Homeomorphisms group of normed vector space: Conjugacy problems and the Koopman operator

This article is concerned with conjugacy problems arising in homeomorphisms group, Hom($F$), of unbounded subsets $F$ of normed vector spaces $E$. Given two homeomorphisms $f$ and $g$ in Hom($F$), it is shown how the existence of a conjugacy may be related to the existence of a common generalized eigenfunction of the associated Koopman operators. This common eigenfunction serves to build a topology on Hom($F$), where the conjugacy is obtained as limit of a sequence generated by the conjugacy operator, when this limit exists. The main conjugacy theorem is presented in a class of generalized Lipeomorphisms.


Introduction
In this article we consider the conjugacy problem in the homeormophsims group of a finite dimensional normed vector space E. It is out of the scope of the present work to review the problem of conjugacy in general, and the reader may consult for instance [12,15,27,30,24,37,43] and references therein, to get a partial survey of the question from a dynamical point of view. The present work raises the problem of conjugacy in the group Hom(F ) constituted by homeomorphisms of a non-compact subset F of E and is intended to demonstrate how the conjugacy problem, in such a case, may be related to spectral properties of the associated Koopman operators. In this sense, this paper provides new insights on the relationships between the spectral theory of dynamical systems and the geometrical problem of conjugacy.
More specifically, given two homeomorphisms f and g of F , we show here that the conjugacy problem in Hom(F ) is subject to the existence of a common generalized eigenfunction for the associated Koopman operators U f and U g (cf. Definition 2.2), i.e. a function Φ satisfying, for some λ, µ > 0, where Φ lives within some cone K of the set of continuous real-valued functions on F . The elements of this cone present the particularity to exhibit a behavior at infinity prescribed by a subadditive function R; see Section 2. More precisely, when such a Φ exists, it is shown how Φ can be used to build a topology such that the sequence of iterates {L n f,g (h 0 )} n∈N , of the conjugacy operator 1 initiated to some h 0 ∈ Hom(F ) close enough to L f,g (h 0 ) in that topology, converges to the conjugacy h satisfying f • h = h • g, provided that {L n f,g (h 0 )} is bounded on every compact of F ; cf. Theorem 4.3. The topology built from Φ, relies on a premetric on Hom(F ) where Φ serves to weight the distance to the identity of any homeomorphism of F , with respect to the composition law.
The plan of this article is as follows. Section 2 sets up the functional framework used in this article, where in particular the main properties of the topology built from any member Φ ∈ K are derived with a particular attention to closure properties and convergence in that topology of sequences in Hom(F ); cf Propositions 2.7 and 2.10. Section 3 establishes a fixed point theorem, Theorem 3.1, for mappings acting on Hom(F ), when this last one is endowed with the topology discussed in Section 2. In section 4 the main theorem of conjugacy, Theorem 4.3, is proved based on Theorem 3.1 applied to the conjugacy operator, where the contraction property is shown to be conditioned to a generalized eigenvalue problem of type (1.1). This related generalized eigenvalue problem for the Koopman operators associated with the conjugacy problem is then discussed in Section 4.3 where in particular connections with relatively recent results about Schröder equations and Abel equation are established. The results obtained here were motivated in part by [13] where some of the results derived in that work where conditioned to a conjugacy problem on non necessarily compact manifold.

A functional framework on the homeomorphisms group
In this section we introduce family of subgroups of homeomorphisms for the composition law. These subgroups associated with the framework from which they are derived, will be used in the analysis of the conjugacy problem in the homeomorphism group itself. The topology with which they are endowed is also introduced here and the main properties are derived. The extension of these topologies to the whole group of homeomorphisms is also presented and the related closure properties and convergence of sequences in the homeomorphisms group are discussed.
2.1. Notations and preliminaries. In this article E denotes a d-dimensional normed vector space (d ∈ N * ), endowed with a norm denoted by · and F denotes a noncompact subset of E. The following class of functions serves to specify some behavior at infinity of homeomorphisms and to build topologies that will be central in our approach; cf Proposition 2.5.
(a) It is interesting to note that the closure K := E R F , is a closed cone with non-empty interior in the Banach space X = C(F, R) of continuous functions Ψ : F → R, endowed with the compact-open topology [23], i.e. K + K ⊂ K, tK ⊂ K for every t ≥ 0, K ∩ (−K) = {0 X } and Int K = ∅. (b) Note that the results obtained in this article could be derived with weaker assumptions than in (G 3 ), such as relaxing (2.1) for x ≥ ν for some ν > 0, and assuming measurability on R and Φ (with respect to the Borel σ-algebras of R + and F respectively) instead of continuity. However, further properties have to be derived in order to extend appropriately the approach developed in this paper. For instance assuming only measurability of R, it can be proved, since R is assumed to be subadditive, that R is bounded on compact subsets of R + , see e.g. [22], lemma 1, p. 167; a property that would appear to be important for extending the results of this article in such a context. We leave for the interested reader these possible extensions of the results presented hereafter. (c) Other generalization about R could be also considered, such as R(u + v) ≤ C(R(u) + R(v)), ∀ u, v ∈ R + , for some C > 0, allowing the fact that any positive power of a subadditive function is subadditive in that sense; but this condition would add complications in the proof of Theorem 3.1 for instance. We do not enter in all these generalities to make the expository less technical.
We need also to consider a function r : R + → R + , verifying the following assumptions.
Assumptions on the function r. We assume that r(x) = 0 if and only if x = 0, r is continuous at 0, r is nondecreasing, subadditive and for some statements we will assume further that, (A r ) r is lower semi-continuous for the simple convergence on F , i.e., for any sequence {f n } n∈N of self-mappings of F .
Cross condition. Finally, we will consider the following cross condition between the growth function R and the function r, As simple example of functions Φ, R, and r satisfying the above conditions (including (A r )), we can cite r(u) = u, Φ(x) = R( x ) = x + 1, that will be used to illustrate the main theorem of this article later on; see subsection 4.2.
Hereafter in this subsection, condition (A r ) is not required. We introduce now the following functional on Hom(F ) with possible infinite values, .
Note that, |f | Φ,r = 0 if and only if f = Id F (separation condition), where Id F denotes the identity map of F .
Definition 2.2. The Koopman operator on E R F associated to f ∈ Hom(F ) is defined as the operator U f given by: Remark 2.2. Classically, the Koopman operator is given with other domain such as L p (F ) [21] (generally p = 2) and arises naturally with the Frobenius-Perron operator in the study of ergodicity and mixing properties of measure-preserving transformations [16]. The Koopman operator addresses the evolution of phase space functions (observables), such as Φ above, described by the linear operator U f rather than addressing a direct study of the dynamics generated by f . This idea has been introduced by Koopman and von Neumann in the early 30's [29], and has paved the road of what is called today the spectral analysis of dynamical systems [16]. We propose here more specifically to link this spectral analysis with the geometrical problem of conjugacy.
We can now state the following proposition. Proposition 2.3. Consider R given as in Definition 2.1, and Φ ∈ E R F . Let r satisfy the above assumptions except (A r ), and such that (C r,R ) is satisfied. Introduce the following subset of Hom(F ), Then (H Φ,r , •) is a subgroup of (Hom(F ),•) and, for any f ∈ H Φ,r , the Koopman operator associated to f satisfies, and where the constants appearing here are as introduced above.
Proof. We first prove the subgroup property. Let x be arbitrary in F , and f , g in H Φ,r . Then, From (G 3 ) and the subadditivity of R, and since R( x ) ≤ γ −1 Φ(x), we get by using (C r,R ) and (G 1 ), with C finite since |g −1 | Φ,r exists by definition of H Φ,r . Going back to (2.6) we deduce that, which concludes that f • g −1 ∈ H Φ,r , and H Φ,r is a subgroup of Hom(F ). The proof of (2.5) consists then just in a reinterpretation of (2.7).
Remark 2.4. Fairly general homeomorphisms are encompassed by the groups, H Φ,r , introduced above. For instance, in the special case Φ(x) = R( x ) := x + 1, and r(x) = x, denoting by H 0 , the group H Φ,r , and | · | Φ,r by | · | 0 for that particular choice of Φ, and r, the following two classes of homeomorphisms belong to H 0 and exhibit non-trivial dynamics.
(a) Mapping f of R d which are perturbation of linear mapping in the following sense: with T a linear automorphism of R d and ϕ a C 1 map which is globally Lipschitz with Lipschitz constant, Lip(ϕ), satisfying Lip(ϕ)< ||T −1 || −1 L(R d ) -that ensures f to be an homeomorphism of R d from the Lipschitz inverse mapping theorem (cf. e.g. [24] p. 244) -and ϕ(x) ≤ C( x + 1) (that ensures |f | 0 < ∞) for some positive constant; and such that the inverse of the differential of f has a uniform upper bound M > 0 in the operator norm, i.e., [Df (u)] −1 L(R d ) ≤ M for every u ∈ R d , which ensures |f −1 | 0 < ∞ by the mean value theorem. For instance, f (x) = x + 1 2 log(1 + x 2 ) provides such an homeomorphism of R. Note that every ϕ that is a C 1 map of R d with compact support and with appropriate control on its differential leads to an homeomorphism of type (2.9) that belongs to H 0 . (b) Extensions of the preceding examples to the class of non-smooth (non C 1 ) perturbations of linear automorphisms can be considered; exhibiting non-trivial homeomorphisms of H 0 . For instance, the two-parameter family of homeomorphisms, {L a,b , b ∈ R\{0}, a ∈ R}, known as the Lozi maps family [35]: where |·| denotes the absolute value here; constitutes a family of elements of H 0 . Indeed, it is not difficult to show that L a,b and its inverse, This family shares similar properties with the Hénon maps family. For instance there exists an open set in the parameter space for which generalized hyperbolic attractors exists [36].
We introduce now the following functional on H Φ,r × H Φ,r , which is well-defined by Proposition 2.3 and non-symmetric. Since obviously ρ Φ,r (f, g) ≥ 0 whatever f, and g, and ρ Φ,r (f, g) = 0 if and only if f = g, then ρ Φ,r is in fact a premetric on H Φ,r . Note that hereafter, we will simply denotes f • g by f g. Due to the non-symmetric property, two natural type of "balls" can be defined with respect to the premetric ρ Φ,r . More precisely, 3. An open ρ Φ,r -ball of center f to the right (resp. left) and radius α > 0 is the Proposition 2.5. Consider R given as in Definition 2.1, and Φ ∈ E R F . Let r satisfy the above assumptions except (A r ), and such that (C r,R ) is satisfied. Then, the premetric as defined in (2.11) satisfies the following properties.
(i) For every f, g, h, in H Φ,r , the following relaxed triangle inequality holds, (ii) The following families of subsets of H Φ,r , (iii) For all f ∈ H Φ,r , for all α * > 0, and for all g ∈ B − ρ Φ,r (f, α * ), the following property holds: and thus for all f ∈ H Φ,r , α>0 B − ρ Φ,r (f, α) is a fundamental system of neighborhoods of f , which renders T − (ρ Φ,r ) first-countable. Analogue statement holds with "+" instead of "-".
(iv) Let H Φ,r (−) denote the closure of H Φ,r for the topology T − (ρ Φ,r ), then Remark 2.6. Proof of (iii) below shows that an arbitrary open ρ Φ,r −ball (centered to the right or left) is not necessarily open in the sense of not being an element H of T +/− (ρ Φ,r ), since b > 0 and γ < β.
Proof. We first prove (i). Using the triangle inequality for · and subadditivity of r, it is easy to note that for all x ∈ F , From the following trivial equality, we deduce from Proposition 2.3, that, which reported in (2.13) gives, leading to (2.12), by re-writing appropriately (2.14) and repeating the computations with the substitutions f ← f −1 , g −1 ← g and h −1 ← h for the estimation of |f −1 g| Φ,r . The proof of (ii) is just a classical game with the axioms of a topology and is left to the reader. We prove now (iii), only for T − (ρ Φ,r ); the proof for T + (ρ Φ,r ) being a repetition. Let f ∈ H Φ,r and α * > 0. Let g ∈ B − ρ Φ,r (f, α * ), then from (2.12) we get for all h ∈ H Φ,r , We seek now the existence of α > 0 such that Denoting ρ Φ,r (f, g) by α ′ , such a problem of existence is then reduced from Eq. (2.15) to the existence of a solution α > 0 of, A necessary condition of existence is, that turns out to be sufficient since any α > 0 satisfying, is a solution because the RHS is positive. The second part of (iii) is a reinterpretation of the result just obtained. Indeed, we have proved that for all f ∈ H Φ,r , and for all α * > 0 there exists 0 < α * * < α * (since γ < β and b > 0 by definition), such that is open from what precedes. The first-countability holds then naturally and the proof of (iii) is complete.
We prove now (iv).
Then by definition of ρ Φ,r , we get in particular that |f f −1 n | Φ,r and |f −1 f n | Φ,r exist from which we deduce that |f | Φ,r and |f −1 | Φ,r exist, since H Φ,r is a group.

2.2.
Closure properties for extended premetric on Hom(F ). In this section, we extend the closure property (iv) of Proposition 2.5 to Hom(F ) itself (cf. Proposition 2.7) and prove a cornerstone proposition (Proposition 2.10) concerning the convergence in T − (ρ ′ Φ,r ) of sequences taking values in Hom(F ), where ρ ′ Φ,r denotes the extension of the premetric ρ Φ,r to Hom(F )×Hom(F ).
The result described in Proposition 2.7 will allow us to make precise conditions for which the solution of the fixed point theorem proved in the next section, lives in H Φ,r . This specific result is not fundamental for the proof of the main theorem of conjugacy of this article, Theorem 4.3; whereas Proposition 2.10 will play an essential role in the proof of the fixed point theorem, Theorem 3.1, and by the way in the proof of Theorem 4.3. Important related concepts such as the one of incrementally bounded sequence are also introduced in this subsection.
We define ρ ′ Φ,r as the extension of the premetric ρ Φ,r to Hom(F )×Hom(F ), by classically allowing ρ ′ Φ,r to take values in the extended real line R := R ∪ {∞} instead of R; with the usual extensions of the arithmetic operations 2 .
According to this basic extension procedure, it can be shown that T +/− (ρ ′ Φ,r ) is a topology on Hom(F ) and that Proposition 2.5 can be reformulated for ρ ′ Φ,r with the appropriate modifications. Note that in particular the relaxed triangle inequality (2.12) holds for ρ ′ Φ,r and any f, g and h in Hom(F ). Indeed, either f g −1 and f −1 g belong to H Φ,r and (2.12) obviously holds; either f g −1 or f −1 g does not belong to H Φ,r and then -because of the group structure of H Φ,r -for all h ∈ Hom(F ), we get that necessarily at least one element of the following list {f h −1 , f −1 h, hg −1 , h −1 g} does not belong to H Φ,r ; which still leads in all the cases to a validation of analogues to (2.12) for ρ ′ Φ,r . We are now in position to introduce contingent conditions to our framework that are required to obtain closure type results in Hom(F ). These conditions present the particularity to hold in T + (ρ ′ Φ,r ) for any sequences involved in the closure problem related to the topology T − (ρ ′ Φ,r ). More precisely we introduce the following concepts.
Definition 2.4. We say that a sequence {f n } of elements of Hom(F ) is incrementally bounded in T + (ρ ′ Φ,r ) with respect to q, for some q ∈ N, if and only if: 2 Note that similarly, | · |Φ,r may be extended in such a way, but it is important to have in mind that |f | ′ Φ,r = ∞ and |g| ′ Φ,r = ∞ do not necessarily imply that ρ ′ Φ,r (f, g) = ∞. This for instance the case for When no precision on q is given, we just say that the sequence is incrementally bounded. Furthermore, the sequence is said to be uniformly bounded in T + (ρ ′ Φ,r ) if and only if: . The "(-)-statements" consist of changing the role of p and q in the above statements. We denote by IB + the set of incrementally bounded sequences in T + (ρ ′ Φ,r ) and by IB + u its subset constituted only by uniformly incrementally bounded sequences.
to be the set constituted by the adherence values in We have then the important proposition that completely characterizes the limits in T − (ρ ′ Φ,r ) (that leaves Hom(F ) stable) of sequences of Hom(F ) which are incrementally bounded in T + (ρ ′ Φ,r ) with respect to some n 0 for which f n 0 belongs to H Φ,r .
be as introduced in Definition 2.5, then, Proof. Let f ∈ Hom(F ) (2.20) Since {f n } is incrementally bounded in T + (ρ ′ Φ,r ) with respect to n 0 , then from (2.20), we deduce that the real-valued sequence {|f n | Φ,r } n≥n 0 is bounded.
Besides, for any x ∈ F , and any n ≥ n 0 , , and therefore by using the estimate (2.5) in Proposition 2.3, since Λ(|f n | Φ,r ) is well defined for n ≥ n 0 because |f n | Φ,r exist for such n. We get then triv- by assumption, we then deduce that |f | Φ,r exist by taking n sufficiently large in (2.21). To conclude it suffices to note that (2.20) shows thanks to the incrementally bounded property, that {|f −1 n | Φ,r } n≥n 0 is bounded as well, leading to the boundedness of {Λ(|f −1 n | Φ,r )} n≥n 0 which by repeating similar estimates leads to the existence of |f −1 | Φ,r by using the assumption: In the sequel we will need sometimes to use the following property verified by the function r. Let r satisfy the condition of the preceding subsection and let G denote a continuous function G : F → E, then for any K compact of F , there exists x K ∈ K, such that r sup x∈K G(x) = r( G(x K ) ) = sup x∈K r( G(x) ), since r is increasing. Since we will need this property of r later, we make it precise as the condition, for every continuous function G : F → E, which holds therefore for r as defined above.
In what follows, ρ Φ,r will refer for both ρ Φ,r when applied to elements of H Φ,r , and to ρ ′ Φ,r when applied to elements of Hom(F ), without any sort of confusion.
Let us now introduce the following concept of Cauchy sequence in H Φ,r or more generally in Hom(F ), which is adapted to our framework.
Remark 2.8. Note that, since ρ Φ,r is not symmetric, the role of p and q are not symmetric as well, to the contrary of the classical definition of a Cauchy sequence in a metric space.
. However, a sequence {f n } in Hom(F ) which is ρ + Φ,r −Cauchy is not an element of IB + u in general but is an element of IB + . We are now in position to prove the following cornerstone proposition concerning the convergence in T − (ρ Φ,r ) of ρ + Φ,r -Cauchy sequences in Hom(F ). Proposition 2.10. Assume that F is a non-compact subset of E which is locally connected, σ-compact and locally compact 3 . Consider R given as in Definition 2.1, and Φ ∈ E R F . Let r satisfy the above assumptions including (A r ), and such that (C r,R ) is satisfied. Let {f n } be a sequence in Hom(F ). If the following conditions hold: (C 1 ) For every compact K ⊂ F , the sequence of the restriction of f n to K, {f n | K }, is bounded, as well as the sequence of the restrictions of the inverses, {f −1 n | K }, If furthermore, either {f n } is a sequence of homeomorphisms living in H Φ,r , or more generally {f n } is such that f n 0 ∈ H Φ,r for some n 0 , then {f n } converges in T − (ρ Φ,r ) towards an element of H Φ,r .
Proof. The proof is divided in several steps.
Step 1. Let {f n } be a sequence of homeomorphisms of F fulfilling the conditions of the proposition. Let ǫ > 0 be fixed. Then from (C 2 ), there exists an integer N such that p ≥ q ≥ N implies, (2.22) which for every compact K ⊂ F , leads to, by assumption (C 1 ) and the continuity of Φ. Similarly, p ≥ q ≥ N , implies, which, for every compact K ⊂ F , can be summarize with (2.23), as, by using (S) and labeling still by M K the greater constant. Now from the continuity of r at 0 and r(0) = 0 if and only if x = 0, we deduce that the sequences {f n } and {f −1 n } converge uniformly on each compact towards respectively a map f : F → F and a map g : F → F . Note that by choosing appropriately a family of compact subsets of F , covering F , we get furthermore that f and g can be chosen continuous on F .
Step 2. Our main objective here is to show that f = g −1 , i.e. f ∈ Hom(F ). Since F is assumed to be σ-compact and locally compact, there exists an exhaustive sequence of compacts sets {K k } k∈N of F ; see e.g. corollary 2.77 in [2]. From step 1, (2.25) is valid for any p ≥ q ≥ N, with N which does not depend on the compact K, and therefore we get that {f n } is Cauchy for the metric ∆ on Hom(F ), given by, any compact K k , and any homeomorphisms of F , φ and ψ. Indeed, it suffices to note that for a given ǫ ′ > 0, from (2.25) there exists l, ǫ and N ǫ such that, which leads to δ(f p , f q ) ≤ ǫ ′ /2, and similarly to δ(f −1 p , f −1 q ) ≤ ǫ ′ /2, for any p ≥ q ≥ N ǫ . Now, since F is locally compact and locally connected from a famous result of Arens (see theorem 4 of [3]), Hom(F ) is complete for ∆ which is a metric compatible with the compact-open topology [23], making Hom(F ) a Polish group 4 . Therefore {f n } converges in the compact-open topology towards an element h ∈ Hom(F ). By recalling that the compact-open topology is here equivalent to the topology of compact convergence [38], we obtain by uniqueness of the limit that f = h ∈ Hom(F ); f being the limit of {f n } in the topology of compact convergence from step 1.
Step 3. Let us summarize what has been proved. We have shown under the assumptions (C 1 ) and (C 2 ), that we can "exhibit" from any sequence {f n } which is ρ + Φ,r -Cauchy, an element f ∈ Hom(F), such that {f n } converges uniformly to it on each compact subset of F , and {f −1 n } does the same towards f −1 . In fact we can say more with respect to the topology of convergence.
Indeed, going back to (2.22), we have that, 4 Note that this reasoning is independent from the choice of the metric rendering Hom(F ) complete, since it is known that Hom(F ) as a unique Polish group structure (up to isomorphism); see [26] i.e. by making p → +∞ and using (A r ), . From similar estimates, we get, We have thus shown the convergence of {f n } in T − (ρ Φ,r ), that is, At this stage, we have proved that {f n } converges in T − (ρ Φ,r ) towards an element of Hom(F ).
Step 4. This last step is devoted to the proof of the last statement of the theorem concerning the membership of the limit of {f n } to H Φ,r . This fact is simply a consequence of Proposition 2.5-(iv) in the case {f n } ∈ H N Φ,r , and a consequence of Remark 2.9, and Proposition 2.7 in the case {f n } ∈ (Hom(F)) N , which gives in all the cases that the limit in T − (ρ Φ,r ) of {f n } lives in H Φ,r . The proof is therefore complete.
Lastly, it is worth to note that it is only in step 4 of the preceding proof that was needed assumption (A r ), but since Proposition 2.10 will be used in the proof of Theorems 3.1 and 4.3, we will make a systematic use of this assumption in the sequel.

A fixed point theorem in the homeomorphisms group
In this section we state and prove a new fixed point theorem valid for self-mappings acting on Hom(F ), which holds within the functional framework developed in the preceding section. This fixed point theorem uses a contraction mapping argument that involves a Picard scheme that has to be controlled appropriately due to the relaxed inequality (2.12). In that section ρ Φ,r will stand for the extended premetric introduced in the last subsection.
Theorem 3.1. Consider R given as in Definition 2.1, and Φ ∈ E R F , with F as in Proposition 2.10. Let r satisfy the above assumptions including (A r ), and such that (C r,R ) is satisfied. Let Υ : Hom(F ) → Hom(F ) be an application. Let {f n } be a sequence in Hom(F ). We assume that there exists h 0 ∈Hom(F ) such that the following conditions hold: is bounded on every compact of F . Assume furthermore that there exists a constant 0 < C < 1, such that, then there exists a unique h ∈ Hom(F ) such that Υ(h) = h, which is obtained as a limit in Proof. From Proposition 2.10, it suffices to show that {Υ n (h 0 )} n∈N is ρ + Φ,r -Cauchy for any h 0 ∈ Hom(F ) satisfying (i). For simplifying the notations we denote by ρ m n the quantity ρ Φ,r (Υ m (h 0 ), Υ n (h 0 )), for m ≥ n. Note that since δ is finite, by recurrence and using the contraction property (3.1) we can show that all the quantities ρ m n are finite as well.
By using the relaxed inequality (2.12), the contraction property (3.1) and the definition of A in condition (i), it's easy to get for any integers n, m ≥ n + 1, and k ≥ 1, which leads to, by using Aδ < 1 from assumption (i) and the notations specified above. Let ǫ > 0 be fixed. For any m and n, we introduce now the two-parameters sequence {F k (m, n)} k∈N defined by recurrence through, When no ambiguity is possible, F k will simply stand for F k (m, n). The role of m and n will be apparent in a moment. Moreover, from (3.3), it's easy to show that for any n and m ≥ n + 1, As we will see, it suffices to consider m = n + 1 to prove the theorem, a choice that we make in what follows. Since C < 1, obviously, and therefore we get that ρ n+1+k n ≤ F k (n + 1, n), from what precedes.
The key idea is now to note that if for all k ≥ 0, F k (n + 1, n) ≤ 2ǫ, for n sufficiently big, then the sequence {Υ n (h 0 )} n∈N is ρ + Φ,r -Cauchy for h 0 ∈ Hom(F ) fulfilling condition (i). In the sequel we prove that it is indeed the case.
To do so, an easy recurrence shows that, ∀ k ∈ N, F k > 0, and {F k } is strictly increasing.
In particular, ∀ k ∈ N, F k ≥ ǫ, and therefore for any k ≥ 1 and m, Thus by using (3.4) and iterating (3.5), we obtain for any k ≥ 1, with the convention k=1 l=2 C m+l−2 1 + 1 ǫ + 1 ≡ 1, making valid (3.6) for k = 1. Since C < 1, then for any l ∈ {2, ..., k}, and k ≥ 2, C m+l−2 ≤ C m , which leads from (3.6) to, which is also valid for k = 1, by simply computing F 1 − F 0 . Besides, which shows for m ≥ N 2 , since C < 1 and a m < 1. Now it can be shown, Fixing m = n + 1, we conclude from (3.9) and the trivial identity which shows in particular that, We have thus proved that {Υ n (h 0 )} n∈N is ρ + Φ,r -Cauchy for any h 0 ∈ Hom(F ) fulfilling conditions (i) and (ii), and thus by Proposition 2.10, h := lim It can be shown furthermore that, for every h 0 ∈ Hom(F ), {Υ n (h 0 )} n∈N is incrementally bounded with respect to any n 0 ∈ N, due to the contraction property and the fact that {Υ n (h 0 )} n∈N is ρ + Φ,r -Cauchy. In consequence, if there exists n 0 such that Υ n 0 (h 0 ) ∈ H Φ,r then by applying Proposition 2.10 again (last part) we obtain h ∈ H Φ,r .

A conjugacy theorem and the generalized spectrum of the Koopman operator
4.1. The conjugacy theorem. We prove in this section the main result of this article, i.e the conjugacy Theorem 4.3. To do so, we need further preliminary tools and notations that we describe hereafter. In that section we assume the previous assumptions on r (see Section 2) including the condition (A r ). As in Section 3 the premetric ρ Φ,r will stand for the extended premetric introduced in subsection 2.2. In what follows, we endow again Hom(F ) with a topology T − (ρ Φ,r ) where r and R satisfy the condition (C r,R ) of Section 2, except that here Φ belonging to E R F is not arbitrary and has to solve a generalized eigenvalue problem to handle the conjugacy problem; cf Theorem 4.3.
For any self-mapping f of F , we introduce the following r-Lipschitz constant, which can takes infinite values. This quantity is clearly the direct extension assessed in the map r( · ), of the Lipschitz constant Lip(f ) classically assessed in · .
Definition 4.1. We denote by L r (F ) the set of homeomorphisms of F such that λ r (f ), and λ r (f −1 ) exist. Such an homeomorphism is called an r-Lipeomorphim of F .
We will also need the following concept of generalized eigenvalue of the Koopman operator, outlined in the introduction. Definition 4.2. Let f ∈ Hom(F ), and U f its Koopman operator with domain E R F . A generalized eigenvalue of U f is any λ ∈ R such that, where in case of existence, Φ is the corresponding generalized eigenfunction.
Lastly, given f and g in Hom(F ), we introduce the following classical conjugacy operator, We are now in position to state and prove the main theorem of this article, a conjugacy theorem which is conditioned to a generalized eigenvalue problem of the related Koopman operators. (a) f, g ∈ L r (F ), (b) there exists α > 1 and a common generalized eigenfunction Φ ∈ E R F of the Koopman operators U f and U g , which solves the following generalized eigenvalue problem, Under these conditions, for any Φ solving (4.3), assume further that there exists an homeomorphism h 0 of F satisfying the following properties: (i) δ := ρ Φ,r (L f,g (h 0 ), h 0 ) < min(1, A −1 ), where A = max(aβ, bm −1 β + βγ −1 ), (ii) {L n f,g (h 0 )} n∈Z is bounded on every compact of F . Then f and g are conjugated by a unique element h of Hom(F ), which is the limit in T − (ρ Φ,r ) of {L n f,g (h 0 )} n∈N . Furthermore, if there exists n 0 ∈ N such that L n 0 f,g (h 0 ) ∈ H Φ,r , then h ∈ H Φ,r . Proof. Let f and g be two homeomorphisms of F . Let the function r and the growth function R be such that the conditions (a) and (b) of Theorem 4.3 are satisfied. Let Φ ∈ E R F be a solution of (4.3). We endow then Hom(F ) with the topology T − (ρ Φ,r ) for such a Φ.
Since the existence of a solution in Hom(F ) to the conjugacy problem is equivalent to the existence of a fixed point in Hom(F ) of L f,g , it suffices from Theorem 3.1 to examine if L f,g is a contraction in the premetric ρ Φ,r . To do so, we need to estimate which leads to, Similar computations show, and since Φ solves the generalized eigenvalue problem P α , we get for all u ∈ F , which allows us to conclude that, for all h 1 , and h 2 in Hom(F ), i.e. the conjugacy operator L f,g is a contraction for the premetric ρ Φ,r . The rest of the assumptions (i) and (ii) of Theorem 4.3 are just a translation of the ones used in Theorem 3.1, and thus by using this last theorem the proof of the present one is easily achieved.
Remark 4.4. This theorem presents the particularity to provide conditions under which, when it exists, the conjugacy h, lives in H Φ,r . This means in such a case that h satisfies some behavior at infinity prescribed by Φ, which has in turn to solve P α , a spectral problem related to the Koopman operators U f , and U g which involves structural constants of f and g: λ r (f ) and λ r (g) as introduced above. This aspect could be of interest in control theory.
Remark 4.5. Condition of type (i) in Theorem 4.3 is often met in a stronger form when dealing with (local) conjugacy problems that arise around a fixed point. Indeed it is often required that the convergence of the sequence {f n g −n } holds in C 0 provided that f and g are tangent to sufficiently high order (cf. e.g. [12], Lemma 3 p. 95).

4.2.
An illustrative example. To simplify we set E = R and we consider F = [0, +∞) which fulfills the conditions of Theorem 4.3. We consider furthermore r(x) = x and Φ(x) = R(x) = √ x + 1 (for x ∈ F ) that are subadditive function on F . Note that such an R satisfies the condition of Section 2, and that Φ ∈ E R F with β = 2 and γ = 1/2 for instance. Note also that r satisfies (A r ) (and (S)) and fulfill all the other standing assumptions. The cross condition (C r,R ) of Section 2 is fulfilled since, We consider the following dynamics on F : f = ηx with 0 < η < 1 and g(x) = ηx + ϕ(x), where ϕ ∈ C 1 (R + , R + ) with compact support is such that g is an homeomorphism of F ; ϕ will be further characterized in a moment. Other conditions on ϕ will be imposed below. Lastly note that λ r (f ) = Lip(f ) for r(x) = x. Since η < 1, there exists ǫ > 0 such that, with α = 1 + ǫ, which shows that Φ is a generalized eigenfunction in E R F of U f with eigenvalue λ = (1 + ǫ)Lip(f ) in that particular context. From (4.9), we get, (4.10) ∃ ǫ 2 > 0, : √ η η + ǫ 2 > (1 + ǫ), and ǫ 2 < η.
Besides, for such an ǫ 2 , there exists a function ϕ ∈ C 1 (R + , R + ) with compact support such that, (4.11) Lip(g) = max x∈F |η + ϕ ′ (x)| = η + ǫ 2 , and such that g is still an homeomorphism of F (since ǫ 2 < η). From (4.11) and (4.10) we get now, which shows that Φ is a generalized eigenfunction in E R F of U g with λ = (1 + ǫ)Lip(g). We are then left with a common eigenfunction of U f and U g satisfying P α with α = 1 + ǫ. From our assumptions, it's easy to check furthermore that f and g belong to L r (F ), and therefore conditions (a) and (b) of Theorem 4.3 are satisfied in that particular setting. Recall from the proof of Theorem 4.3 that these conditions ensure the contraction of the conjugacy operator L f,g for the premetric ρ Φ,r .
To apply Theorem 4.3 we have now to check the remaining conditions (i) and (ii). We check first condition (i) for h 0 = g. Note that there exists ψ ∈ C 1 (R + , R − ) such that g −1 (x) = x/η + ψ(x), and ψ is with compact support. Furthermore in that particular setting, A = 9/4 for A as defined in condition (i) of Theorem 4.3. We have to estimate Note that we can find ϕ and thus ψ such that, without violating (4.11) and thus having Φ still satisfying P α . For such a choice, ην < η 2 A −1 < A −1 since η < 1, and in particular, and, (4.14) which allows us to conclude that condition (i) is checked with h 0 = g.
Since ϕ and ψ are with compact support then it can be shown that for h 0 = g, the sequence {L n f,g (h 0 )} n∈Z is bounded on every compact of F . We leave the details to the reader. We can thus apply Theorem 4.3 to conclude that f and g are conjugated which was of course obvious for Lip(ϕ) sufficiently small, from a trivial application of the global Hartman-Grobman theorem, cf. for instance [24].
This modest example is just intended to illustrate some mechanisms of the approach developed in this article. Of course, further investigations are needed with respect to the existence of solutions of the generalized eigenvalue problem P α in spaces of type E R F , for more general homeomorphisms. We postpone this difficult task for a future work, discussing in the next subsection some related issues.

4.3.
Generalized eigenvalues of the Koopman operator. We describe here two possible approaches to examine the generalized eigenvalue problem P α , the first one is based on Schröder equations and the second one is based on cohomological equations. The point of view retained is based on functional equations techniques coming from different part of that literature where we emphasize the overlapping. In both cases, by shortly reviewing the existing results, we provide hereafter conditions under which the generalized eigenvalue problem P α may possess continuous solutions, without being able to specify -in a general setting -if Φ can live in E R F . We make precise these aspects below, in the perspective to gather some results that were connected to our problem, and that we have found to be sparsely distributed in the literature. Note that in the sequel, we will focus more precisely on the generalized eigenvalue problem for U f and not P α itself, in order to exhibit already the main issues for the associated single existence problem of a generalized eigenfunction.  [28]. The reader may consult [42] or [14] with references therein, for a recent account about this part of the literature. This functional problem has also been considered historically for map of the half-line [31] or more general Banach spaces in the past decades [44]. We mention lastly, that the Schröder equation is sometimes encountered under the form of the Poincaré functional equation [31,25] and arises in various applications such that iterated function theory [19,32], branching process [41] or dynamical systems theory (see for instance [9] or [45]). Naturally, the generalized eigenvalue problem P α can be related to Schröder equations. Indeed, if the following Schröder equation, has a solution Ψ : F → F for α > 1, then the generalized eigenvalue problem for U f has an obvious solution, provided that Φ(·) = Ψ(·) ∈ E R F and f ∈ L r (F ). It is known that such an equation can be solved for particular domain F and particular space of function over F such as Hardy spaces; see [17]. It is interesting to note that most of the results typically demand some compacity assumptions of the Koopman operator U f which involve that f possesses at least a fixed point in F ; cf. [11], cf. also Theorem 5.1 in [14] for extensions of results of [11].
There exist other techniques coming from functional analysis rather than complex analysis, to deal with P α from the point of view of Schröder equations. It consists of considering a more general type of Schröder equations where the unknown is a map Ψ : F → E aiming to satisfy, where A is a linear map of E. If we assume that A is invertible, and that there exist a C 0functional N : E → R with a constant m > 0 satisfying, then Theorem 2.1 of [44] permits to conclude to the existence of a continuous nonzero solution of (4.17). Note that if F ⊂ E\{0} and f (x) ≤ κ x on F , with κ ∈ (0, 1) then the functional inequality (4.18) is satisfied on F by simply taking N (x) := − log x and m := − log(κ). Now by assuming A invertible, and thus if, and there exists a couple (m, N ) satisfying (4.18), we obtain that Φ(·) := Ψ(·) with Ψ a solution of (4.17), is a solution of U f (Φ) ≥ αλ r (f ) · Φ. Thus in order to have a solution of the generalized eigenvalue problem for U f , we only need to know about the asymptotic behavior of Φ. However, the examination of the growth of an eigenfunction Ψ solution of (4.17) is a difficult task in general (see e.g. [10] or [4] for particular cases more related to situations like the "basic" Schröder equation (4.16)), which renders the generalized eigenvalue problem for U f and thus the functional problem P α introduced here non-trivial to solve in general.
To conclude, we emphasize that the Schröder equation is related to Abel's functional equation [1], which is a well known functional equation often presented into the form ϕ(f (x)) = ϕ(x) + 1, where ϕ : X → C is an unknown function and f : X → X is a given continuous mapping of a topological space X [31]. 6 With respect to our purpose, if the Abel equation possesses a continuous solution, then it provides a continuous solution to (4.18) that attributes to Abel equation a central role in our approach. In fact, the relationships between both equations with respect to our functional problem are deeper as illustrates the following subsection.

4.3.2.
Approach based on cohomology equations. Even if, to the best of the knowledge of the authors, Theorem 4.3 exhibits new relationships between the existence of a conjugacy between two homeomorphisms and the spectrum of the related Koopman operators, relationships between conjugacy problems and functional equations are far to be new. They arise classically under the form of the Livshitz cohomology equation [33,34], φ = Φ • f − Φ, where f : M → M is a dynamical system of some manifold M , φ : M → R, a given function, and Φ maps M into R or a multidimensional space; see for instance [5,6,27,18]. As pointed by Livshitz [33,34] the existence of a continuous solution Φ strictly depends on the dynamics generated by f and the topological as well as geometrical properties of M . For instance if we consider the particular case of the Abel equation, Φ(f (x)) = Φ(x) + 1, there is no continuous solution if M is compact, since if such solution exists, Φ(f n (x)) = Φ(x) + n, which is impossible.
In the case of non-compact topological manifold, Belitskii and Lyubich have proved the following theorem (see [7]; Corollary 1.6), that we present in a slightly less general setting, adapting their statements with respect to our purpose:  (c) Every compact subset of M is wandering for f .
In the above theorem, a compact set K ⊂ M , is qualified to be wandering if there exists an integer ν ≥ 1 such that f n (K) ∩ f m (K) = ∅ (n − m ≥ ν), in particular such a dynamical system f is fixed-point free and periodic-point free, which might be consistent with dynamical restriction imposed by any solution to the functional problem P α in the case λ r (f ) ≥ 1. 7 This powerful theorem provides however an incomplete solution to the problem P α . For instance, Theorem 4.6 shows that for f satisfying condition (c) above, there exits a solution of the equation ϕ(f (x)) = λϕ(x) with |λ| ≥ αλ r (f ), and therefore a solution of the generalized eigenvalue problem for U f ; obtained by taking its module, Φ(·) := |ϕ(·)|. The missing step for having a full solution of that problem is still the knowledge of the asymptotic behavior of Φ, which is a difficult task also for cohomology equations; see e.g. [6].
Remark 4.7. It is worth mentioning that in the framework developed in this article, the subset F plays a central role in the existence of solutions of functional problem of type P α . The reader may consult for instance [9], §5.2, where it is shown that the Abel equation associated to a contracting mapping of the cut plane R 2 \(−∞, 0], has a real-analytic solution, whereas the same map considered on the whole plane leads to an Abel equation without any continuous solution, since such a map possesses obviously a fixed point which is excluded by Theorem 4.6. Whatever F and the approach retained, we can conclude that the main issue concerns therefore the asymptotic behavior of a possible eigenfunction of the generalized eigenvalue problem P α . The regularity of the common eigenfunction is also an important aspect of the problem, in that respect, the continuity assumption on R and thus on Φ could be relaxed using Remark 2.1. Dynamical properties such as condition (c) of Theorem 4.6 might play also a role in the existence of such eigenfunctions. Note also, that since the closure of E R F in the compact-open topology [23] is a closed cone with non empty interior in C(F, R) (cf. Remark 2.1-(a)), it would be interesting to study the spectral properties of U f and U g within an approach of type Krein-Rutman theorem [39]. However, the task is more difficult than usually since F is non-compact, σ-compact and locally compact, and therefore C(F, R) has a Fréchet structure [2](and not a Banach one), which renders non straightforward an extension of the classical Krein-Rutman theorem in that context to analyze the existence of a principal eigenfunction in E R F . Finally, we have intentionally not considered important dynamical properties such as hyperbolicity [27] that could lead to other spectral problem than P α ; the purpose of the present work was indeed more to set up, in a general context, a framework that makes apparent certain relationships between the spectral theory of dynamical systems and the geometrical problem of conjugacy. Likewise, the idea of using some observable -here a common eigenfunction of the Koopman operators -to build a specific topology to deal with the conjugacy problem does not seem to be limited to the case of non-compact phase-space. We leave open all these interesting fields of investigations. 7 Indeed, if there exists a generalized eigenfunction Φ of U f and a periodic orbit of f of period p emanating from some x * , then by repeating p-times the change of variable x ← f (x) in Φ(f (x)) ≥ αλr(f )Φ(x), we deduce necessarily that (αλr(f )) p ≤ 1 (since Φ > 0), which imposes that f cannot possess such a periodic orbit in the case λr(f ) ≥ 1.