Propriedades erg��dicas de semifluxos impulsivos

Propriedades erg´odicas de semifluxos impulsivos
Jos´e Ferreira Alves
Impulsive dynamical systems
Dynamical systems with impulse effects seem to be an adequate
mathematical model to describe real world phenomena that exhibit
sudden changes in their states.
An impulsive dynamical system is prescribed by three ingredients:
a continuous semiflow ϕ on a space X which governs the state of the
system between impulses;
a set D ⊂ X where the flow undergoes some abrupt perturbations;
a function I : D → X which specifies how a jump event happens each
time a trajectory of the flow hits D.
Jos´
e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
2 / 32
The major developments so far on the theory of impulsive dynamical
systems have been to extend the classical theorem on existence and
uniqueness of solutions and to establish sufficient conditions to ensure a
characterization and some asymptotic stability of the limit sets [Ba˘ınov,
Bonotto, Ciesielski, Federson, Kaul, Lakshmikantham, Simeonov...].
Meanwhile, a significant progress in the study of dynamical systems has
been achieved due to a remarkable sample of the so-called ergodic
theorems which concern the connection between the time and the spatial
averages of observable measurable maps along orbits, and whose
fundamental request is the existence of an invariant probability measure.
The results we present here give a first approach to the theory of impulsive
dynamical systems from a probabilistic point of view, providing:
conditions for the existence of invariant probability measures;
a suitable notion of topological entropy;
conditions for the validity of a variational principle.
.
Jos´
e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
3 / 32
Semiflows
Let X be a compact metric space. We say that ψ : R+
0 × X → X is a
semiflow if for all x ∈ X and all s, t ∈ R+
we
have
0
1
ψ0 (x) = x,
2
ψt+s (x) = ψt (ψs (x)),
where ψt (x) stands for ψ(t, x). The curve defined for t ≥ 0 by ψt (x) is
called the ψ-trajectory of the point x ∈ X .
Given a continuous semiflow ϕ : R+
0 × X → X , a compact set D ⊂ X and
a continuous function I : D → X , define τ1 : X → [0, +∞] as
(
inf {t > 0 : ϕt (x) ∈ D} , if ϕt (x) ∈ D for some t > 0;
τ1 (x) =
+∞,
otherwise.
τ1 is called the first impulsive time.
Jos´
e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
4 / 32
Assume τ1 (x) > 0 for all x ∈ X . We define the impulsive trajectory γx (t)
and the subsequent impulsive times as follows:
For 0 ≤ t < τ1 (x), define
γx (t) = ϕt (x).
If τ1 (x) < ∞, set
γx (τ1 (x)) = I (ϕτ2 (x) (x)).
Defining for τ1 (x) < t < τ2 (x)
τ2 (x) = τ1 (x) + τ1 (γx (τ1 (x))),
let
γx (t) = ϕt−τ1 (x) (γx (τ1 (x))).
Proceed inductively...
The duration of the trajectory of x is
T (x) = sup{τn (x) : n ≥ 1}
Jos´
e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
5 / 32
Impulsive semiflows
In general, T (x) < +∞ or T (x) = +∞ are possible for x ∈ X . Under the
condition I (D) ∩ D = ∅, for instance, we have T (x) = ∞ for all x ∈ X .
We say that (X , ϕ, D, I ) is an impulsive dynamical system if
τ1 (x) > 0
and T (x) = +∞,
for all x ∈ X .
The semiflow ψ of an impulsive dynamical system (X , ϕ, D, I ) is defined as
ψ : R+
0 ×X
(t, x)
−→
X
7−→ γx (t),
where γx (t) is the impulsive trajectory of x determined by (X , ϕ, D, I ).
It is straightforward to check that ψ is indeed a semiflow.
Jos´
e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
6 / 32
Invariant probability measures
A map between two topological spaces is called measurable if the
pre-image of any Borel set is a Borel set. An invertible map is called
bimeasurable if both the map and its inverse are measurable.
Notice that the measurability of a semiflow ψ : R+ × X → X gives in
particular that ψt is measurable for each t ≥ 0.
A probability measure µ on the Borel sets of a topological space X is said
to be invariant by a semiflow ψ (or ψ-invariant) if ψ is measurable and
µ(ψt−1 (A)) = µ(A),
for every Borel set A ⊂ X and every t ≥ 0.
Let M(X ) be the set of all probability measures on the Borel sets of X
and Mψ (X ) the set of those measures in M(X ) which are ψ-invariant.
Jos´
e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
7 / 32
Given a measurable map f : X → Y between two topological spaces X
and Y we introduce the push-forward map
f∗ : M(X ) −→ M(Y )
µ
7−→
f∗ µ
with f∗ µ defined for any µ ∈ M(X ) and any Borel set B ⊂ Y as
f∗ µ(B) = µ(f −1 (B)).
We clearly have for µ ∈ M(X )
µ ∈ Mψ (X )
Jos´
e F. Alves (CMUP)
⇐⇒
(ψt )∗ µ = µ, for all t ≥ 0.
Ergodic properties
Impulsive semiflows
8 / 32
Non-wandering sets
A point x ∈ X is said to be non-wandering for a semiflow ψ if, for every
neighborhood U of x and any T > 0, there exists t ≥ T such that
ψt−1 (U) ∩ U 6= ∅.
The non-wandering set of ψ is defined as
Ωψ = {x ∈ X : x is non-wandering for ψ }.
It follows that Ωψ is closed. Moreover, Ωψ contains the set of limit points
of the semiflow, which is clearly nonempty when X is compact. Therefore,
Ωψ is always nonempty and compact.
The support of a measure µ ∈ M(X ) is defined as the set of points x ∈ X
such that µ(U) > 0 for any neighborhood U of x. It’s a general fact that
any invariant probability measure for a semiflow ψ necessarily has its
support contained in the non-wandering set Ωψ .
Jos´
e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
9 / 32
Example: system with no invariant probability measure
Consider the phase space
X = (r cos θ, r sin θ) ∈ R2 : 1 ≤ r ≤ 2, θ ∈ [0, 2π] ,
and define ϕ as the flow associated
to the vector field in X given by
(
r 0 = f (r )
f (r ) = 1 − r , 1 ≤ r ≤ 2.
θ0 = 1,
D
.
I(D)
.
Trajectories of ϕ are curves spiraling
to the inner border circle of X . Take
D = {(1, 0)} and I (1, 0) = (2, 0) .
Jos´
e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
10 / 32
Let ψ be the semiflow of the impulsive dynamical system (X , ϕ, D, I ).
It is not difficult to see that
Ωψ = {(cos θ, sin θ) : 0 ≤ θ ≤ 2π}
and that this set is not forward
invariant under ψ. We claim
that ψ has no invariant probability
measure. Actually, if µ were such
a measure, then we would have
D
.
I(D)
.
−1
1 = µ(Ωψ ) = µ(ψ2π
(Ωψ )).
However,
−1
ψ2π
(Ωψ ) = ∅.
Remark
The non-wandering set of an impulsive semiflow may not be forward
invariant, as the previous example illustrates.
Jos´
e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
11 / 32
We introduce a function τΩ : Ωψ → [0, +∞], defined as
(
τ1 (x), if x ∈ Ωψ \ D;
τΩ (x) =
0,
if x ∈ Ωψ ∩ D.
Theorem 1 (A.-Carvalho, 14)
Let ψ be the semiflow of an impulsive dynamical system (X , ϕ, D, I ) s.t.
1
I (Ωψ ∩ D) ⊂ Ωψ \ D;
2
τΩ is continuous.
Then ψ has some invariant probability measure.
If Ωψ ∩ D = ∅, then ψ has some invariant probability measure.
The continuity of τΩ essentially means that there are no points in Ωψ
on the “right hand side” of D.
In the previous example we have τΩ : Ωψ → [0, 2π] given by
τΩ (cos θ, sin θ) = 2π − θ,
clearly discontinuous at the point (1, 0).
Jos´
e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
12 / 32
Example: system with an invariant probability measure
Consider the phase space X as the annulus
X = (r cos θ, r sin θ) ∈ R2 : 1 ≤ r ≤ 2, θ ∈ [0, 2π]
and define ϕ as the
flow of the vector field in X given by
(
r0 = 0
θ0 = 1.
I(D)
D
Trajectories of ϕ are circles
spinning around zero. Take
D = {(r , 0) ∈ X : 1 ≤ r ≤ 2}
and define I : D → X by
1 1
I (r , 0) = − − r , 0 .
2 2
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e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
13 / 32
Example: system with an invariant probability measure
Let ψ be the semiflow of the impulsive dynamical system (X , ϕ, D, I ).
We have
Ωψ = {(cos θ, sin θ) : π ≤ θ ≤ 2π} .
Ωψ is not forward invariant
under ψ: the trajectory of (1, 0) ∈ Ωψ
is not contained in Ωψ . Still, we have
D
I(D)
I (Ωψ ∩D) = I ({(1, 0)}) = {(−1, 0)} ⊂ Ωψ \D.
Moreover,
τΩ (cos θ, sin θ) = 2π − θ,
which is clearly continuous in Ωψ .
By Theorem 1, ψ has an invariant probability measure.
Jos´
e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
14 / 32
Theorem 2 (A.-Carvalho, 14)
Let ψ be the semiflow of an impulsive dynamical system (X , ϕ, D, I ) s.t.
1
I (Ωψ ∩ D) ⊂ Ωψ \ D;
2
τΩ is continuous.
e , a continuous semiflow ψ˜ in X
e
Then there are a compact metric space X
e such that
and a continuous bimeasurable map g : Ωψ \ D → X
˜t ◦ g |Ω \D = g ◦ ψt |Ω \D for all t ≥ 0;
1 ψ
ψ
2
ψ
g −1 )
e
(ι ◦
∗ : Mψ˜ (X ) → Mψ (X ) is a bijection, where ι : Ωψ \ D → X
is the inclusion map.
Theorem 1 is a corollary of Theorem 2. In fact, as ψ˜ is continuous
e is a compact metric space, then ψ˜ has some invariant
and X
probability measure, by Kryloff-Bogoliouboff Theorem.
Under our assumptions we have µ(D) = 0 for any ψ-invariant
measure µ, and so the second conclusion follows from the first one.
Jos´
e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
15 / 32
Quotient dynamics
Given an impulsive dynamical system (X , ϕ, D, I ), we consider the
quotient space X /∼ , where ∼ is the equivalence relation given by
x ∼y
⇔
x = y,
y = I (x),
x = I (y )
or
I (x) = I (y ).
We shall use x˜ to represent the equivalence class of x ∈ X .
Consider X /∼ with the quotient space and the natural projection
π : X → X /∼ .
In general, as Ωψ is compact, π(Ωψ ) is a compact pseudometric space.
Lemma 1
π(Ωψ ) is a compact metric space.
It’s enough to prove that π(Ωψ ) is a T0 space: given two distinct points in
π(Ωψ ), there is an open set containing one and not the other.
Jos´
e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
16 / 32
The next proposition gives the first conclusion of Theorem 2 with
e = π(Ωψ ) and g = π.
X
Proposition
Assume that I (Ωψ ∩ D) ⊂ Ωψ \ D and τΩ is continuous. Then π|Ωψ \D is a
continuous bimeasurable bijection onto π(Ωψ ) and there exists a
continuous semiflow ψ˜ : R+
0 × π(Ωψ ) → π(Ωψ ) such that for all t ≥ 0
ψ˜t ◦ π|Ω \D = π ◦ ψt |Ω \D .
ψ
ψ
Assuming that I (Ωψ ∩ D) ⊂ Ωψ \ D, from the definition of the equivalence
relation one easily deduces that
π(Ωψ \ D) = π(Ωψ ).
Additionally, for any x, y ∈ Ωψ \ D we have x ∼ y if and only if x = y .
This shows that π|Ωψ \D is a continuous bijection (not necessarily a
homeomorphism) from Ωψ \ D onto π(Ωψ ).
Jos´
e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
17 / 32
Setting for each x ∈ Ωψ \ D and t ≥ 0
˜ x˜) = π(ψ(t, x)),
ψ(t,
then ψ˜ : R+ × π(Ωψ ) → π(Ωψ ) satisfies for all t ≥ 0
ψ˜t ◦ π|Ωψ \D = π ◦ ψt |Ωψ \D .
We are left to prove that ψ˜ is continuous.
Consider for each x˜ ∈ π(Ωψ ) the map ψ˜x˜ : R+
0 → π(Ωψ ) defined by
˜ x˜).
ψ˜x˜ (t) = ψ(t,
It is enough to prove that the maps ψ˜x˜ and ψ˜t are continuous.
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e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
18 / 32
Case 1. ψ˜x˜ is continuous for each x ∈ Ωψ \ D.
Assume first that t0 ≥ 0 is not an impulsive time for x.
We have for t in a small neighborhood of t0 in R+
0,
ψ˜x˜ (t) = π(ϕ(t, x))
As ϕ is continuous, this obviously gives the continuity of ψ˜x˜ at t0 .
If t0 is an impulsive time for x, then we have
lim ψ˜x˜ (t) = lim π(ψ(t, x)) = lim π(ϕ(t, x)) = π(ϕ(t0 , x)).
t→t0−
t→t0−
t→t0−
As ϕ(t0 , x) ∈ D, it follows from the definition of ψ(t0 , x) and the
equivalence relation that
π(ϕ(t0 , x)) = π(I (ϕ(t0 , x))) = π(ψ(t0 , x)) = ψ˜x˜ (t0 ).
Continuity on the right hand side of t0 follows easily from the fact that the
impulsive trajectories are continuous on the right hand side.
Jos´
e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
19 / 32
Case 2. ψ˜t is continuous for each t ≥ 0.
As we are considering the quotient topology on π(Ωψ \ D), then
ψ˜t continuous
⇐⇒
ψ˜t ◦ π|Ωψ \D continuous.
By definition
ψ˜t ◦ π|Ωψ \D = π ◦ ψt |Ωψ \D .
The continuity of ψ˜t ◦ π|Ωψ \D is an immediate consequence of Lemma 2.
Lemma 2
Assume that I (Ωψ ∩ D) ⊂ Ωψ \ D and τΩ is continuous.
Then π ◦ ψt |Ωψ \D is continuous for all t ≥ 0.
D
..
x
y
Jos´
e F. Alves (CMUP)
.
Ergodic properties
I(D)
.
Impulsive semiflows
20 / 32
Variational principle
The continuous bimeasurable map g given by Theorem 2 allows us to
e and ψ in Ωψ .
exchange information between the semiflows ψe in X
e
For instance, the topological entropy of ψ is given by
n
o
e = htop (ψe1 ) = sup hν (ψe1 ) : ν ∈ M e(X
e) ,
htop (ψ)
ψ
where hν (ψe1 ) stands for the measure-theoretic entropy of the map ψe1 with
respect to the probability ν. It follows from our second theorem that
e
sup {hµ (ψ1 ) : µ ∈ Mψ (X )} = htop (ψ).
Problems
1
Variational principle: sup {hµ (ψ1 ) : µ ∈ Mψ (X )} = htop (ψ)?
2
What is the definition of htop (ψ)?
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e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
21 / 32
Topological entropy: classical definition
Let ϕ be a continuous semiflow on a compact metric space X .
Given x ∈ X , T > 0 and > 0 we define the dynamic ball
B(x, ϕ, T , ) = {y ∈ X : dist(ϕt (x), ϕt (y )) < , for every t ∈ [0, T ]} .
The continuity of ϕ implies that B(x, ϕ, T , ) is an open set of X since it
is the open ball centered at x of radius for the metric
distϕ
T (x, y ) = max {dist(ϕt (x), ϕt (y ))}.
0≤t≤T
A set E ⊆ X is said to be (ϕ, T , )-separated if, for each x ∈ E there is no
other points of E inside the ball B(x, ϕ, T , ) besides x.
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e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
22 / 32
As a consequence of the compactness of X and the continuity of ϕ, any
set E ⊆ X which is (ϕ, T , )-separated is finite.
If we denote by |E | the cardinality of E , then we define
s(ϕ, T , ) = max{|E | : E is (ϕ, T , )-separated},
and the growth rate of this number as
h(ϕ, ) = lim sup
T →+∞
1
log s(ϕ, T , ).
T
The topological entropy of ϕ is then given by
htop (ϕ) = lim+ h(ϕ, ).
→0
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e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
23 / 32
Topological entropy: a new definition
Let X be a compact metric space and ψ : R+
0 × X → X a (not necessarily
continuous) semiflow. Consider a function τ
X 3 x 7−→ (τn (x))n∈A(x) ,
where
either A(x) = {1, . . . , `} for some ` ∈ N or A(x) = N;
(τn (x))n∈A(x) is a strictly increasing (possibly finite) sequence in R+ .
We say that τ is admissible with respect to Z ⊂ X if there exists η > 0
such that for all x ∈ Z
τ1 (x) ≥ η;
and for all x ∈ X
τn (ψs (x)) = τn (x) − s, for all n ∈ N and all s ≥ 0;
τn+1 (x) − τn (x) ≥ η, for all n ∈ N with n + 1 ∈ A(x).
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e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
24 / 32
For each admissible function τ , x ∈ X , T > 0 and 0 < δ < η/2, we define


nT (x)
[
JTτ ,δ (x) = (0, T ] \ 
]τj (x) − δ, τj (x) + δ [ ,
j=1
where
nT (x) = max{n ≥ 1 : τn (x) ≤ T }.
The τ -dynamical ball of radius > 0 centered at x is the set
B τ (x, ψ, T , , δ) = y ∈ X : dist(ψt (x), ψt (y )) < , ∀t ∈ JTτ ,δ (x) .
A set E ⊆ X is called (ψ, τ, T , , δ)-separated if, for each x ∈ E , we have
y∈
/ B τ (x, ψ, T , , δ),
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e F. Alves (CMUP)
∀y ∈ E \ {x}.
Ergodic properties
Impulsive semiflows
25 / 32
As before, define
s τ (ψ, T , , δ) = sup{|E | : E is a finite (ψ, τ, T , , δ)-separated set},
and the growth rate
hτ (ψ, , δ) = lim sup
T →+∞
As the function 7→
hτ (ψ, , δ)
1
log s τ (ψ, T , , δ),
T
is decreasing, the following limit exists
hτ (ψ, δ) = lim+ hτ (ψ, , δ).
→0
Finally, as the function δ 7→
τ -topological entropy of ψ
hτ (ψ, δ)
is also decreasing, we define the
τ
htop
(ψ) = lim+ hτ (ψ, δ).
δ→0
Theorem 3 (A.-Carvalho-V´asquez, 14)
Let ϕ : R+
0 × X → X be a continuous semiflow on a compact metric
τ (ϕ) = h
space X and τ an admissible function on X . Then htop
top (ϕ).
Jos´
e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
26 / 32
Impulsive semiflows
Let now ψ be a semiflow of an impulsive dynamical system (X , ϕ, D, I ).
Given ξ > 0, define
[
Dξ =
{ϕt (x) : 0 ≤ t < ξ}.
x∈D
We say that D satisfies a half-tube condition if there is ξ0 > 0 such that:
⇒
∃0 ≤ t 0 < t with ϕt 0 (x) ∈ D;
1
ϕt (x) ∈ Dξ0
2
for all x1 , x2 ∈ D with x1 6= x2
{ϕt (x1 ) : 0 < t < ξ0 } ∩ {ϕt (x2 ) : 0 < t < ξ0 } = ∅.
This is to ensure that removing Dξ from X we have a ψ-invariant region
on which ψ has the same topological entropy.
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e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
27 / 32
For small enough ξ > 0 we define
Xξ = X \ Dξ .
Lemma
If D satisfies a half-tube condition, then ψt (Xξ ) ⊆ Xξ for all t ≥ 0 and
τ
τ
htop
(ψ) = htop
(ψ|Xξ ).
We say that I (D) is transverse if there are s0 > 0 and ξ0 > 0 such that
ϕt (x) ∈ I (D)
⇒
ϕt+s (x) ∈
/ I (D), ∀ 0 < s < s0 ;
for all x1 , x2 ∈ I (D) with x1 6= x2
{ϕt (x1 ) : 0 < t < ξ0 } ∩ {ϕt (x2 ) : 0 < t < ξ0 } = ∅.
This property holds, for instance, when ϕ is a C 1 semiflow and I (D) is
transversal to the flow direction.
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e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
28 / 32
Quotient dynamics
Given an impulsive dynamical system (X , ϕ, D, I ), consider the quotient
space X /∼ with the quotient topology, where ∼ is as before
x ∼y
⇔
x = y,
y = I (x),
x = I (y )
or
I (x) = I (y ).
Let π : X → X /∼ be the natural projection. If d denotes the metric on
X , the metric d˜ in π(X ) that induces the quotient topology is given by
d˜ (˜
x , y˜ ) = inf {d (p1 , q1 ) + d (p2 , q2 ) + · · · + d (pn , qn )},
where p1 , q1 , . . . , pn , qn is any chain of points in X such that p1 ∼ x,
q1 ∼ p2 , q2 ∼ p3 , ... qn ∼ y . In particular, we have
d˜ (˜
x , y˜ ) ≤ d (x, y ),
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e F. Alves (CMUP)
Ergodic properties
∀x, y ∈ X .
Impulsive semiflows
29 / 32
The length n of chains needed to evaluate d˜ (˜
x , y˜ ) may be arbitrarily large,
˜
preventing us from comparing d (˜
x , y˜ ) with d (p, q) for p ∼ x and q ∼ y .
This difficulty can be overcome if the map I does not expand distances, for
instance: I is called 1-Lipschitz if
dist (I (x), I (y )) ≤ dist (x, y ),
for all x, y ∈ D.
Lemma
If I is 1-Lipschitz, then for all x˜, y˜ ∈ π(X ) there exist p, q ∈ X such that
p ∼ x,
q∼y
and
d(p, q) ≤ 2 d˜ (˜
x , y˜ ).
Consider now τξ : Xξ ∪ D → [0, +∞] defined by
(
τ1 (x), if x ∈ Xξ ;
τξ (x) =
0,
if x ∈ D
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e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
30 / 32
Theorem 4 (A.-Carvalho-V´asquez, 14)
Let ψ be the semiflow of an impulsive dynamical system (X , ϕ, D, I ) such
that D satisfies a half-tube condition, I is 1-Lipschitz, I (D) ∩ D = ∅, I (D)
is transverse and τξ is continuous. Then there exists a continuous semiflow
ψ˜ in π(Xξ ) and a continuous invertible bimeasurable map h : Xξ → π(Xξ )
such that ψ˜t ◦ h = h ◦ ψt for all t ≥ 0 and
τ
˜
htop
(ψ) = htop (ψ).
Corollary
Let ψ be the semiflow of an impulsive dynamical system (X , ϕ, D, I )
satisfying the assumptions of Theorem 4. Then
τ
htop
(ψ) = sup {hµ (ψ1 ) : µ ∈ Mψ (X )}.
Jos´
e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
31 / 32
References
J. F. Alves, M. Carvalho
Invariant probability measures and non-wandering sets for impulsive
semiflows
J. Stat. Phys. 157 (2014), no. 6, 1097–1113
J. F. Alves, M. Carvalho, C. V´asquez
A variational principle for impulsive semiflows
arxiv.org/pdf/1410.2372
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e F. Alves (CMUP)
Ergodic properties
Impulsive semiflows
32 / 32