### Ultrafilter-limit points in metric dynamical systems

S. Garc´ıa-Ferreira, M. Sanchis

Abstract. Given a free ultrafilter p on N and a space X, we say that x ∈ X is the
p-limit point of a sequence (xn)n∈N inX (in symbols,x=p-limn→∞xn) if for every
neighborhood V ofx,{n∈N:xn ∈V} ∈p. By usingp-limit points from a suitable
metric space, we characterize the selective ultrafilters on Nand the P-points ofN^{∗}=
β(N)\N. In this paper, we only consider dynamical systems (X, f), whereX is a
compact metric space. For a free ultrafilterponN^{∗}, the functionf^{p}:X→Xis defined
by f^{p}(x) = p-limn→∞f^{n}(x) for each x ∈ X. These functions are not continuous
in general. For a dynamical system (X, f), where X is a compact metric space, the
following statements are shown:

1. IfX is countable,p∈N^{∗}is aP-point andf^{p}is continuous atx∈X, then there
isA∈psuch thatf^{q} is continuous atx, for everyq∈A^{∗}.

2. Letp∈N^{∗}. If the family{f^{p+n}:n∈N}is uniformly equicontinuous atx∈X,
thenf^{p+q}is continuous atx, for allq∈β(N).

3. Let us consider the functionF :N^{∗}×X→Xgiven byF(p, x) =f^{p}(x), for every
(p, x)∈N^{∗}×X. Then, the following conditions are equivalent.

(1) f^{p}is continuous onX, for everyp∈N^{∗}.

(2) There is a denseGδ-subsetDofN^{∗}such thatF|D×Xis continuous.

(3) There is a dense subsetDofN^{∗}such thatF|D×X is continuous.

Keywords: ultrafilter, P-limit point, dynamical system, selective ultrafilter, P-point, compact metric

Classification: Primary 54G20, 54D80, 22A99: secondary 54H11

1. Preliminaries and notation

All the spaces are assumed to be Tychonoff (= completely regular and Haus-
dorff). Iff :X →Y is a continuous function, thenf :β(X)→β(Y) will stand
for the Stone extension off. For a metric space X and ǫ > 0, B(x, ǫ) = {y ∈
X :d(x, y)< ǫ}. For short,xn→xmeans that the sequence (xn)_{n∈}Nconverges
to x. The Stone- ˇCech compactificationβ(N) of the natural numbers Nwith the
discrete topology will be identified with the set of all ultrafilters on N, and its
remainder N^{∗} = β(N)\N with the set of all free ultrafilters on N. If A ⊆ N,

Research of the second-named author was supported by Generalitat Valenciana, grant num- ber CTESIN/2005/015, CONACYT grant no. 40057-F and PAPIIT grant no. IN-106103.

Hospitality and financial support received from the Department of Mathematics of Jaume I University(Spain) where this research was essentially performed are gratefully acknowledged.

then ˆA = cl_{β(}N)A = {p∈ β(N) :A ∈ p} is a basic clopen subset of β(N), and
A^{∗} = ˆA\A={p∈N^{∗} :A∈p} is a basic clopen subset ofN^{∗}. IfA, B⊆N, then
A⊆^{∗}B means thatA\B is finite. In this paper, we shall use the following fact:

If {An : n ∈ N} is a family of subsets of Nwith the infinite finite intersection
property, then there is an infinite subset B of Nsuch that B ⊆^{∗} An, for every
n∈N. The set of real numbers will be denoted byRand the set of positive inte-
gers will be denoted byN^{+}. A pair (X, f) is called adynamical system ifX is a
Tychonoff space andf :X→X is a continuous function. If (X, f) is a dynamical
system, then theorbit of a pointx∈X is the set O_{f}(x) ={f^{n}(x) :n∈N}. For
an infinite setX, we let [X]^{ω}={A⊆X :|A|=ω}.

LetX be space. Given p∈N^{∗}, a point x∈X is said to be the p-limit point
of a sequence (xn)_{n∈}N in X (x = p-lim_{n→∞}xn) if for every neighborhood V
ofx, {n∈ N:xn ∈V} ∈p. The notion of p-limit point was introduced, in the
context of non-standard analysis, by R.A. Bernstein [4]. H. Furstenberg [9, p. 179]

and E. Atkin [1, p. 5, 61] considered the F-limit points in Dynamical Systems, where F is a family of nonempty sets with the finite intersection property (for the definition of a F-limit point of a sequence we replacep byF). The p-limit points play a very important role in the study of countably compact spaces. In this paper, we will give some of their applications to Dynamical Systems.

Observe that a pointx∈Xis an adherent point of a countable set{xn:n∈N} iff there isp∈β(N) such thatx=p-limn→∞xn. In other words,xis an adherent point of a countable set{xn:n∈N} iff the set {{n∈N:xn∈V}:V ∈ N(x)}

is a filter base onN. Notice that xn→xiffx=Fr-lim_{n→∞}xn, where Fr is the
Frech´et filter {A ⊆ N: N\A is finite}. Hence, we see that xn → x iff x= p-
lim_{n→∞}xn for all p∈N^{∗}. It is not hard to prove that in a compact space the
p-limit point of a sequence always exists and is unique (for Hausdorff spaces), for
everyp∈N^{∗}.

By usingp-limit points in metric spaces, we characterize theP-points ofN^{∗}and
the selective ultrafilters onN. In the second section, we study the continuity of
the functionsf^{p}(for the definition of this function see the abstract) when (X, f) is
a dynamical system in whichX is a compact metric space. These functions have
been also studied in [5], where the author establishes the connection between the
algebra ofβ(N) and an arbitrary dynamical system. We consider the particular
case when pis aP-point ofN^{∗} and analyze the continuity of the corresponding
function f^{p}. The functions f^{p}’s are very useful to study the limiting behavior
of the iterates of the original function f when X is a metric compact space.

The fourth section is concerning with some applications to actions of compact metrizable semigroups.

2. p-limit points in metric spaces

Suppose thatX is a metric space andp∈N^{∗}. Ifx=p-limn→∞xn, then there
is a subsequence (xnk)_{k∈}N of (xn)_{n∈}N such that xnk → x. In general, xn 6→ x

and {n_{k} : k ∈ N} ∈/ p. Our first task is to use this remark to characterize the
P-points ofN^{∗} and the selective ultrafilters on N. Let us recall a combinatorial
definition of aP-point ofN^{∗}:

An ultrafilter p∈N^{∗} is called P-point iff for every partition{An:n∈N} of
NwithAn∈/p, for eachn∈N, there isA∈psuch thatA∩Anis finite for every
n∈N.

W. Rudin [13] proved that CH implies the existence of 2^{c}-manyP-points inN^{∗},
and years later S. Shelah [6] found a model of ZFC in whichN^{∗} does not have
anyP-point.

Lemma 2.1. Letp∈N^{∗}and let(xn)_{n∈}Nbe a sequence in a spaceX. If there is
a subsequence (xn_{k})_{k∈}Nsuch that {n_{k} :k∈N} ∈pand lim_{k→∞}xn_{k} =x, then
x=p-limn→∞xn.

Proof: Let V ∈ N(x). By assumption, we know that {n_{k} : k ∈ N} ⊆^{∗} {n ∈
N: xn ∈ V}. Hence, we deduce that{n∈ N : xn ∈V} ∈ p. This shows that

x=p-limn→∞xn.

The next lemma was suggested by the referee and simplifies the original proofs of our main results of this section.

Lemma 2.2. Let p∈N^{∗} and let{An:n∈N}be a partition of Ninto infinite
sets such thatAn∈/ p, for alln∈N. Letσ:N→N×Nbe a bijection such that
σ[An] ={n}×N, for everyn∈N. If for everyk∈Nwe have thatx_{k}= ^{1}_{n}+_{a}_{n}^{1}_{+m},
whereσ(k) = (n, m)andn≤an∈N, then0 =p-lim_{k→∞}x_{k}.

Proof: Letǫ >0 and assume thatA={k∈N:x_{k}> ǫ} ∈p. SinceAn∈/p, for
eachn∈N, we must have that{n∈N:A∩An6=∅}is infinite. Hence, we can
findn > ^{2}_{ǫ} such thatA∩An6=∅. Pickk∈A∩An. Then,σ(k) = (n, m) for some
m∈Nand we have thatx_{k}= _{n}^{1} +_{a}_{n}^{1}_{+m} <_{n}^{2} < ǫ, but this is a contradiction.

Theorem 2.3. For a pointp∈N^{∗}, the following are equivalent.

(1) pis aP-point of N^{∗}.

(2) In every metric space X, for every sequence (xn)_{n∈}N in X and every
x∈X, we have thatx=p-lim_{n→∞}xniff there is a subsequence(xnk)_{k∈}N

such that{n_{k}:k∈N} ∈pandlim_{k→∞}xnk =x.

(3) For every sequence (xn)_{n∈}N of real numbers and for every x ∈ R, we
have thatx=p-lim_{n→∞}xniff there is a subsequence(xnk)_{k∈}Nsuch that
{n_{k}:k∈N} ∈pandlim_{k→∞}xnk =x.

Proof: (1)⇒(2). Necessity. Let An={i∈N:xi ∈B(x,_{n}^{1})}. By assumption,
An∈pfor everyn∈N. Then, we can findA∈pso thatA⊆^{∗}A_{k}for everyk∈N.
If we enumerateA as{xn_{k}:k∈N}, then (xn_{k})_{k∈}Nis the desired subsequence.

Sufficiency. This follows directly from Lemma 2.1.

(2)⇒(3). This is trivial.

(3)⇒(1). Suppose that pis not aP-point ofN^{∗}. Then, there is a partition
{A_{n}:n∈N} ofNsuch that An∈/ p, for everyn∈N, and for everyA∈pthere
is n ∈ N for which A∩An is infinite. Fix a bijection σ : N → N×N so that
σ[An] ={n}×N, for alln∈N. For eachk∈N, we definex_{k}=_{n}^{1}+_{n+m}^{1} provided
that σ(k) = (n, m). Then, by Lemma 2.2, we know that 0 = p-lim_{k→∞}x_{k}.
By assumption, we can find a subsequence (xnk)_{k∈}N such that B = {n_{k} : k ∈
N} ∈p and 0 = lim_{k→∞}xnk. Pick l ∈ Nso that B∩A_{l} is infinite. Then, the
sequence (xn)_{n∈B∩A}_{l} must converge to ^{1}_{l} and as a subsequence of (xnk)_{k∈}N it

must converge to 0, which is impossible.

An ultrafilter p∈N^{∗} is calledselective if for every partition {An :n∈N} of
NwithAn ∈/ p, for eachn∈N, there isA∈psuch that|A∩An| ≤ 1, for every
n∈N. Every selective ultrafilter is aP-point and under CH we can find 2^{c}-many
selective ultrafilters (see [7]).

Theorem 2.4. For a pointp∈N^{∗}, the following are equivalent.

(1) pis selective.

(2) In every metric space X, for every sequence (xn)_{n∈}N in X and every
x ∈ X \ {x_{n} : n ∈ N}, we have that x = p-limn→∞xn iff there are
a subsequence (xn_{k})_{k∈}N and an increasing sequence of integers(mk)_{k∈}N

such that{n_{k}:k∈N} ∈pand _{m}^{1}

k+1 ≤d(xnk, x)<_{m}^{1}

k, for everyk∈N.
(3) For every sequence of real numbers(xn)_{n∈}N and everyx∈R\ {xn:n∈
N}, we have thatx=p-lim_{n→∞}xn iff there are a subsequence(xnk)_{k∈}N

and an increasing sequence of integers(m_{k})_{k∈}Nsuch that{n_{k}:k∈N} ∈p
and _{m}^{1}

k+1 ≤ |xnk−x|<_{m}^{1}

k, for everyk∈N.

Proof: (1)⇒(2). Necessity. Define A_{0} ={i∈N: 1≤d(x_{i}, x)} and for every
1≤n∈N, we letAn={i∈N: _{n+1}^{1} ≤d(x_{i}, x)<_{n}^{1}}. It is evident thatAn∈/ p,
for eachn∈N. Then, we can find A∈pso that |A∩An| ≤1, for everyn∈N.
EnumerateA as{xn_{k} :k∈N}. Then, for everyk∈Nthere is a uniquem_{k}∈N
such thatn_{k}∈Amk. Without loss of generality we may assume that the sequence
(m_{k})_{k∈}Nis increasing. It clear that(xnk)_{k∈}Nis the desired subsequence.

Sufficiency. It is a consequence of Lemma 2.1.

(2)⇒(3). It is evident.

(3) ⇒ (1). Assume that p is not selective. Then there is a partition{An :
n ∈ N^{+}} of N such that for all n ∈ N^{+}, An ∈/ p and for every A ∈ p there
is n ∈ N^{+} with |A∩An| ≥ 2. Let σ : N^{+} → N^{+}×N^{+} be a bijection such
that σ[An] = {n} × N^{+}, for each n ∈ N^{+}. Put a_{1} = 1 and for n > 1, we let
an=n^{2}−n. Observe that ifn >1, then ^{1}_{n}+_{a}^{1}_{n} =_{n−1}^{1} . Now, for eachk∈N^{+},
we definex_{k}= ^{1}_{n}+_{a} ^{1}

n+m provided thatσ(k) = (n, m). By Lemma 2.2, we know
that 0 = p-lim_{k→∞}x_{k}. Then, by hypothesis, there is a subsequence (xnk)_{k∈}N

and an increasing sequence of integers (m_{k})_{k∈}N such thatB ={n_{k}:k∈N} ∈p
and _{m}^{1}

k+1 ≤ xnk < _{m}^{1}

k, for every k ∈ N. Notice that B\A_{1} ∈ p. We know
that there is r ∈ N with r > 1 such that |B ∩Ar| = |(B \A_{1})∩Ar| ≥ 2.

Choose k, l ∈N such that k < l and n_{k}, n_{l} ∈ B∩Ar. Putσ(n_{k}) = (r, s) and
σ(n_{l}) = (r, t) for somes, t∈N^{+}. Then, we have that ^{1}_{r} < xnl= ^{1}_{r}+_{a}^{1}

r+t < _{m}^{1}

l

and _{r−1}^{1} = ^{1}_{r}+_{a}^{1}_{r} > ^{1}_{r} +_{a}_{r}^{1}_{+s} =xn_{k} ≥ _{m}^{1}

k+1. Hence, r−1< m_{k+1} ≤ml< r,
which is impossible sincerandm_{l} are natural numbers.

3. p-limit points and dynamical systems

This section is devoted to study the continuity and discontinuity of the function
f^{p}:X →X, forp∈N^{∗}.

Definition 3.1. Let (X, f) be a dynamical system, whereX is a compact space.

For a free ultrafilter ponN, the functionf^{p} :X →X is defined byf^{p}(x) =p-
limn→∞f^{n}(x), for everyx ∈ X. For a point x ∈ X, the function fx := p 7→

f^{p}(x) :β(N)→X is the Stone extension of the continuous functionn7→f^{n}(x) :
N→X.

We remark that the function fx : β(N) → X is continuous for every x ∈
X. Observe thatfx[β(N)] = cl_{X}(O_{f}(x)). But, the functions f^{p} are not always
continuous as we shall see in the next example:

Example 3.2. LetX ={0} ∪ {^{1}_{n}:n∈N^{+}} and definef :X →X as follows:

f(x) =

x if x∈ {0,1}

1

n if x= _{n+1}^{1} and 1≤n∈N.

It is easy to see that ifp∈ N^{∗}, then f^{p}(x) = 1 for everyx > 0 and f^{p}(0) = 0.

Thus, f^{p} is discontinuous at 0, for allp ∈ N^{∗}. For a connected example, take
X = [0,1] and definef : [0,1]→[0,1] as follows:

f(x) =

x if x∈[0,^{1}_{2}]

(n+1)(n+2)x−(2n+1)

n(n+1) if x∈[_{n+1}^{n} ,^{n+1}_{n+2}] and 1≤n∈N

1 if x= 1.

Observe thatf is a homeomorphism between the closed intervals [_{n+1}^{n} ,^{n+1}_{n+2}] and
[^{n−1}_{n} ,_{n+1}^{n} ], for each 1 ≤ n ∈ N. Then, we have that f^{p}[[0,1)] = [0,^{1}_{2}] and
f^{p}(1) = 1, for everyp ∈N^{∗}. This implies that f^{p} is discontinuous at 1, for all
p∈N^{∗}.

Let us explain one way to extend the ordinary addition on the set of natural numbers to the whole β(N) and how to apply this extension to the Theory of Dynamical Systems:

Forp∈β(N) andn∈N, we definep+n=p-lim_{m→∞}(m+n) and ifp, q∈β(N),
then we definep+q=q-lim_{n→∞}p+n.

The following theorem is taken from [5].

Theorem 3.3. Let(X, f)be a dynamical system where X is a compact space.

Then

f^{p}◦f^{q}(x) =f^{q+p}(x),
for everyx∈X and for everyp, q∈β(N).

Thus, if f^{q} is continuous at x and f^{p} is continuous at f^{q}(x), then f^{q+p} is
continuous atx, for p, q∈β(N).

The following two theorems are characterizations of the continuity of the func-
tionf^{p} at some point of the given space.

Theorem 3.4. Let(X, f)be a dynamical system, whereX is a compact metric
space, and letp∈N^{∗}. For a pointx∈X, the following are equivalent.

(1) f^{p} is continuous atx.

(2) For allǫ > 0 there is δ >0 such that for all y ∈X ifd(x, y)< δ, then
{n∈N:d(f^{n}(x), f^{n}(y))< ǫ} ∈p.

Proof: (1) ⇒ (2). Let ǫ > 0. So, there is δ > 0 such that if y ∈ X and
d(x, y)< δ, thend(f^{p}(x), f^{p}(y))< ^{ǫ}_{3}. Suppose thaty∈X satisfies thatd(x, y)<

δ. By definition, we have that A ={n ∈N : d(f^{n}(x), f^{p}(x)) < _{3}^{ǫ}} ∩ {n∈ N :
d(f^{n}(y), f^{p}(y))< _{3}^{ǫ}} ∈p. Hence,

d(f^{n}(x), f^{n}(y))≤d(f^{n}(x), f^{p}(x)) +d(f^{p}(x), f^{p}(y)) +d(f^{p}(y), f^{n}(y))

≤ ǫ 3 +ǫ

3 + ǫ 3 =ǫ, for everyn∈A.

(2)⇒(1). Letǫ >0 and letδ >0 be satisfy the conditions of our hypothesis.

Fixy∈X withd(x, y)< δ. Then, we have thatA={n∈N:d(f^{n}(x), f^{n}(y))<

ǫ

3} ∈p. Thus,

d(f^{p}(x), f^{p}(y))≤d(f^{p}(x), f^{n}(x)) +d(f^{n}(x), f^{n}(y)) +d(f^{n}(y), f^{p}(y))

≤d(f^{p}(x), f^{n}(x)) +ǫ

3+d(f^{n}(y), f^{p}(y)).

We know thatn∈Acan be chosen so thatd(f^{p}(x), f^{n}(x))< ^{ǫ}_{3},d(f^{n}(x), f^{n}(y))<

ǫ3 and d(f^{n}(y), f^{p}(y)) < ^{ǫ}_{3}. Therefore, d(f^{p}(x), f^{p}(y)) < ǫ. This shows the

continuity off^{p} atx.

Definition 3.5. Let (X, f) be a dynamical system, whereX is a metric space,
and letp∈N^{∗}. We say that a sequence (x_{k})_{k∈}N in X is p-proximal to a point
x if lim_{k→∞}x_{k} = x and for every ǫ > 0 there is N ∈ N such that {n ∈ N :
d(f^{n}(x), f^{n}(x_{k}))< ǫ} ∈p, for everyk∈Nwithk≥N. Two pointsx, y∈X are
said to bep-proximal if for everyǫ >0,{n∈N:d(f^{n}(x), f^{n}(y))< ǫ} ∈p.

Theorem 3.6. Let(X, f)be a dynamical system, whereX is a compact metric
space, and letp∈N^{∗}. For a pointx∈X the following are equivalent.

(1) f^{p} is continuous atx.

(2) Every sequence(x_{k})_{k∈}N that converges toxisp-proximal tox.

Proof: (1) ⇒ (2). Let (x_{k})_{k∈}N be a sequence converging to x. Given ǫ > 0,
by Theorem 3.4, we can findδ > 0 such that for ally ∈ X, if d(x, y)< δ, then
{n∈ N:d(f^{n}(x), f^{n}(y))< ǫ} ∈p. Let N ∈Nsuch that d(x_{k}, x) < δ for every
N ≤k ∈ N. Then, we have that {n∈ N :d(f^{n}(x), f^{n}(x_{k})) < ǫ} ∈ pfor every
k∈Nwithk≥N.

(2)⇒(1). Let us assume thatf^{p}is not continuous atx. Then, by Theorem 3.4,
there isǫ >0 such that for everyk∈Nthere isx_{k}∈X such thatd(x, x_{k})<_{k+1}^{1}
and{n∈N:d(f^{n}(x), f^{n}(x_{k}))< ǫ}∈/ p. It is evident that the sequence (x_{k})_{k∈}N

converges toxand it is notp-proximal tox.

Next we state a classical notion in Dynamical Systems and establish its relation with the concept introduced in Definition 3.5.

Definition 3.7. Let (X, f) be a dynamical system where X is a metric space.

We say that two points x, y ∈ X are proximal if for every ǫ > 0, {n ∈ N :
d(f^{n}(x), f^{n}(y))< ǫ} is infinite.

The following result shows that the standard notion “proximal” is included in Definition 3.5.

Theorem 3.8. Let (X, f)be a dynamical system, where X is a metric space, and letx, y∈X. The following conditions are equivalent.

(1) xandy are proximal.

(2) There isp∈N^{∗} such thatf^{p}(x) =f^{p}(y).

(3) xandy arep-proximal for somep∈N^{∗}.

Proof: The equivalence (1)⇔ (2) is stated, for a general case, in [3] and it is proved in [5]. The implication (3)⇒(1) is trivial.

(1) ⇒ (3). For everyǫ > 0, we define Aǫ ={n ∈ N: d(f^{n}(x), f^{n}(y)) < ǫ}.

Since the family {Aǫ : ǫ > 0} has the finite intersection property, we can find
p∈N^{∗} such that{A_{ǫ}:ǫ >0} ⊆p. It is then evident thatxandyarep-proximal.

The equivalence (2)⇔(3) of the previous theorem can be rewritten as follows.

Theorem 3.9. Let(X, f)be a dynamical system, whereX is a metric space, let
x, y∈X and letp∈N^{∗}. The following conditions are equivalent.

(1) xandy arep-proximal.

(2) f^{p}(x) =f^{p}(y).

It follows from Theorem 3.9 that ifp∈N^{∗}is an idempotent (that is,p+p=p),
then everyx∈X isp-proximal tof^{p}(x). Indeed, f^{p}(x) =f^{p+p}(x) =f^{p}(f^{p}(x)).

Theorem 3.10. Let (X, f)be a dynamical system, whereX is a metric space,
and let x, y ∈ X. Then, {p ∈ N^{∗} : x and y are p-proximal} is a closed subset
of N^{∗}.

Proof: PutD={p∈N^{∗}:xandyarep-proximal}and letq∈clN^{∗}D. Suppose
that xand y are not q-proximal. Then, there is ǫ >0 such that A ={n∈ N:
d(f^{n}(x), f^{n}(y))≥ǫ} ∈ q. Choose p∈ A^{∗}∩D. By assumption, B ={n ∈N :
d(f^{n}(x), f^{n}(y)) < ǫ} ∈ p. But this is impossible since A∩B = ∅. Therefore,

D= clN^{∗}D.

We remark that the pointsxandy arep-proximal, for allp∈N^{∗}, iff

n→∞lim d(f^{n}(x), f^{n}(y)) = 0.

The next example shows that the notion of p-proximally could distinguish, in some sense, two proximal points.

Example 3.11. Let (an)_{n∈}N be a sequence of positive real numbers such that
lim_{n→∞}an= 0,a_{0}= 1 and a_{n+1}< an, for eachn∈N. For everyn∈N, choose
a strictly decreasing sequence (an,m)_{m∈}N such that

(1) limm→∞an,m=an, for eachn∈N, and

(2) an< an,m< an−1, for alln, m∈N; here,a−1= 2.

Consider the subspaceX ={0} ∪ {an:n∈N} ∪ {an,m:n, m∈N}ofR. Then,X is a compact metric space. Now, we shall define a functionf :X →X as follows.

a. f(a_{0}) = 0 andf(0) = 0.

b. f(an) =a_{n−1}, for eachn∈N.
c. f(a_{n,0}) =a_{n+1,0}, for eachn∈N.
d. f(a_{0,n}) =a_{n,1}, for each 1≤n∈N.

e. f(a_{n−m,m+1}) =a_{n−m−1,m+2}, for eachm < n∈N.

It is not difficult to prove that f is continuous. Let x=a_{0,0} and y =a_{0,1}. We
definei_{0} = 1,j_{0}= 2,i_{1} = 3,j_{1} = 5 and if 2≤k∈N, then we definei_{k}=j_{k−1}+ 1
and jk = j_{k−1} +k+ 2. We know from the definition that f^{i}^{0}(a0,1) = a1,1,
f^{j}^{0}(a_{0,1}) = a_{0,2}, f^{i}^{1}(a_{0,1}) = a_{2,1} y f^{j}^{1}(a_{0,1}) = a_{0,3}. By induction, we can
establish that

f^{i}^{k}(a_{0,1}) =f^{j}^{k−1}^{+1}(a_{0,1}) =f(f^{j}^{k−1}(a_{0,1})) =f(a_{0,k+1}) =a_{k+1,1},
f^{j}^{k}(a_{0,1}) =f^{j}^{k−1}^{+k+2}(a_{0,1}) =f^{k+1}(f^{i}^{k}(a_{0,1})) =f^{k+1}(a_{k+1,1}) =a_{0,k+2},

and

f^{i}(a_{k,1}) =a_{k−i,i+1},

for every k ∈ N and for each 1 ≤i ≤k. Let us define A ={i_{k} : k ∈ N} and
B={j_{k}:k∈N}. Then, we have that

k→∞lim |f^{i}^{k}(a_{0,0})−f^{i}^{k}(a_{0,1})|= lim

k→∞|a_{i}_{k}_{+1,0}−a_{k+1,0}|= 0.

On the other hand,

k→∞lim |f^{j}^{k}(a_{0,0})−f^{j}^{k}(a_{0,1})|= lim

k→∞|a_{j}_{k}_{+1,0}−a_{0,k+1}|= 1.

These two conditions imply thatxand yare p-proximal for allp∈A^{∗} and they
are notq-proximal for anyq∈B^{∗}.

When the functionf^{p} is continuous on the whole space we have the following
uniform property:

Theorem 3.12. Let(X, f)be a dynamical system whereX is a compact metric
space and let p∈N^{∗}. Then, f^{p} is continuous iff for every ǫ > 0 there is δ > 0
such that for allx, y∈X, if d(x, y)< δ, then {n∈N:d(f^{n}(x), f^{n}(y))< ǫ} ∈p.

Proof: Necessity. Iff^{p} is continuous onX, thenf^{p} is uniformly continuous on
X and then we follow the proof of Theorem 3.4.

Sufficiency. This follows directly from Theorem 3.4.

Now, let us study the behavior of the functionfx around aP-point ofN^{∗}.
Theorem 3.13. Let(X, f) be a dynamical system and letx ∈X, where X is
a compact metric space. If p∈N^{∗} is a P-point, then there is A∈ psuch that
fx(p) =fx(q), for everyq∈A^{∗}.

Proof: By the continuity offx, for everyk∈Nthere isAk∈psuch that d(fx(p), fx(q))< 1

k+ 1,

for all q∈A^{∗}_{k}. Since pis aP-point there is A∈psuch thatA ⊆^{∗} Ak, for each
k∈N. Thus, if q∈A^{∗} andk∈N, thenq∈A^{∗}_{k}and henced(fx(p), fx(q))< _{k+1}^{1} .
This implies thatfx(p) =fx(q), for everyq∈A^{∗}.

For an arbitrary free ultrafilterponNwe have the following property.

Theorem 3.14. Let(X, f) be a dynamical system and letx ∈X, where X is
a compact metric space. Then, for every p ∈ N^{∗}, there is A ∈ [N]^{ω} such that
fx(p) =fx(q)for everyq∈A^{∗}.

Proof: We know that fx(p)∈cl_{X}({f^{n}(x) :n∈ N}). First suppose that fx(p)
is not an accumulation point of O_{f}(x). Then, fx(p) =f^{p}(x) = f^{n}(x) for some
n∈N and there is ǫ >0 such thatB(f^{n}(x), ǫ)∩ O_{f}(x) ={f^{n}(x)}. Sincefx is
continuous, there isA∈psuch that fx(q)∈B(f^{n}(x), ǫ) for allq∈A^{∗}. That is,
fx(p) =fx(q) =f^{n}(x) for everyq∈A^{∗}. Now, assume that there is a non-trivial
sequence (f^{n}^{k}(x))_{k∈}Nfor which lim_{k→∞}f^{n}^{k}(x) =fx(p) and we also assume that
f^{n}^{i}(x) 6= f^{n}^{j}(x) for distinct i, j ∈ N. Put A = {n_{k} : k ∈ N} and fix q ∈ A^{∗}.
According to Lemma 2.1, we obtain thatfx(p) =fx(q).

The proof of Theorem 3.14 with small changes establishes the next result.

Theorem 3.15. Let(X, f) be a dynamical system and letx ∈X, where X is
a compact metric space. Then, for everyA ∈[N]^{ω}, there isB ∈[A]^{ω} such that
fx(p) =fx(q), for everyp, q∈B^{∗}.

Now, let us study the continuity of the functionf^{p} whenpis aP-point ofN^{∗}
andX is a countable metric space.

Theorem 3.16. Let(X, f)be a dynamical system, whereX is a compact metric
countable space. If f^{p} is continuous at x ∈X, for someP-point p∈ N^{∗}, then
for everyǫ >0 there areδ >0 andA∈pso that for y ∈X if d(x, y)< δ, then
d(f^{p}(y), f^{n}(y))< ǫ, for alln∈Aexcept finitely many.

Proof: By definition, we know that f^{p}(x) = p-limn→∞f^{n}(x). Since X is a
metric space, by Theorem 2.3, there is a sequence (nk)_{k∈}N of natural numbers
such thatf^{p}(x) = lim_{k→∞}f^{n}^{k}(x) and B ={n_{k} :k ∈N} ∈p. Given ǫ >0, by
Theorem 3.4, we may findδ >0 such thatCy={n∈N:d(f^{n}(x), f^{n}(y))<_{3}^{ǫ}} ∈
pandd(f^{p}(x), f^{p}(y))< _{3}^{ǫ}, provided thatd(x, y)< δ. Aspis aP-point, there is
A ∈p such thatA ⊆^{∗} Cy∩B for all y ∈X withd(x, y)< δ. Fix y ∈ X with
d(x, y)< δ andm∈Nsuch thatA\ {0,1, . . . , m} ⊆Cy andd(f^{n}(x), f^{p}(x))< ^{ǫ}_{3},
for everyn∈A\ {0,1, . . . , m}. Then, forn∈A\ {0,1, . . . , m} we have that

d(f^{p}(y), f^{n}(y))< d(f^{p}(y), f^{p}(x)) +d(f^{p}(x), f^{n}(x)) +d(f^{n}(x), f^{n}(y))

< ǫ 3 +ǫ

3 + ǫ 3 =ǫ,

as required.

Lemma 3.17. Let(X, f)be a dynamical system, whereX is a compact metric
countable space. Suppose thatf^{p} is continuous atx∈X for aP-point pof N^{∗}.
Then, for everyǫ >0 there areδ >0 andA∈psuch that ify ∈X satisfies that
d(x, y)< δ, then d(f^{n}(x), f^{n}(y))< ǫfor alln∈Aexcept finitely many.

Proof: According to Theorem 3.16, we can find δ > 0 and B ∈ p so that if
y∈X andd(x, y)< δ, thend(f^{p}(x), f^{p}(y))< ^{ǫ}_{3} andd(f^{p}(y), f^{n}(y))< ^{ǫ}_{3} for all
n∈ B except finitely many. Put A={n∈ B :d(f^{p}(x), f^{n}(x)) < ^{ǫ}_{3}}. Assume

thaty∈Xsatisfies the inequalityd(x, y)< δ. By assumption, we can findm∈N
such that d(f^{p}(y), f^{n}(y))< _{3}^{ǫ}, for eachn∈A\m. Thus, if n∈A\m, then we
obtain that

d(f^{n}(x), f^{n}(y))≤d(f^{n}(x), f^{p}(x)) +d(f^{p}(x), f^{p}(y)) +d(f^{p}(y), f^{n}(y))

< ǫ 3 +ǫ

3 + ǫ 3 =ǫ.

Theorem 3.18. Let(X, f)be a dynamical system, whereX is a compact metric
countable space, and let x∈X. Suppose that f^{p} is continuous atx∈ X for a
P-pointpof N^{∗}. Then, there isA∈psuch thatf^{q} is continuous atx, for every
q∈A^{∗}.

Proof: By Theorem 3.13, we know that there isB ∈psuch thatf^{p}(x) =f^{q}(x)
for eachq∈B^{∗}. From the previous lemma, for everyn∈N, we can find δn>0
andAn⊆Bsuch that ifd(x, y)< δn, thend(f^{k}(x), f^{k}(y))< _{n+1}^{1} for allk∈An

except finitely many. For everyn∈N, letCn={k∈N:d(f^{p}(x), f^{k}(x))< _{n+1}^{1} }.

We know that Cn ∈ p for all n ∈ N. Since p is a P-point, we can find A ∈ p
so that A⊆^{∗} An∩Cn, for each n∈N. Now, fixq∈A^{∗} and let ǫ >0. Choose
n∈Nsuch that _{n+1}^{1} < ^{ǫ}_{3}. Suppose that y∈X satisfies thatd(x, y)< δn. Since
D={i∈N:d(f^{i}(y), f^{q}(y))<_{n+1}^{1} } ∈q, we can findk∈D∩Cn∩Anfor which
d(f^{k}(x), f^{k}(y))<_{n+1}^{1} . Then, we have that

d(f^{q}(x), f^{q}(y)) =d(f^{p}(x), f^{q}(y))

≤d(f^{p}(x), f^{k}(x)) +d(f^{k}(x), f^{k}(y)) +d(f^{k}(y), f^{q}(y))

< 1

n+ 1 + 1

n+ 1+ 1 n+ 1

< ǫ 3+ ǫ

3 +ǫ 3 =ǫ.

Therefore,f^{q} is continuous atx.

The next corollary is a direct application of Theorem 3.18.

Corollary 3.19. Let(X, f)be a dynamical system, whereX is a compact metric
countable space. If p∈N^{∗} is aP-point andf^{p} is continuous onX, then there is
A∈psuch thatf^{q} is continuous onX, for everyq∈A^{∗}.

Proof: According to Theorem 3.18, for everyx∈X, there isAx∈psuch that
f^{q} is continuous at x, for everyq∈A^{∗}_{x}. ChooseA∈p so thatA⊆^{∗} Ax, for all
x∈X. Then, it is evident thatf^{q}is continuous onX, for eachq∈A^{∗}.

In the general case, we have the following statement:

Theorem 3.20. Let(X, f)be a dynamical system, whereX is a compact metric
space, and letp∈N^{∗}. Suppose that there existA∈pandx∈X such that

(1) fx(s) =fx(t)for eachs, t∈A^{∗}; and
(2) f^{p} is continuous atx.

If x= lim_{n→∞}xn, then there is B∈[A]^{ω} such thatf^{q}(x) = lim_{n→∞}f^{q}(xn)for
everyq∈B^{∗}.

Proof: Since f^{p} is continuous atx, by Theorem 3.4, for every i ∈ N there is
K_{i} ∈Nsuch thatB_{k,i} ={n∈N:d(f^{n}(x), f^{n}(x_{k}))< _{i+1}^{1} } ∈p, for allk ≥K_{i}.
For eachi∈N, letCi={n∈N:d(f^{n}(x), f^{p}(x))<_{i+1}^{1} }. By definition, we know
thatCi ∈pfor eachi∈N. ChooseB∈[A]^{ω}so thatB⊆^{∗}B_{k,i}∩Ci, for everyi∈N
and for everyk≥K_{i}. Letq∈B^{∗} and letǫ >0. Pickj ∈Nsuch that _{j+1}^{1} < ^{ǫ}_{3}.
Fix k ≥K_{j}. We know that D = {n ∈N : d(f^{n}(x_{k}), f^{q}(x_{k})) < _{j+1}^{1} } ∈q. Let
h∈D∩B_{k,j}∩C_{j}. Then, we have that

d(f^{q}(x_{k}), f^{q}(x)) =d(f^{q}(x_{k}), f^{p}(x))

≤d(f^{q}(x_{k}), f^{h}(x_{k})) +d(f^{h}(x_{k}), f^{h}(x)) +d(f^{h}(x), f^{p}(x))

< 1

j+ 1+ 1

j+ 1 + 1 j+ 1 < ǫ.

Next, we shall study the continuity properties of various functionsf^{p}’s at the
same time.

Lemma 3.21. Let(X, f)be a dynamical system, whereX is a compact metric,
x, y ∈ X and p ∈ N^{∗}. If d(f^{p}(x), f^{p}(y)) < _{3}^{ǫ} for some ǫ > 0, then {n ∈ N :
d(f^{n}(x), f^{n}(y))< ǫ} ∈p.

Proof: We know thatA={n∈N:d(f^{p}(x), f^{n}(x))< ^{ǫ}_{3}} ∈pandB ={n∈N:
d(f^{p}(y), f^{n}(y))< _{3}^{ǫ}} ∈p. Then, we have thatA∩B∈pand ifn∈A∩B, then

d(f^{n}(x), f^{n}(y))≤d(f^{n}(x), f^{p}(x)) +d(f^{p}(x), f^{p}(y)) +d(f^{p}(y), f^{n}(y))

< ǫ 3 +ǫ

3 + ǫ 3 =ǫ.

Theorem 3.22. Let(X, f)be a dynamical system, whereX is a compact metric
space, and let x ∈ X. Let {p_{n} : n ∈ N} ⊆ β(N) and assume that the family
{f^{p}^{n}:n∈N} is uniformly equicontinuous atx. Then,f^{q} is continuous atx, for
eachq∈clN^{∗}({p_{n}:n∈N}).

Proof: Fix q∈ clN^{∗}({p_{n} :n∈ N}). We know that q=p-limn→∞pn for some
p ∈ N^{∗}. Suppose that f^{q} is not continuous at x. According to Theorem 3.4,

there is ǫ > 0 and a sequence (x_{k})_{k∈}N in X converging to x such that A_{k} =
{m∈N:d(f^{m}(x), f^{m}(x_{k}))≥ǫ} ∈q, for eachk∈N. We know thatB_{k} ={n∈
N : A_{k} ∈ pn} ∈ p, for all k ∈ N. By assumption, there is δ > 0 such that if
y∈X and d(x, y)< δ, thend(f^{p}^{n}(x), f^{p}^{n}(y))< ^{ǫ}_{3}, for alln∈N. Choosel ∈N
such that d(x, x_{k}) < δ for eachk ∈ N with l ≤ k. Fix k ∈ N with l ≤ k. So,
d(f^{p}^{n}(x), f^{p}^{n}(x_{k}))< _{3}^{ǫ} for alln∈N. By Lemma 3.21, we have that

Cn={m∈N:d(f^{m}(x), f^{m}(xk))< ǫ} ∈pn,

for every n ∈ N. Pick n ∈ B_{k}. It then follows that A_{k}∩Cn ∈ pn, which is

impossible.

Corollary 3.23. Let(X, f)be a dynamical system, whereX is a compact metric
space, and letp∈N^{∗}. If {f^{p+n}:n∈N} is uniformly equicontinuous atx∈X,
thenf^{p+q} is continuous atx, for allq∈β(N).

Proof: Let p ∈ N^{∗}. We know that the function λp : β(N) → β(N) given by
λp(q) = p+q is continuous (see [11]). Hence, we obtain that λp[cl_{β(}N)N] =
{p+q:q∈β(N)} = cl_{β(}N)(λp[N]). By Theorem 3.22, we conclude that f^{p+q} is

continuous atx, for eachq∈β(N).

Theorem 3.24. Let(X, f)be a dynamical system, whereX is a compact metric
space, andx∈X. If {q∈N^{∗}:f^{q} is continuous atx} is dense inN^{∗}, thenf^{p} is
continuous atxfor allp∈N^{∗}.

Proof: Put D = {q ∈ N^{∗} : f^{q} is continuous at x}. Suppose that f^{p} is not
continuous atxfor somep∈N^{∗}\D. Then, by Theorem 3.4, there isǫ >0 and
for every k ∈N there isx_{k} ∈ X such that x= lim_{k→∞}x_{k} and A_{k} ={n∈ N:
d(f^{n}(x), f^{n}(x_{k})) ≥ ǫ} ∈ p, for each k ∈ N. We can find A ∈ [N]^{ω} such that
A ⊆^{∗} Ak for all k ∈ N. By assumption, there is q ∈ A^{∗} ∩D for which f^{q} is
continuous atx. Hence, we may choseN ∈Nsuch thatd(f^{q}(x), f^{q}(x_{k}))<_{3}^{ǫ}, for
allk∈Nwithk≥N. It then follows from Lemma 3.21 that

B_{k}={m∈A:d(f^{m}(x), f^{m}(x_{k}))< ǫ} ∈q,

for allk ≥ N. Fix N ≤ i ∈N. We know that Bi ⊆^{∗} Ai. So, if m ∈Bi∩Ai,
then d(f^{m}(x), f^{m}(xi)) < ǫ and d(f^{m}(x), f^{m}(xi)) ≥ ǫ, but this is impossible.

Therefore,f^{p} is continuous atx, for allp∈N^{∗}.
Theorem 3.25. Let(X, f)be a dynamical system, whereX is a compact metric
space, and x ∈ X. Let 1 < k ∈ N. For i < k, we define Ai = {n ∈ N : n ∼=
imod(k)}. If there isj < k such that f^{q} is continuous at xfor allq∈A^{∗}_{j}, then
f^{p} is continuous atxfor everyp∈N^{∗}.

Proof: First, observe that N^{∗} =S

i<kA^{∗}_{i}. Letj 6=i < k. We defineφi :N→N
byφi(n) =|n+i−j|for everyn∈N. It is not hard to see that φi is a bijection

betweenA_{j} and A_{i} module a finite set. Hence, if p∈A^{∗}_{i}, then there is q∈ A^{∗}_{j}
such thatφ_{i}(q) =q+i−j =p. Thus, if i > j andf^{q} is continuous at x, then
f^{q+i−j} =f^{p}=f^{i−j}◦f^{q}is continuous atx. Ifi < j, then we consider the function
φ_{k+i} which is also a bijection betweenAj andAi module a finite set. Thus, for
a given p∈A^{∗}_{i} there is q∈A^{∗}_{j} such thatφ_{k+i}(q) =q+k+i−j =pand then
f^{q+k+i−j}=f^{p}=f^{k+i−j}◦f^{q}is continuous atxwheneverf^{q} is continuous atx.

Let (X, f) be a dynamical system, where X is a metric compact space, and
letx ∈ X. The previous corollary assures that if f^{p} is continuous at x, for all
p∈ {an:n∈N}^{∗}, wherea∈N, thenf^{p} is continuous atx, for allp∈N^{∗}.
Lemma 3.26. Let(X, f)be a dynamical system, whereX is a compact metric
space, and letx∈Xbe a fixed point of f. Suppose that there isǫ >0such that for
everyk∈Nthere arex_{k}, y_{k}∈X such thatd(x, x_{k})< _{k+1}^{1} ,O_{f}(y_{k})∩B(x, ǫ) =∅
andO_{f}(y_{k})∩ O_{f}(x_{k})6=∅. Then,f^{p} is discontinuous atxfor everyp∈N^{∗}.
Proof: Fix k∈N. We know that f^{l}(x_{k}) =f^{m}(y_{k}), for somel, m∈N. Then,
f^{l+a}(x_{k}) =f^{m+a}(y_{k})∈ O_{f}(y_{k}), for all a∈N. Hence,{n∈N:d(f^{n}(x_{k}), x)≥ǫ}

is a cofinite subset ofNand so

{n∈N:d(f^{n}(x_{k}), f^{n}(x))≥ǫ}={n∈N:d(f^{n}(x_{k}), x)≥ǫ} ∈p,

for eachp∈N^{∗}. Therefore,f^{p} is discontinuous atxfor everyp∈N^{∗}.
Theorem 3.27. Let (X, f) be a dynamical system such that X is a compact
metric space with only one non-isolated point. Then, eitherf^{p} is continuous for
allp∈N^{∗} or f^{p} is discontinuous for allp∈N^{∗}.

Proof: Letxbe the unique non-isolated point ofX. First, suppose thatf(x)6=

x. Then, we have that A = {y ∈ X : f(y) = f(x)} is cofinite. If y ∈ A
and n ∈ N, then f^{n}(y) = f^{n}(x); hence, we deduce that f^{p}(y) = f^{p}(x) for all
y ∈ A and for all p ∈ N^{∗}. Thus, f^{p} is continuous, for all p ∈ N^{∗}. Now, we
assume that f(x) = x. Let ǫ > 0 and let X \B(x, ǫ) = {x_{0}, . . . , xm}. Put
F = {i ≤ m : O_{f}(x_{i}) is finite} and I = m\F. We may also assume that
x /∈ O_{f}(x_{i}) for every i∈F. Suppose that the conditions of the previous lemma
fail. Then, we can find δ > 0 such thatB(x, δ)∩ O_{f}(x_{i}) = ∅, for each i ≤F,
and if d(x, y) < δ, then O_{f}(y)∩ O_{f}(z) = ∅, whenever O_{f}(z)∩B(x, ǫ) = ∅.

Let y ∈ X such that d(x, y) < δ. If O_{f}(y)∩ O_{f}(xi) 6=∅ for some i ∈ I, then
limn→∞f^{n}(y) = x and hence {n ∈ N : d(x, f^{n}(y)) < ǫ} ∈ p for all p ∈ N^{∗}.
Suppose thatO_{f}(y) does not intersect anyO_{f}(xi), for alli≤m. Then,O_{f}(y)⊆
B(x, ǫ). So,N={n ∈N:d(x, f^{n}(y))< ǫ} ∈pfor all p∈N^{∗}. This shows that

f^{p}:X →X is continuous, for eachp∈N^{∗}.

Theorem 3.28. Let (X, f) be a dynamical system such that X is a compact
metric space and letx∈X be a fixed point of f. Suppose that there ism∈N
such that|O_{f}(y)| ≤m, for ally ∈X. Then, eitherf^{p} is continuous atxfor all
p∈N^{∗}, orf^{p} is discontinuous atxfor allp∈N^{∗}.

Proof: Suppose that f^{p} is continuous at x and f^{q} is discontinuous at x, for
somep, q ∈N^{∗}. Then there areǫ >0 and a sequence (x_{k})_{k∈}N in X converging
to xsuch that{n∈N:d(x, f^{n}(x_{k}))≥ǫ} ∈q, for allk ∈N. By the continuity
of f^{p} and Theorem 3.4, there is δ > 0 such that δ < ǫ and if y ∈ X and
d(x, y) < δ, then{n∈N:d(x, f^{n}(y))< ǫ} ∈p. We know that there isM ∈N
such that d(x, x_{k})< δ for all M ≤ k ∈N. Then, for each k ∈N with k ≥M,
there is 0 < m_{k} ≤ m so that d(x, f^{m}^{k}^{+1}(x_{k})) ≥ ǫ and m_{k} is the minimum
positive integer with this property. Without loss of generality, we may assume
that there isl≤mfor whichm_{k}=l, for each k∈N\M. Since f is continuous
we can find 0 < δ_{l} < δ_{l−1} < · · · < δ_{0} < ǫ such that if d(x, y) < δ_{i}, then
d(x, f(y)) < δ_{i−1}, for every 0≤ i < l, and if d(x, y)< δ_{0}, then d(x, f(x)) < ǫ.

ChooseN ∈Nsuch that M < N andd(x, x_{k})< δ_{l}, for everyN ≤k∈N. Then,
we have thatd(x, f^{l}(x_{k}))< δ_{0}, for eachN ≤k∈N. But, this is impossible since

d(x, f^{l+1}(x_{k}))≥ǫ, for everyN≤k∈N.

We finish this section with some conditions that are equivalent to the continuity
of all functionsf^{p}’s.

Theorem 3.29. Let(X, f)be a dynamical system, whereX is a compact metric
space. Let us consider the functionF^{∗} :N^{∗}×X →X given byF^{∗}(p, x) =f^{p}(x),
for every(p, x)∈N^{p}×X. Then, the following conditions are equivalent.

(1) f^{p} is continuous onX, for everyp∈N^{∗} (that is,F^{∗} is separately contin-
uous).

(2) There is a denseG_{δ}-subsetD of N^{∗} such thatF^{∗}|_{D×X} is continuous.

(3) There is a dense subsetD of N^{∗} such thatF^{∗}|_{D×X} is continuous.

Proof: The implication (1)⇒(2) follows directly from Namioka’s Theorem ([2, Theorem III.5.5], [12]), the implication (2)⇒ (3) is trivial and the implication

(3)⇒(1) follows directly from Theorem 3.24.

4. Dynamical systems and actions of metrizable semigroups

Throughout this section, (X, f) will stand for a dynamical system where X
is a compact metric space. From now on to avoid trivial situations we assume
thatX is infinite and that for every couple of natural numbers (n, m) there exists
x∈X such that f^{n}(x)6=f^{m}(x). Our main goal is to establish that the action
F :β(N)×X →X induced by (X, f) is (in some sense) equivalent to the action
of a metrizable semigroup onX. To do this, let us define an equivalent relation

∼onβ(N) by letting p∼qif and only iff^{p}(x) =f^{q}(x) for everyx∈X. Ifdis a

compatible metric onX, the real-valued function onβ(N)×β(N) defined by d(p, q) = sup¯

x∈X

d(f^{p}(x), f^{q}(x)) p, q∈β(N),

is a pseudometric on β(N) (notice that being X compact, d is bounded). It is clear that ¯dinduces a metric (also denoted by ¯d) on the quotient spaceβ(N)/∼.

The following result follows from Theorem 3.3.

Proposition 4.1. β(N)/∼is a semigroup with the addition+defined as [p] + [q] = [p+q],

for eachp, q∈β(N).

As we deal with actions on metrizable semigroups, a natural question is when the semigroup (β(N)/∼,+) equipped with the topology induced by the metric ¯d is a topological semigroup; that is, when the operation defined in Proposition 4.1 is continuous. A useful sufficient condition is given in Theorem 4.3 below. Before the statement of this theorem, we prove a lemma.

Lemma 4.2. Let (X, f)be a dynamical system, where X is a compact metric
space. If the family of functions {f^{n}:n∈N} is uniformly equicontinuous, then
the family{f^{p}:p∈N^{∗}}is also uniformly equicontinuous.

Proof: By assumption, givenǫ >0 we can findδ >0 such that ifx, y∈Xsatisfy
that d(x, y)< δ, thend(f^{n}(x), f^{n}(y))< _{3}^{ǫ} for all n∈N. Letx, y ∈X. Assume
that d(x, y)< δ and fixp∈N^{∗}. Choose n∈Nso thatd(f^{p}(x), f^{n}(x))< _{3}^{ǫ} and
d(f^{p}(y), f^{n}(y))< _{3}^{ǫ}. Then, we obtain that

d(f^{p}(x), f^{p}(y))≤d(f^{p}(x), f^{n}(x)) +d(f^{n}(x), f^{n}(y)) +d(f^{n}(x), f^{p}(y))

< ǫ 3 +ǫ

3+ǫ 3 =ǫ.

Therefore, the family{f^{p}:p∈N^{∗}}is also uniformly equicontinuous.

Theorem 4.3. Assume that the family{f^{n}:n∈N}is uniformly equicontinuous,
thenβ(N)/∼is a topological semigroup with the topology induced by the metricd.¯
Proof: Let [p],[q] ∈ β(N)/∼. We know from Lemma 4.2 that the family
of functions {f^{t} : t ∈ β(N)} is also uniformly equicontinuous. Hence, given
ǫ > 0 there is δ > 0 such that δ < _{2}^{ǫ} and if x, y ∈ X and d(x, y) < δ, then
d(f^{t}(x), f^{t}(y))< ^{ǫ}_{2} for allt∈β(N). Suppose thatr, s∈β(N) satisfy that

d(p, r) = sup{d(f¯ ^{p}(x), f^{r}(x)) :x∈X}< δ
and d(q, s) = sup{d(f¯ ^{q}(x), f^{s}(x)) :x∈X}< δ.

Then, d(f^{s}(f^{p}(x)), f^{s}(f^{r}(x))) < _{2}^{ǫ} and d(f^{q}(f^{p}(x)), f^{s}(f^{p}(x))) < _{2}^{ǫ}, for allx∈
X. Thus,

d(f^{q}(f^{p}(x)), f^{s}(f^{r}(x)))≤d(f^{q}(f^{p}(x)), f^{s}(f^{p}(x))) +d(f^{s}(f^{p}(x)), f^{s}(f^{r}(x)))

< ǫ 2+ ǫ

2 =ǫ, for allx∈X. Therefore,

d(p¯ +q, r+s) = sup{d(f^{q}(f^{p}(x)), f^{s}(f^{r}(x))) :x∈X} ≤ǫ.

This shows the theorem.

Given a dynamical system (X, f) whereX is a compact metric space, and an
ultrafilter p ∈ β(N), f^{[p}^{]} stands for the function from X into itself defined by
f^{[p}^{]}(x) = f^{p}(x), for every x ∈ X. Let F : β(N)×X −→ X be defined by
F(p, x) =f^{p}(x), for all (p, x)∈β(N)×X. We observe that this actionF induces
a natural actionFb: (β(N)/∼)×X −→X defined as

F([b p], x) =f^{[}^{p}^{]}(x) for each ([p], x)∈(β(N)/∼)×X.

Although the authors could not find a specific reference, the following result is
probably well known. We include a proof for reader convenience. Given a function
f:X×Y →Z we shall denote byfx(respectively, byf^{y}) the functionfx:Y →Z
defined by the rule fx(y) = f(x, y) for every y ∈ Y (respectively, by the rule
f^{y}(x) =f(x, y) for everyx∈X). We recall that, if X, Y andZ are topological
spaces, then f is said to be separately continuous if every fx and every f^{y} are
continuous functions.

Theorem 4.4. Let(X, d^{1}),(Y, d^{2})and(Z, d^{3})be three compact metric spaces. If
f :X×Y →Z is a separately continuous function, then the following conditions
are equivalent.

(1) f is continuous.

(2) The family {f_{x}|x∈X}is uniformly equicontinuous.

(3) The family {f^{y} |y∈Y}is uniformly equicontinuous.

Proof: Obviously we only need to prove that the clauses (1) and (2) are equiv- alent.

(1)⇒(2) Consider the space (C(Y, Z),k·k) wherek·kstands for the supremum norm. It is a well-known fact thatf continuous implies that the functiong:X → (C(Y, Z),k · k) defined as g(x) = fx is continuous (for a more general result the reader can consult [14, Theorem 3.3]). Let ε > 0. Since X is compact, the