis 08’, 13’ ] [Ma, Wu 11’] [Miles, Ward 11’] 9 > > > > > > > > > > = > > > > > > > > > > ; Some of these notions require abelianity or amenability
[Ruelle 73’] Zd-expansive actions with (very strong) specification .
[Ghys, Langevin, Walczak 88’] Entropy for pseudo-groups and foliations .
[Bufetov 99’] Entropy free semigroup actions . . . .
I.1 Topological pressure:
Ptop((G, G1), ', E) := lim
ε→0 lim sup n→∞
1 n log ⇣ 1 mn X
|g|=n
sup
F
n X
x∈F
e
Pn−1
i=0 ϕ(g i(x))o⌘
supremum over all (g, n, ")-separated sets F = Fg,n," ⇢ E
htop((G, G1), E) := lim
ε→0 lim sup n→∞
1 n log ⇣ 1 mn X
|g|=n
s(g, n, ") ⌘
I.2 Entropy:
h((G, G1), E) = lim
ε→0 lim sup n→∞
1 n log s(n, ", E)
where s(n, ") is maximal cardinality of (n, ")-separated sets in E. Entropy taking the compact set E = X.
Simple illustration: g1 : S1 ! S1 g2(x) = 2x( mod1) g2 : S1 ! S1 g2(x) = 3x( mod1) g3 : S1 ! S1 g3(x) = 5x( mod1) I.1 Topological pressure:
htop((G, G1), S1) = lim
ε→0 lim sup n→∞
1 n log ⇣ 1 3n X
|g|=n
s(g, n, ") ⌘ = log(10 3 )
I.2 Entropy:
h((G, G1), S1) = lim
ε→0 lim sup n→∞
1 n log s(n, ", S1) = log 3
II.1 Entropy point x0 2 X is an entropy point for htop((G, G1), ·) if htop((G, G1), U) = htop((G, G1), X) for any open nhood U of x0 II.2 Entropy point x0 2 X is an entropy point for h((G, G1), ·) if h((G, G1), U) = h((G, G1), X) for any open nhood U of x0 Rmk: II.2 was introduced by [Bis 13’] which proved that the set of entropy points is non-emtpy provided X is compact.
III.1 Orbital specification
III.1 Orbital specification Rmk 1: Similar notion is studied on the space of push-forwards Rmk 2: Each element in G ∗
1 must satisfy specification
III.2 Weak orbital specification
III.2 Weak orbital specification Rmk 3: Other notions of specification for semigroups / groups can be defined similarly (not needed for this talk!)
T : G ⇥ X ! X satisfies the weak orbital specification property if: for any " > 0 there exists p(") > 0 so that for any p p("), there exists a set ˜ Gp ⇢ G ∗
p satisfying limp→∞ ] ˜ Gp ]G ∗
p = 1 for which: for any
hpj 2 ˜ Gpj with pj p("), any points x1, . . . , xk 2 X, any natural numbers n1, . . . , nk and any concatenations gnj,j = ginj ,j . . . gi2,j gi1,j 2 Gnj with 1 j k there exists x 2 X so that d(g`,1(x) , g`,1(x1)) < " for every ` = 1 . . . n1 and d( g`,j hpj−1 . . . gn2,2 hp1 gn1,1(x) , g`,j(xj) ) < " for every j = 2 . . . k and ` = 1 . . . nj.
Theorem: Let G be a finitely generated semigroup with generators
space X is strongly ∗-expansive and the potentials ', : X ! R are continuous and satisfy the bounded distortion property then:
limit of differentiable maps. Moreover, R 3 7! Ptop((G, G1), ', X) is differentiable Leb-a.e.
Theorem: Let G ⇥ X ! X be a continuous finitely generated continuous semigroup action.
lim sup
p→∞
|G ∗
p \ ˜
Gp| mp < 1, 80 < < 1 9 = ; ) htop((G, G1), X) > 0
Theorem: Let G ⇥ X ! X be a continuous finitely generated semigroup action s.t. every element g 2 G1 is a local homeomorphism. 1. weak orbital specification ) every x 2 X is an entropy point for h((G, G1), ·) 2. strong orbital specification ) every x 2 X is an entropy point for htop((G, G1), ·)
Theorem: Let G ⇥ X ! X be a continuous finitely generated semigroup action s.t. every element g 2 G1 is a local homeomorphism. 1. weak orbital specification ) every x 2 X is an entropy point for h((G, G1), ·) 2. strong orbital specification ) every x 2 X is an entropy point for htop((G, G1), ·) Rmk: h((G, G1), ¯ U) h((G, G1), ¯ U) for every ¯ U ⇢ X htop((G, G1), ¯ U) htop((G, G1), X) h((G, G1), ¯ U) h((G, G1), X) Although involve similar ideas, 1. and 2. are independent
Theorem: Let G be the semigroup generated by a set G1 = {g1, . . . , gk} of uniformly expanding maps. Then: (a) G satisfies the periodic orbital specification property, (b) ’periodic loops’ Per(G) are dense in X, and (c) the mean growth of periodic points is bounded from below as 0 < htop((G, G1), X) lim sup
n→∞
1 n log ⇣ 1 mn X
|g|=n
]Fix(g) ⌘ . Rmk: Similarly, the exponential growth rate of ’periodic loops’ is larger than the entropy h((G, G1), X): h((G, G1), X) lim sup
n→∞
1 n log ]Per(Gn).
p ’not suitable’ for orbital
specification is fastly convergent to zero as p ! 1