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VARIATIONAL PRINCIPLES FOR THE TOPOLOGICAL PRESSURE OF MEASURABLE POTENTIALS

. We introduce notions of topological pressure for measurable potentials and prove corresponding variational principles. The formalism is then used to establish a Bowen formula for the Hausdorﬀ dimension of cookie-cutters with discontinuous geometric potentials.


1.
Introduction. Let (X, d) be a compact metric space and T : X → X be a continuous transformation. Throughout this paper we consider (X, T ) to be a timediscrete dynamical system. An important notion in the field of dynamical systems and its associated thermodynamic formalism is the topological pressure. For a function ϕ : X → R, the topological pressure with respect to (X, T ) on a given subset Z ⊆ X is defined to be P Z (T, ϕ) := lim where the supremum is taken over all ( , n)-separated sets E in Z. Above definition was introduced and discussed for Z = X and ϕ ∈ C(X, R) in [18]. The variational principle was also proven there: One has where the supremum is taken over all ergodic T -invariant Borel probability measures µ on X. The aim of this paper is to extend the definition of pressure to not necessarily continuous functions ϕ, and to prove a corresponding variational principle. Up to now there seem to be at least two systematic attempts to treat this task. In [15] dynamical systems were considered, where the invariant set under study can be exhausted by an increasing sequence of subsets, such that ϕ is continuous on the closure of each subset. Consequently, the topological pressure is then defined to be the supremum of the topological pressures of ϕ on those sets. A corresponding variational principle holds under some integrability assumptions. A variational principle for sub-additive, upper semi-continuous sequences of functions was established in [4] and [2], as well as in [13] for Z d + -actions. A generalization was recently given in [11] for weighted topological pressure on systems with upper semi-continuous entropy mapping. We note, that in [15] and [11] Carathéodory dimension type definitions of pressure (as introduced and discussed in [17] and [16]) were used, whereas [4] and [13] extended the classical topological pressure, defined via separated sets.
In this paper we also stick to the original pressure definitions given in [18]. More precisely, we extend those definitions of pressure to discontinuous ϕ, and compare them to the classical ones. Furthermore we determine several classes of functions, which admit variational inequalities and principles. Various examples are given. In particular we construct an example for a potential, which is not upper semicontinuous and for which the formalism of [15] cannot be applied, but which admits a variational principle for the pressure considered here.
As an application, we establish a Bowen formula for the Hausdorff dimension of attractors of cookie-cutters with discontinuous geometric potentials. This is done by connecting Hofbauer's Bowen formula [12] to the pressure defined in (1) (see Theorem 7.4 and Remark 21 (a)). We then use this relation to show a continuity property of the Hausdorff dimensions of a sequence of cookie-cutters (see Theorem 8.2 and Remark 23).
2. Main results. Let (X, T ) be a dynamical system and ϕ : X → R be a measurable function. For every subset Z ⊆ X, define P Z (T, ϕ) as in (1).
Theorem A (Mass Distribution Principle, Theorem 5.2). Let Z ⊆ X be a Borel set. If µ is a Borel probability measure on X satisfying µ(Z) > 0, one has Above result is well known for ϕ ∈ C(X, R), and we show in the present paper, that the method of proof works well in the more general setting of measurable functions. Using Brin-Katok's theorem (see [3]), we can derive the following variational inequality: Theorem B (Variational Inequality, Theorem 5.3). Let h top (T ) < ∞ and µ be a T -invariant ergodic Borel probability measure. If ϕ : X → R is quasi-integrable with respect to µ, then there exists a Borel set G ⊆ X such that µ(G) = 1 and In particular, if ϕ : X → R is quasi-integrable with respect to T (see Definition 5.4), then where the supremum is taken over all T -invariant Borel probability measures µ on X.
Inequality (4) was already proven for upper semi-continuous functions in [4] and [13] (see Remark 15). In both proofs it is used, that ϕ is bounded from above. Definition 3.6. Let (Ω, A, µ) be a probability space and ϕ : Ω → R be a measurable function. We call ϕ to be quasi-integrable with respect to µ, if either Quasi-integrable functions share some important properties with integrable functions. Some of them are recalled in the next two lemmas: Lemma 3.8. Let f, g : Ω → R be quasi-integrable functions with respect to µ, such that Ω f dµ+ Ω g dµ is well-defined. Then f +g is well-defined µ-almost everywhere and quasi-integrable with respect to µ, and The ergodic theorem of Birkhoff and the ergodic decomposition theorem can be restated for quasi-integrable functions. Assume (X, T ) to be a dynamical system for the rest of this section. Theorem 3.9. Let µ ∈ M T (X) and ϕ : X → R be quasi-integrable with respect to µ. Then there exists some quasi-integrable function ψ : X → R with respect to µ, such that ψ • T = ψ µ-almost everywhere and and in particular, if µ is ergodic, one has Theorem 3.10. Fix some µ ∈ M T (X) and denote by m µ the ergodic decomposition of µ, that is µ = E T (X) ν dm µ (ν). If ϕ : X → R is quasi-integrable with respect to µ, then one has In particular, E T (X) ν → X ϕ dν is a quasi-integrable with respect to m µ .

4.
Topological pressure for arbitrary potentials. In this section we introduce three notions of topological pressure for not necessarily continuous potentials. Let (X, T ) be a dynamical system.
1. An function ϕ : X → R is called potential. For given potential ϕ, ∅ = Z ⊆ X, > 0 and n ∈ N define where the supremum is taken over all ( , n)-separated sets E in Z. Likewise, define where the infimum is taken over all (δ, n)-covers F of Z. Both definitions make sense: The set { z } is ( , n)-separated for all z ∈ Z, > 0 and n ∈ N, and by Lemma 3.4 there exists always a (δ, n)-cover for Z. Both limits exist, as every ( , n)-separating set is also ( , n)-separating for 0 < < , and every (δ , n)-cover is a (δ, n)-cover for 0 < δ < δ. The quantity P Z (T, ϕ) is called upper topological pressure of ϕ on Z with respect to T , and Q Z (T, ϕ) is called lower topological pressure of ϕ on Z with respect to T .
Remark 1. In case ϕ : X → R is continuous, the definitions of Q X (T, ϕ) and P X (T, ϕ) coincide with the classical definitions given in [19]. In particular, one has then by [19] Theorem 9.1 Q X (T, ϕ) = P X (T, ϕ). However, as we shall see in Remark 5, above equality does not hold in general for discontinuous ϕ.
Remark 2. By definition, the quantity M Z (T, ϕ, , n) is always finite. In contrast, M Z (T, ϕ, , n) may take values in [0, +∞]. We also have the estimate So if ϕ is bounded from above, M Z (T, ϕ, , n) < ∞ follows from [19] 7.2 Remark (5). We also note that Q Z (T, ϕ, δ), P Z (T, ϕ, ) ∈ [−∞, +∞] for all Z ⊆ X and Q ∅ (T, ϕ) = P ∅ (T, ϕ) = −∞. If ϕ is not bounded from above, but from below on one trajectory, we have the following: Proof. Choose for every n ≥ 1 some k n ≥ 0 such that ϕ(T kn x 0 ) ≥ 2 n + (n + 1) · C. Set x n := T kn x 0 . Then Thus, as { x n } is a ( , n)-separated set in X for every > 0, The next lemma follows readily from the definitions of the pressure: For every x ∈ X one has in addition where the infimum is taken over all (δ, n)-covers F of Z. By Lemma 3.4 above quantity is well-defined and finite for all ∅ = Z, and we set M ∅ (T, ϕ, δ, α, n) := 0 Next define for all Z ⊆ X, δ > 0 and α ∈ R M Z (T, ϕ, δ, α) := lim sup n→∞ M Z (T, ϕ, δ, α, n).
Proof. Denote ϕ n (x) := n−1 i=0 ϕ(T i x). Let C ∈ R and F n a (δ, n)-cover of Z such that x∈Fn exp − α · n + ϕ n (x) < M Z (T, ϕ, δ, α) + 1 < C for all n > N 0 , N 0 large enough. Then The last lemma justifies the next definition: Definition 4.5. Let Z ⊆ X and δ > 0. Then As every (δ , n)-cover is a (δ, n)-cover for 0 < δ < δ, the limit exists and is called topological capacity pressure of ϕ on Z with respect to T .

Remark 4.
Although the quantity CP Z (T, ϕ) is called capacity pressure and its definition looks like a lower Carathéodory capacity (see [16] for a detailed introduction and discussion of this subject), it is important to emphasize, that it is not a proper Carathéodory construction for general ϕ. In particular, one cannot hope monotonicity of Z → CP Z (T, ϕ), if ϕ is discontinuous. However, the next theorem shows that CP Z (T, ϕ) recovers Q Z (T, ϕ), and gives a monotonicity relation between the pressures defined so far. This is of importance in the proof of Theorem 5.2.
Proof. We may assume CP Z (T, ϕ) > −∞, which implies Z = ∅. Denote ϕ n (x) := n−1 i=0 ϕ(T i x). Fix some δ 0 > 0 such that CP Z (T, ϕ, δ) > −∞ for all 0 < δ < δ 0 . Fix furthermore an −∞ < α < CP Z (T, ϕ, δ). Then there exists some sequence { n l } l∈N , which depends on δ and α, such that where the infimum is taken over all (δ, n l )-covers F of Z. Hence there has to be a l 0 ∈ N large enough, such that for all l > l 0 . As α < CP Z (T, ϕ, δ) was arbitrarily chosen, letting α → CP Z (T, ϕ, δ) yields Repeating above argument for CP Z (T, ϕ, δ) < ∞, one obtains in addition for all δ > 0. Next we pick by Z = ∅ and Lemma 3.2 some maximal (δ, n)-separated set E n ⊆ Z. That means Z ⊆ z∈En B dn (z, δ), hence for all n ∈ N. Thus, for all δ < δ 0 , and letting δ → 0 we finally obtain Remark 5. In general, it can happen that To see this, recall the example of Remark 3. It yields Other examples for differing lower and upper pressures were given in [7].

5.
Mass distribution principle and variational pressure. Let (X, T ) be a dynamical system and ϕ : X → R be a potential. We introduce various measuretheoretic notions of pressure for discontinuous potentials. The main goal of this section is then to establish analogs of the classical mass distribution principle (see [8]) for those pressures.
Definition 5.1. Given some Borel probability measure µ on X, define for x ∈ X and δ > 0 exists. It is called measure-theoretic pressure of ϕ on x with respect to µ and T . In case ϕ : X → R is measurable and bounded from below, the function is also measurable and bounded from below. Thus x → P µ (T, ϕ, x, δ) is quasiintegrable with respect to µ for every δ > 0. Denote by M(X) the set of all Borel probability measures on X, and by B(X) the set of all measurable functions ϕ : X → R bounded from below. For µ ∈ M(X) and ϕ ∈ B(X) the quantity is called mean measure-theoretic pressure of ϕ with respect to T and µ. Immediately by monotone convergence follows.
We state now three versions of the so-called mass distribution principle, beginning with the most important: Assume that µ(Z δ,N ) = 0 for all δ > 0, N ∈ N. By definition of L, for every z ∈ Z there exists a 0 < δ z and an N z ∈ N such that for all 0 < δ < δ z and n ≥ N z . This shows Z = n≥1 N ≥1 Z 1/n,N and µ(Z) = 0, which is a contradiction. Hence we can choose As the cover F was arbitrarily chosen, letting n → ∞ results in Hence for all 0 < δ < δ . Letting δ → 0 and using Theorem 4.6 yields Now letting → 0 gives us P Z (T, ϕ) ≥ L. The case L = ∞ can be proven in the same way as above by considering the sets If we assume ϕ to be measurable and bounded from below, we immediately obtain the second version of the mass distribution principle: Corollary 1. Let µ ∈ M(X) and ϕ ∈ B(X). Suppose Z to be a Borel set satisfying µ(Z) = 1. Then one has Proof. The proof works in a similar way like the proof of Theorem 1.2 (i) in [10]. Assume −∞ < P µ (T, ϕ) < ∞ and fix > 0. Clearly Taking the limit → 0 yields P Z (T, ϕ) ≥ P µ (T, ϕ). The case P µ (T, ϕ) = ∞ works in same way.
Remark 6. In the proof of Theorem 5.2, ϕ(x) ∈ R for all x ∈ X was the only property we used, whereas Corollary 1 needed more assumptions. Actually, Theorem 5.2 gives us the third important version of the mass distribution principle, if we assume finite topological entropy and quasi-integrable ϕ: for every x ∈ G. By Brin-Katok's theorem [3] there exists another Borel set G 2 ⊆ X such that µ(G 2 ) = 1 and Combining both yields for all x ∈ G =: G 1 ∩ G 2 . Here we used lim inf n→∞ (a n + b n ) = lim inf n→∞ a n + lim n→∞ b n , if b n converges in R and the sum of both limits is well-defined. Hence applying Lemma 4.2 and Theorem 5.
Note that the statement of Theorem 5.3 also holds, if h top (T ) = ∞ and X ϕ dµ > −∞.
Corollary 2. Assume ϕ : X → R to be measurable such that for every N > 0 there exists a µ N ∈ M T (X) with the properties: Then one has P X (T, ϕ) = ∞. In particular above statement holds, if there is a µ ∞ ∈ M T (X) satisfying X ϕ dµ ∞ = ∞.
Proof. By Theorem 3.10 we have where m µ N denotes the ergodic decomposition of µ N . Thus there must exist some We now assume that ϕ : X → R is quasi-integrable for all invariant measures, which allows us to introduce the variational pressure.
which is called variational pressure of ϕ with respect to T . In particular S(T, ϕ) is well-defined for all ϕ ∈ Q T (X) in case of finite topological entropy.
As ϕ is a real-valued function, one sees immediately the following: Remark 8. The variational pressure can be ±∞. Consider for example the unit circle S 1 with irrational rotation R on S 1 . The unique R-invariant measure on S 1 is the normalized Hausdorff measure H 1 , which satisfies h H 1 (R) = 0. One can choose now a Borel measurable partition It turns out that by ergodic decomposition, the variational pressure can be computed as the supremum over all ergodic measures (as in the classical case): Proof. Denote s := sup h µ (T ) + X ϕ dµ : µ ∈ E T (X) . We may assume S(T, ϕ) > −∞, as S(T, ϕ) ≥ s. Let µ ∈ M T (X) and m µ be its ergodic decomposition. As E T (X) ν → h ν (T ) ≥ 0 and E T (X) ν → X ϕ dν are quasi-integrable functions with respect to m µ , one has by Lemma 3.8 and Theorem 3.10 Next assume µ n ∈ M T (X) to be an sequence such that lim n→∞ h µn (T )+ X ϕ dµ n = S(T, ϕ). In case S(T, ϕ) < ∞ we can choose by (6) some ergodic measure ν n ∈ E T (X) such that which shows by n → ∞ the statement. In case S(T, ϕ) = ∞ we can choose in a similar way ergodic measures ν n ∈ E T (X) such that 6. Variational principles for measurable potentials. Let (X, T ) be a dynamical system. We give a first version of the variational principle for quasi-integrable functions: One might suspect that if S(T, ϕ) = −∞, similarly to Theorem 6.1 one has Q X (T, ϕ) = −∞. It is not clear whether this holds in general, but we have the following positive result: for all x, y ∈ X and i ≥ 1. Suppose there exists an ergodic measure ν such that ν(U ) > 0 for all open sets U ⊆ X. Fix a ϕ ∈ Q T (X). If S(T, ϕ) = −∞, one has Q X (T, ϕ) = −∞.
Proof. First note that B dn (x, ) = B d (x, ) for all x ∈ X and > 0. Therefore h top (T ) = 0, and S(T, ϕ) is well-defined. Furthermore X ϕ dµ = −∞ for all µ ∈ M T (X). By [19]  The next theorem gives us one half of the variational principle: Theorem 6.2. In case ϕ ∈ B(X) one has In case h top (T ) < ∞ and ϕ ∈ Q T (X), one has P X (T, ϕ) ≥ S(T, ϕ).
In case h top (T ) < ∞ and ϕ ∈ B(X), one has Proof. The first two statements are consequences of Lemma 5.5, Theorem 5.3 and Corollary 1. By (5), for every ergodic ν ∈ E T (X) we have That means by Lemma 5.5 In view of Proposition 1, the following second version of the variational principle holds: Proof. We show that δ x0 is an equilibrium measure. By − log δ x0 B dn (x 0 , δ) = 0 we have for all δ > 0 Remark 11. Above proof combined with Lemma 4.2 shows that for all ϕ ∈ B(X) and To prove upper estimates for the variational principle, we first introduce a new class of functions: Definition 6.3. Let ϕ ∈ Q T (X). We call ϕ upper semi-continuous with respect to T , if the following holds: If { µ n } n∈N is a sequence of atomic probability measures µ n = kn i=1 λ n i δ x n i , where (λ n i ) kn i=1 are some probability vectors and {x n i } kn i=1 ⊆ X for n ∈ N, such that there exists a µ ∈ M T (X) satisfying µ n → µ in the weak * topology, then The set of all upper semi-continuous functions with respect to T is denoted by U T (X) ⊆ Q T (X).

Remark 12.
The example of Remark 3 is a system, where each function in U T (X) needs to be bounded from above. Assume there is a ϕ ∈ U Id ([0, 1]), which is not bounded from above. Pick a sequence x n ∈ [0, 1], such that lim n→∞ x n = x and lim n→∞ f (x n ) = ∞. Then one has lim sup which is a contradiction. It turns out that many systems exhibit the same behaviour as the above example: Proposition 4. If (X, T ) has a periodic orbit, than each ϕ ∈ U T (X) is bounded from above.
. Assume there is a measurable function ϕ : X → R such that for each n ≥ 1 there is an x n ∈ X satisfying ϕ(x n ) ≥ n. Define Then it is easy to see that µ n → µ as n → ∞. On the other hand one has lim sup n→∞ X ϕ dµ n = lim sup This shows ϕ / ∈ U T (X).
We do not know whether the last statement also holds for systems where each ergodic measure has no atoms. Hence, to cover the full generality, we have to deal with cases where the pressure is infinite: Proposition 5. Suppose ϕ ∈ U T (X) and P X (T, ϕ, ) = ∞ for some > 0. Then there exists a µ ∈ M T (X) such that X ϕ dµ = ∞.
Proof. Let ξ = { A 1 , . . . , A k } be a measurable partition of X such that diam(A i ) < for all i = 1, . . . , k. Denote by If σ is some probability measure on X, define Note that for all n ≥ 1 We first consider subsequences { n j } j∈N such that lim j→∞ 1 n j log M X (T, ϕ, , n j ) = ∞ and −∞ < M X (T, ϕ, , n j ) < ∞ for all j ∈ N. Set ϕ n (x) := n−1 i=0 ϕ(T i x) and choose ( , n j )-separated sets E nj satisfying log x∈En j exp ϕ nj (x) ≥ log M X (T, ϕ, , n j ) − 1.
Next define the probability measure for every j ∈ N. Then by the definition of σ nj and [19] Lemma 9.9 Next define Combining (7), (8) and (9) yields Furthermore there exists a subsequence { n j l } and a µ ∈ M T (X) such that lim l→∞ µ nj l = µ in the weak * topology. Thus using ϕ ∈ U T (X) This implies X ϕ dµ = ∞. Now let { n j } j∈N be a subsequence such that M X (T, ϕ, , n j ) = ∞ for all j ∈ N. We can then choose ( , n j )-separated sets E nj satisfying log x∈En j exp ϕ nj (x) ≥ 2 nj , and the statement follows in the same way as above.
Proposition 5 gives us a third version of the variational principle, which is in some sense the reverse direction of Theorem 6.1: Corollary 4. Let h top (T ) < ∞ and ϕ ∈ U T (X). If there is an > 0 such that P X (T, ϕ, ) = ∞, then one has S(T, ϕ) = P X (T, ϕ).
We are now able to state the variational principle for all ϕ ∈ U T (X): Theorem 6.4. Let h top (T ) < ∞ and ϕ ∈ U T (X). Then one has P X (T, ϕ) = S(T, ϕ).
Proof. By Theorem 6.2 it remains to show that P X (T, ϕ) ≤ S(T, ϕ). Furthermore, by Corollary 4 we may assume P X (T, ϕ, ) < ∞ for all > 0. In this situation, the proof follows the conventional proof of the classical variational principle as given in [19], Theorem 9.10. We are only outlining it here. Denote ϕ n (x) := n−1 i=0 ϕ(T i x). Fix > 0 and choose for all n ∈ N a ( , n)-separated set E n such that log x∈En exp ϕ n (x) ≥ log M X (T, ϕ, , n) − 1. Define Given 1 ≤ q < n j and 0 ≤ m ≤ q − 1, define a(m) := (n − m)/q . One can now decompose where S is a set with cardinality at most 2q. Hence Summing over all m = 0, . . . , q − 1 gives and dividing by n j yields q n j log M X (T, ϕ, , Using ϕ ∈ U T (X) and µ(∂A i ) = 0 for all i = 1, . . . , k we obtain by j → ∞

VARIATIONAL PRINCIPLES FOR MEASURABLE POTENTIALS 383
Finally dividing by q and letting q → ∞ gives that is P X (T, ϕ, ) ≤ S(T, ϕ) for every > 0. This shows the statement.
Next we introduce two non-trivial classes of (dis-)continuous functions, which satisfy Theorem 6.4. Definition 6.5. Let (Y, ρ) an arbitrary metric space and ϕ : Y → R measurable. The set is called set of discontinuity points of ϕ. One can show that D ϕ is Borel measurable. Denote by C T (X) the set of all bounded, Borel measurable functions ϕ : X → R, such that µ(D ϕ ) = 0 for all µ ∈ M T (X).
Proposition 6. Let { µ n } n∈N be a sequence of Borel probability measures with limit measure µ in the weak * topology. Then one has for all ϕ ∈ C T (X) In particular one has C T (X) ⊆ U T (X).
Proof. The first statement is part of the Portmanteau theorem, see for example [14], Theorem 13.16. The second statement follows immediately.
Proof. As ϕ is bounded, the statement follows from Proposition 6, Theorem 6.4 and Remark 13.

Remark 14.
We give an illustration of above corollary. Suppose (X, d) to be a non-empty, compact space and T : X → X to be a contraction. By the Banach fixed-point theorem there exists a unique fixed-point x 0 ∈ X. It is then easy to see that for every continuous ψ : X → R and every x ∈ X In particular by Lemma 4.2 one has P X (T, ϕ) = P { x } (T, ϕ) for every x ∈ X.
is an open set for every c ∈ R. We denote the set of all upper semi-continuous functions ϕ : X → R by U(X). As X is compact, every ϕ ∈ U(X) is bounded from above (see for example [1] Theorem 2.43). This immediately yields U(X) ⊆ Q T (X). In addition, the following holds: Proposition 7. Let { µ n } n∈N be a sequence of Borel probability measures with limit measure µ in the weak * topology. Then one has for ϕ ∈ U(X) In particular one has U(X) ⊆ U T (X) for every continuous mapping T : X → X.
Proof. The statement follows from Proposition 7 and Theorem 6.4.
Remark 15. Corollary 6 was already proven (as a special case) in [4] (1). However, the proof in [4] for the lower bound of the variational principle requires the functions f n := exp i<n ϕ(T i ) to attain a maximum on every compact subset of X. This is clearly the case if ϕ is upper semi-continuous, as this implies the upper semi-continuity of f n . Another proof for Corollary 6 can be found in [13] (see Theorem 4.4.11). Here for the lower bound of the variational principle, the functions f n need to be bounded from above on X.
In the method of proof used in the present paper, above properties are not needed. Instead, the lower estimate is first proven for ergodic measures with the help of an ergodic theorem. After that it can be extended to all invariant measures via ergodic decomposition.

Remark 16. Recall the example given in Remark 14 and consider the indicator function
As one has D χ = { x 0 } in general, we can not apply Corollary 5. One the other hand the function χ is upper semi-continuous, as { x 0 } is closed. Thus again by Corollary 6 P X (T, χ) = χ(x 0 ) = 1.
Remark 14 motivates the following example, which shows that the pressure defined in this paper might be applied to systems, which are unavailable to the pressures and its variational principle derived in [15]. Proof. Define Clearly ϕ α is measurable and continuous in α only. Also one has lim sup y→x ϕ α (x) > 0 = ϕ α (x) for every x ∈ E \ { α }, which proves ϕ α not to be upper semi-continuous.  1 and open interval (a, b) such that (a, b) ⊆ Λ k . But that means ϕ α is not continuous on Λ k . Corollary 7. Let X := [0, 1], α ∈ (0, 1) ∩ Q and T : X → X contracting such that T (α) = α. If ϕ α is the function constructed in Proposition 8, then one has Proof. This follows from Remark 14 and Proposition 8.
Remark 17. Clearly both sets U(X) and C T (X) contain all continuous functions ϕ : X → R. Moreover, Corollary 5 can be seen as variational principle for potentials which are continuous from a measure theoretical point of view. We want to emphasize that the set C T (X) heavily depends on the mapping T : X → X. The set U(X) on the other hand only depends on the metric d on X. Note that in general it may happen that C T (X) U(X) = ∅, which implies U(X) U T (X). This can be seen from the following statement: Proposition 9. Assume that (X, d) has no isolated points, and there exists a nonatomic µ ∈ E T (X). Then there is a function ϕ ∈ C T (X), which is neither upper nor lower semi-continuous on X.
Proof. As µ is non-atomic, there are two distinct points x 1 = x 2 ∈ X such that lim n→∞ i<n δ T i x1 = lim n→∞ i<n δ T i x2 = µ. Next define ϕ := 1 {x1} − 1 {x2} . As X has no isolated points, ϕ is not lower semi-continuous in x 1 , and not upper semicontinuous in x 2 . That means D ϕ = {x 1 , x 2 }, and as µ has no atoms, ϕ ∈ C T (X) follows.
7. Cookie-cutters with discontinuous geometric potentials. In this section we introduce cookie-cutter systems with discontinuous geometric potentials and their corresponding attractors. Following this, we use the topological pressure and its variational principle for discontinuous functions to compute the Hausdorff dimension of those attractors. The definitions and notations are based on the classical treatment of cookie-cutters in [9]. We define the derivative of T on the interval endpoints x i , y i , i = 1, . . . , N to be the left or right derivatives respectively. By definition, T is not well-defined in the set D, and there is also no way to extend T continuously to D. Proof. Fix i ∈ { 1, . . . , N }. Clearly ϕ i : I i := (0, 1) T (D ∩ J i ) → J i is continuously differentiable satisfying c i := sup ξ∈Ii ϕ i (ξ) < 1. Fix x < y ∈ [0, 1] and denote by z 1 < · · · < z m , m ≥ 0, the set of points [x, y]∩T (D ∩J i ). Set z 0 := x and z m+1 := y. As ϕ i is strict monotonic on [0, 1], we have Therefore the number c := max i=1,...,N c i has the desired property. The second part follows for example from [8] Theorem 9.1.
is called attractor of the cookie-cutter T : J → [0, 1] . As T |X is continuous and X = T (X), the tuple (X, T |X) is a dynamical system. Furthermore, T |X D is continuous. Thus, if ϕ : X → R is some function satisfying ϕ|X D = T |X D, one has D ∩ X = D ϕ (see Definition 6.5). We call such a function ϕ to be an extension of T to X. Clearly an extension ϕ of T to X is continuous if and only if one has D ∩ X = ∅. In that case ϕ := T |X by definition is the only possible extension.
If ϕ : X → R is an extension from T to X, the function log |ϕ| : X → R is called geometric potential of T . In case D ∩ X = ∅ the system is called cookiecutter with discontinuous geometric potentials, and every possible extension ϕ defines a corresponding geometric potential.
Remark 18. We shall give some examples to illustrate the notion of cookie-cutters. Let J := [0, 1 3 3 ≤ x ≤ 1. In this case the corresponding attractor X 1 is the middle-third Cantor set. The derivative T 1 is well-defined everywhere. If we change T 1 to we see that T 2 does not exist in 5 6 . One the other hand one has T ( 5 6 ) = 3 5 / ∈ J, which means 5 6 / ∈ X 2 and ϕ := T 2 |X 2 is the continuous extension. To obtain a cookie-cutter satisfying D ϕ = ∅ for every extension ϕ of T to X, one can easily modify the second example in a way that the point of discontinuity is a fixed point (see Figure 1).
The goal is to compute the Hausdorff dimension dim H X of an attractor X. In case D ∩ X = ∅ it is determined by the zero of a certain pressure function, which Proof. As T is continuously differentiable on J D, holds for all x ∈ X, thus a : X → R is continuous. In addition by the third property of Definition 7.1 one has 1 < inf x∈X a(x), which implies Then by [6] Theorem 2.4 it follows thatP X (T |X, −s · log a) = 0 if and only if s = dim H X. HereP Z denotes a Carathéodory dimension type definition of topological pressure, which was first given in [17]. Furthermore, it is well-known that for all continuous ϕ : X → R and all non-empty compact T -invariant subsets Z ⊆ X one hasP Z (T |X, ϕ) = P Z (T |X, ϕ). Hence the statement follows.
Remark 19. Actually Theorem 2.4 in [6] is much more powerful than above proof suggests. It basically states, that the Hausdorff dimension of every subset of an attractor is the zero of the pressure function s →P Z (T |X, −s · log a), provided the system (X, T |X) is conformal and reasonable expanding. Conformal in this context means that the expression (10) is well-defined and continuous on X. However the theorem cannot be applied anymore, if the limit a(x) in (10) does not exist for even one x ∈ X. In higher dimensions this can happen, if the derivative in a point exists, but has distinct eigenvalues. For a survey of recent research on the topic of non-conformal repellers, see [5].
As indicated in Remark 19, the classical thermodynamic formalism and its celebrated Bowen formula cannot be applied directly to cookie-cutters T : J → [0, 1], as the derivative might not exist in finitely many points on the attractor. Nevertheless its still possible to establish an analogous formula for them: Theorem 7.4. Let T : J → [0, 1] be a cookie-cutter and X its attractor. Then there exists a geometric potential log |ψ| : X → R such that P X (T |X, −s · log |ψ|) = 0 if and only if s = dim H X.
Remark 20. As we shall see, the geometric potential can be constructed with the lower semi-continuous extension of T . It might be considered as the natural choice among all possible geometric potentials.
To prove above theorem, we collect some preparatory results first: x ∈ J D, lim inf y→x T (y), x ∈ D, T |B x > 1, lim sup y→x T (y), x ∈ D, T |B x < −1.
Thus x → f (x) is lower semi-continuous. As log(·) is strictly monotone increasing, the function x → log f (x) remains lower semi-continuous, whereas x → −s · log f (x) is upper semi-continuous for all s ≥ 0. The remaining parts easily follow.
Lemma 7.6. The dynamical system (X, T |X) is topological transitive and satisfies h top (T |X) = log N .
Fix 0 < < 1 4 . Define . Clearly g(x) = (2 − 4 ) > 1 and 0 < g(x) < 1 for all x ∈ (0, 1). Fix an i ∈ { 1, . . . , N }. Define an affine scaling Φ : (0, 1) → (y i−1 , x i ) by It is easy to see that τ satisfies the properties of an EPM map, if we linearly order all points of D ∪ { x i , y i | i = 1, . . . , N } ∪ { 0, 1 } as demanded in (11). In addition one has τ (x) / ∈ X for every x / ∈ X. This follows from the construction of τ : For every x ∈ [y i−1 , x i ] one has T i (x) ∈ (y i−1 , x i ). Furthermore every x ∈ J X will eventually be mapped by T into one of the intervals (y i−1 , x i ), where it cannot escape into X. Hence X is a closed completely invariant subset of the EPM mapping τ : [0, 1] → [0, 1] (see [12]). Clearly τ |X = T |X. Next take the function f constructed in Lemma 7.5 and define We then observe: On the other hand the corresponding pressure functions for log|ψ| might not have a zero, nor a variational principle for the topological pressure has to exist. (c) By Lemma 7.6 the entropy mapping h : is upper semi-continuous. Similarly, by Lemma 7.5 and Proposition 7 the mapping µ → X −s · log|ψ| dµ is upper semi-continuous too. Hence there exists for every s ≥ 0 an equilibrium state µ s ∈ M T |X (X) such that h µs (T |X) + X −s · log|ψ| dµ s = P X (T |X, −s · log |ψ|).
In particular there is a µ s0 ∈ M T |X (X) such that Thus · X induces a norm on the R-vector space I(X), which is defined as the set of all measurable, bounded functions ϕ : X → R.
Proposition 11. Let (X, T ) be a dynamical system such that h top (T ) < ∞. Then one has for all ϕ ∈ I(X) hence P X (T, ϕ) is finite. In particular one has for all ϕ 1 , ϕ 2 ∈ I(X), that is P X (T, ·) : I(X) → R, ϕ → P X (T, ϕ), is Lipschitz continuous.
Proof. The first statement follows from the definition of pressure. By Remark 2 we have M X (T, ϕ, , n) to be finite for all n ∈ N, > 0. Thus the second statement can be similarly proven like [19] Theorem 9.7 (iv). Remark 22. Above theorem basically states, that if one has a classical, smooth cookie-cutter T : J → [0, 1], the Hausdorff dimension of its attractor changes only slightly, if one adds some tiny corners to T . Another way to view this theorem is that in terms of the dimension, a smooth cookie-cutter can be approximated by cookiecutters with discontinuous geometric potentials. Note that the theorem cannot be used the other way around, i.e. to approximate a cookie-cutter with discontinuous geometric potentials by smooth cookie-cutters, as lim n→∞ f n − f ∞ J = 0 cannot hold in this case. Figure 3. The cookie-cutters T n approaching the limit cookiecutter T ∞ .
Before we prove the theorem, we recall the following lemma: Clearly one has f n ∈ I(J) for all n ∈ N ∪ { ∞ }. Assume sup n∈N f n J = ∞, then there exist some n k ∈ N, x k ∈ J such that lim k→∞ f n k (x k ) = ∞. Thus lim k→∞ f n k (x k ) − f ∞ (x k ) = ∞, which is a contradiction. Hence there is a constant C > 1 such that for all x ∈ J, n ∈ N ∪ { ∞ }. As x → log(x) is Lipschitz continuous on [1, C] with Lipschitz constant 1, we in addition have lim n→∞ − log |f ∞ | + log |f n | J ≤ lim n→∞ |f n | − |f ∞ | J = 0.
This means P Σ + N (σ N , λ s n ) = sup h µ (T n |X n ) + Xn −s · log f n |X n dµ : µ ∈ M Tn|Xn (X n ) , hence, using the variational principle again, P Σ + N (σ N , λ s n ) = P Xn T n |X n , −s · log f n |X n (15) for all n ∈ N ∪ { ∞ }, s ≥ 0. Now denote s n := dim H X n for all n ∈ N ∪ { ∞ }. Let s n k be some convergent subsequence with limit s * . Recall that by (15) and Theorem 7.4 each s n k is the unique zero of s → P Σ + N (σ N , λ s n k ). Thus one has by Proposition 11,(13) and (14) P ≤ s * · − log |f ∞ • π ∞ | + log f n k • π n k Σ + N + s * − s n k · log f n k • π n k Σ + N ≤ s * · − log |f ∞ | + log f n k J + s * − s n k · log f n k J → 0 as k → ∞. This means P Σ + N (σ N , λ s * ∞ ) = 0, hence As s n ∈ [0, 1] for all n ∈ N, there exists at least one convergent subsequence. This shows by (16) that the limit of s n exists and one has lim n→∞ dim H X n = dim H X ∞ .
Remark 23. The two key observations for the proof of Theorem 8.2 are: (α) The operator ϕ → P X (T, ϕ) is Lipschitz continuous.
(β) For each n ∈ N ∪ { ∞ } there is a topological conjugation Σ + N → X n . To apply (α), one has to relate the variational pressure and the topological pressure via a variational principle. For this, we used Corollary 6. As mentioned in Remark 21, there is also the option to change the underlying EPM system of each X n into a new system, where dynamics and the potential are continuous again. Then one would be able to use the classical variational principle. However, by changing the system, one has to introduce new symbolic spaces F n ⊆ Σ + Nn , where the numbers N n depend on the discontinuities of the functions f n , and each F n is a subshift of the full shift on Σ + Nn . Using this approach together with our method of proof, only a weaker version of Theorem 8.2 can be proven: To satisfy (β), one has to assume in addition that all F n are pairwise topological conjugated for n ∈ N ∪ { ∞ }.