Geometric pressure in real and complex 1-dimensional dynamics via trees of preimages and via spanning sets

We consider $f:\hat I\to \R$ being a $C^3$ (or $C^2$ with bounded distortion) real-valued multimodal map with non-flat critical points, defined on $\hat I$ being the union of closed intervals, and its restriction to the maximal forward invariant subset $K\subset I$. We assume that $f|_K$ is topologically transitive. We call this setting the (generalized multimodal) real case. We consider also $f:\C\to \C$ a rational map on the Riemann sphere and its restriction to $K=J(f)$ being Julia set (the complex case). We consider topological pressure $P_{\spanning}(t)$ for the potential function $\varphi_t=-t\log |f'|$ for $t>0$ and iteration of $f$ defined in a standard way using $(n,\e)$-spanning sets. Despite of $\phi_t=\infty$ at critical points of $f$, this definition makes sense (unlike the standard definition using $(n,\e)$-separated sets) and we prove that $P_{\spanning}(t)$ is equal to other pressure quantities, called for this potential {\it geometric pressure}, in the real case under mild additional assumptions, and in the complex case provided there is at most one critical point with forward trajectory accumulating in $J(f)$. $P_{\spanning}(t)$ is proved to be finite for general rational maps, but it may occur infinite in the real case. We also prove that geometric tree pressure in the real case is the same for trees rooted at all `safe' points, in particular at all points except the set of Hausdorff dimension 0.


Introduction
Let us start with the classical Definition 1.1 (Topological pressure via separated sets). Let f : X → X be a continuous map of a compact metric space (X, ρ) and φ : X → R be a real continuous function. For every positive integer n and x ∈ X denote S n φ(x) = n−1 j=0 φ(f j (x)). For every integer n ≥ 0 define the metric ρ n (x, y) = max{ρ(f j (x), f j (y)) : j = 0, ..., n}. For every ε > 0 a set Y ⊂ X is said to be (n, ε)-separated if for every y 1 , y 2 ∈ Y such that y 1 = y 2 it holds ρ n (y 1 , y 2 ) ≥ ε. Define infimum taken over all (n, ε)-spanning sets Y ⊂ X, i.e. such that for every x ∈ X there exists y ∈ Y such that ρ n (x, y) < ε, in other words such that y∈Y B ρn (y, ε) = X, and P spanning (f, φ) = lim ε→0 P spanning (f, φ, ε).
It is easy to prove Theorem 1.2 (see e.g. [18]). For every continuous φ : This pressure depends on topology, but does not depend on metric.
In this paper we shall discuss φ = φ t = −t log |f | for all parameters t > 0. This is slightly different from the previous situation in dimension 1 if f is differentiable and has critical points, i.e. points c ∈ X where the derivative of f is 0. At these points φ is not continuous. We assign to φ the value +∞ there. The notion P sep does not make much sense in this case, as this quantity is equal to +∞, by taking Y containing some critical points, so it is replaced by the notion of tree pressure, see (2.1). However P spanning defined as above happens to make sense and a part of this paper is devoted to explaining this. We discuss two settings: 1. (Complex) f is a rational mapping of degree at least 2 of the Riemann sphere C usually with the spherical metric. We consider f on its Julia set K = J(f ). 2.
(Real) f is a real generalized multimodal map. Namely it is defined on a neighbourhood U ⊂ R of its invariant set K. We assume f ∈ C 2 , is nonflat at all its turning and inflection critical points, has bounded distortion property for its iterates, and is topologically transitive and has positive topological entropy on K.
We assume that K is a maximal invariant subset on a finite union of pairwise disjoint closed intervals I = j I j ⊂ U whose ends are in K. (This maximality corresponds to Darboux property.) By adjusting I and U we can assume there are no critical points outside K, no attracting periodic orbits in U and no parabolic periodic orbits in U \ K.
We write (f, K) ∈ A BD + . The subscript + is to mark positive topological entropy. Sometimes we write (f, K, I, U). (In place of BD one can assume C 3 and all periodic orbits in K hyperbolic repelling, denote the related class by A 3 + .) For this real setting see [13]. Examples: sets in the spectral decomposition [8].
In both settings the set of all critical points will be denoted by Crit(f ).
The function φ t is sometimes called geometric potential and the pressure is called geometric pressure, see e.g. [16]. This name is justified by where B n (z) := Comp z (f −n (B(f n (z), ∆))) for a constant ∆ > 0 and appropriate t 0 (with pressure P (φ t 0 ) = 0). Here Comp z means the component in C or R containing z, called also a pull-back. We consider only pull-backs intersecting K.
There are several equivalent definitions of geometric pressure P (φ t ), see [15], [16] or [13] in the interval case. One of them useful in this paper is Definition 1.3 ( hyperbolic pressure).
supremum taken over all compact f -invariant (that is f (X) ⊂ X) isolated (Cantor) uniformly hyperbolic subsets of K.
Isolated (or forward locally maximal ), means that there is a neighbourhood U of X such that f n (x) ∈ U for all n ≥ 0 implies x ∈ X.
A set X is said to be hyperbolic or expanding if there is a constant λ X > 1 such that for all n large enough and all x ∈ X we have |(f n ) (x)| ≥ λ n X .
2. Tree-pressure 2.1. Definitions. We devote this section to studying a modified definition of pressure by separated sets which may not have sense, called tree-pressure, see e.g. [16].
In the real and complex settings we define Definition 2.1. A point z ∈ K is said to be safe, or Crit(f )-safe if for every δ > 0 and all n ≥ n(δ > 0 large enough It immediately follows from the definitions that Hausdorff dimension of the set of points which are not safe is equal to 0. In the complex setting the following holds Theorem 2.2 ( [12], [15]). For every rational f : C → C of degree at least 2 and for every z ∈ K = J(f ) safe and t > 0 it holds In particular in the complex case P tree (z, t) does not depend on z safe; it is constant except z in a set of Hausdorff dimension 0. We denote this tree-pressure for z safe by P tree (t).
2.2. The real case: independence of a safe point. In the generalized multimodal setting the above equality was known for z being safe, safe forward (in case K is not weakly isolated) and hyperbolic, see [13][ Lemma 4.4]. We remind the definitions mentioned here, compare [13]: Definition 2.4 (safe forward). A point z ∈ K is called safe forward if there exists ∆ > 0 such that dist(f j (z), ∂ I K ) ≥ ∆ for all j = 0, 1, .... Definition 2.5 (weak isolation). A compact set K ⊂ R is said to be weakly isolated for a continuous mapping on a neighbourhood of K to R for which K is forward invariant, if there exists ε > 0 such that every f -periodic orbit O(p) ⊂ B(K, ε) must be in K.
Though the set of all expanding points has Hausdorff dimension equal to the hyperbolic dimension of K, i.e. supremum of Hausdorff dimensions of isolated uniformly hyperbolic subsets of K, being the first zero of the hyperbolic pressure, see the definition above, the complementary set can also be large.
One of aims of this paper is to prove Theorem 2.6. For (f, K) ∈ A BD + without parabolic periodic orbits (or for f ∈ A 3 + ), if K is weakly isolated and t > 0, the tree pressure P tree (z, t) does not depend on z ∈ K safe. In particular Moreover limsup can be replaced by lim in the definition of tree pressure, i.e the limit exists.
As in the complex case we denote this tree-pressure for safe points by P tree (t).
Before proving this theorem let us recall the following definition valid in the real and complex cases Definition 2.7 (backward Lyapunov stable). f is said to be backward Lyapunov stable if for every ε > 0 there exists δ > 0 such that for every z ∈ K, n ≥ 0 and W = Comp z f −n (B(f n (z), δ) (the balls and components in R or C) diam W < ε.
In the real case this property holds for B(f n (z), δ) not containing any parabolic periodic point, see [13][Lemma 2.10].
In the sequel we call W a pull-back of the interval B(f n (z), δ) for f n containing z.
Proof of Theorem 2.6. The inequality P tree (z, t) ≥ P hyp (f, φ t ) is obviously true for every z ∈ K, under a mild non-exceptionality condition, weaker than safe, see [13][ Lemma 4.4].
Fix an arbitrary 0 < δ ≤ δ, where δ is small, chosen to ε in Definition 2.7, and ε satisfies the weak isolation condition 2.5.
In this case, if δ is small enough, as in Definition of backward Lyapunov stability, all pull-backs of W 0 = [z 0 , z 0 ] are shorter than ε. Now we use a procedure by Rivera-Letelier [17], see also [3][Appendix C]. We consider pull-backs of W 0 = 2W 0 (the twice longer interval with the same origin) for f i along a backward trajectory until for certain i 1 the pull-back W 1 i 1 for f i 1 captures a critical point. Next we consider pull-backs of 2W i 1 , where W i 1 ⊂ W 1 i 1 is a pull-back of W 0 . We arrive after time i 2 −i 1 at W 2 i 2 containing a critical point, etc.
Using bounded distortion between consecutive captures of critical points, more precisely for , and the inequality for every x ∈ W i true for any pull-back of W 0 for f i we prove that for z n ∈ ∂W n and α > 0 arbitrarily close to 0 for δ small enough that the differences of times of consecutive captures of each critical points are bounded below by a constant arbitrarily large (possible due to absence of attracting periodic orbits), Since w is safe we get also, replacing W n by the appropriate pull-back Hence Summing over all n'th preimages w n of w in K, taking in account that the number of w n 's can be at most exp ε 2 n in each pull-back W n , gives the demanded estimate. More precisely: for L = sup |f |. Acting with 1 n log and passing with n to ∞ in limsup we get the inequality (2.4) for w and z hence after the interchanging their roles, the equality. Similarly we obtain the equality of lower limits, writing Q n (z, t) ≥ 1 4 L −N t Q n−N (w, t). But they coincide with P hyp (t) for z safe, safe forward and hyperbolic, see the beginning of this section. Hence limsup and liminf coincide and are equal to P hyp (t) for every z safe.
It is a priori not clear whether the points z n belong to K. This trouble can be dealt with as follows, compare the proof of [3][ Lemma 3.2]. Take an arbitrary repelling periodic not post-critical orbit O ⊂ K and a backward trajectory of a point p ∈ O in K accumulating at z 0 (and z 0 ). To simplify notation we can assume that p is a fixed point for f .
Consider w n ∈ [z n , z n ] = W n as above. Choose w n+Np ∈ f −Np (w n ) ∩ B(p, r p ), where r p is such that there exists a branch g of f −1 with g(p), mapping B(p, r p ) into itself, with its iterates converging to p. Since z is safe, z 0 (and z 0 ) is not postcritical.
Next choose a backward trajectory (y 0 , y 1 , ...) of p so that z 0 , z 0 are its limit points. Choose the intervals B = B(z 0 , ξ) and B = B(z 0 , ξ) so short that the pull-back B n+Np of the one of them containing z n+Np is in B(p, r p ) and f n+Np has no turning critical points in it. Next choose r and n such that a pull-back W of B(p, r ) for f n is in B (or B ). Finally choose m such that g m (B(p, r p )) ⊂ B(p, r ). So the adequate branch G of f −(n+Np+m+n ) maps B (or B ) into itself, so a corresponding fixed point p ξ for f n+Np+m+n exists in B (or B ). Write f n (z n ) = z 0 . A part of the periodic trajectory of p ξ shadows the backward trajectory (z 0 , ..., z n+Nr ). By the weak isolation property p ξ ∈ K. The shadowing error tends to 0 as ξ → 0. Thus z n ∈ K.
Case 2. The safe point w ∈ K is not between two points z 0 , z 0 , in notation of Case 1. We assume δ ≤ δ/4. Then the interval implying x∈∂G,f (x) / ∈∂G B(f (x), δ ))∩B(∂G, δ ) = ∅, we conclude that either for some m : 0 < m ≤ 2Γ δ , f m (w) satisfies the assumption of Case 1 (is between z 0 , z 0 ), or all f j (w) for 0 ≤ j < 2Γ δ are δ close to ∂G hence w is pre-periodic w := f j 1 (w) = f j 2 (w), and the length of its forward orbit is bounded by 2Γ δ .
Then use z ∈ f −κn (z) which is exp −ηn close to w i.e. in a "safe" ball, for 0 < η < κχ, where χ is Lyapunov exponent at w. Taking κ arbitrarily small (positive) we can replace z 0 by z when comparing the tree pressure at z and w. We use |f | ≤ L and t ≥ 0. We use also the fact that by the safety condition the distortion (This allows not to use z ∈ f −k (z) on the other side of w maybe not existing for k of order at most κn.)

2.3.
On the weak isolation condition. Notice that proving that z n ∈ K above, we use the existence of z = z n+Nr such that f Nr (z) = z n . A priori we cannot exclude that only z n happens in the images. An example is Then points slightly to the left of f 2 (1/2) may have no preimages.
Fortunately in the proof above we use only those z i which are boundary points of pull-backs of [z 0 , z 0 ] containing w i ∈ K.
In the above proof to know that z n , z n belong to K we could refer to [3][Corollary 3.3] in the form of the proposition below (interesting in itself), true under the additional assumption, see [3][Subsection 1.4], that no point in ∂ I is weakly Σ-exceptional, for Σ being the set of all turning critical points.
Definition 2.8. Given an arbitrary finite set Σ ⊂ K, we call a nonempty set E ⊂ K weakly Σ-exceptional, if E is non-dense in K and satisfies We call x ∈ K weakly Σ-exceptional if it is contained in a weakly Σexceptional set.
Proposition 2.9 (On K-homeomorphisms). Let (f, K) ∈ A + satisfies weak isolation condition. Let W be an arbitrary interval sufficiently short (closed, half-closed or open), not containing in its closure weakly Σ-exceptional points for Σ being the set of turning critical points in ∂ I, such that f is monotone Proof. It is sufficient to consider W closed. The assertion of the Proposition follows for W := W ∩ I = W ∩ I j by the maximality of K (notice that W short enough intersects only one interval I j ). By definition W := W \ W is disjoint from K. For W short enough W has one component or it is empty. Suppose it is non-empty. The case f (W ) intersects K, but f (a) is not a limit point of f (W ) ∩ K can be eliminated by considering W short enough. Here a is the boundary point of I j belonging to cl W . We use the assumption that the family I j is finite. Therefore we need only to consider the case f (a) is an accumulation point of f (W )∩K (in particular f | K is not open at a). In this case however there exists a periodic orbit Q passing through W arbitrarily close to K. The proof is the same as the proof of [3][Lemma 3.2] and similar to the proof of Theorem 2.6. Briefly: we choose a repelling periodic orbit O ⊂ K. Next choose a backward trajectory (y 0 , y 1 , ...) of a point p ∈ O with a limit point in f (W ) ∩ K and a backward trajectory (z 0 , z 1 , ...) of a converging to Q. This allows us to find a backward trajectory of W at a time n approaching to O along z j and next at a time m being in f (W ). So W after the time n + m + 1 enters itself. Hence there exists a branch of f −(m+n+1) mapping W into itself, yielding the existence of Q.
So Q is in K by the weak isolation condition. We obtain a point in K ∩ W , a contradiction. Remark that in the example f (x) = ax(1 − x) discussed above the assumption of the lack of weakly Σ-exceptional points does not hold and the assertion of Proposition 2.9 fails for W = 3. Geometric pressure via spanning sets. The complex case In the real case in the previous section we used the property: backward Lyapunov stability, Definition 2.7. In the complex case this property need not hold. So the following weaker version occurs useful.
Denote P spanning (f | K , φ t ) by P spanning (t), both in the real and complex case. The following is the main theorem in this section Theorem 3.2. For every rational mapping f : C → C of degree at least 2 and for every t > 0 it holds P spanning (t) ≥ P tree (t). If f is weakly backward Lyapunov stable then the opposite inequality holds, hence Proof. I. First we prove P spanning (t) ≤ P tree (t). This is the CONSTRUC-TION part of the proof, where we construct an (n, ε)-spanning set not carrying much more "mass" than f −n ({z 0 }). This corresponds to the right hand side inequality in (1.4), where we can just consider as the (n, ε)-spanning sets maximal (n, ε)-separated sets. Fixed an arbitrary ε > 0 and δ > 0, by the property wbls (Definition 3.1) we have for n large enough for every x ∈ J(f ) and every pull-back V of B(x, exp(−nδ/2)) for f j , j = 0, ..., n, diam V < ε.
We can easily find a set X ⊂ J(f ) \ B which is r/2-spanning for ρ the standard metric on the Riemann sphere and #X ≤ Const exp 2nδ.
The set B(X, r/2) covers J(f )\B. By bounded distortion for every x ∈ X and x ∈ B(x, r/2) and branch g of For n large we have also diam B k < exp(−nδ/2). Hence the diameters of all pull-backs of B k for f j , j = 1, ..., n are less than ε.
We have for an arbitrary ξ > 0 for n large enough. This uses the fact that the convergence in (2.1) is uniform for all x safe with the same δ, see Lemma 3.5. (Here we consider x ∈ X depending on n, so we abuse the terminology; we consider safe for each n separately, just satisfying (2.2).) Considering n → ∞ and passing with δ and ε to 0, we end the proof. Now consider the general case, allowing J(f ) being disconnected. Definition 3.3. A compact set X ⊂ C in the complex plane is said to be uniformly perfect if there exists M > 0 such that there is no annulus D ⊂ C of modulus bigger than M , separating X. In other words there is no Proof. We can assume diam X ≥ 2. If X is uniformly perfect with constant M , then for every x ∈ X and i ≥ 0 such that (3i + 1)M < am we can find x i ∈ X such that exp(−m + 3iM ) ≤ |x − x i | ≤ exp(−m + (3i + 1)M ). Now we define X m,a . Each its element will be encoded by a block of 0's and 1's of length I + 1 where I := [am/3M ] − 1 where [·] means the integer part, and denoted x(ν 0 , ..., ν I ), where ν j = 0 or 1. We define these points by induction using codings of length 1, 2, ... and finally I + 1 which will be our final choice.
In fact we shall prove P spanning (t) ≥ P hyp (t) which is enough due to Theorem (2.2). By [15, Proposition 2.1] for every ξ > 0 and t > 0 there exists an Then for every ε > 0 small enough for every n ≥ 0 large enough there exists an (n, 2ε)-separated set X n ⊂ X such that y∈Xn |(f n ) (y) Therefore for every (n, ε)-spanning set Y n ⊂ J(f ) for every y ∈ X n , there exists y ∈ Y n which is (n, ε)-close to y.
Hence by triangle inequality the selection y → y is injective. By the hyperbolicity of X, if ε is small enough, there is a constant C such that for every n and y ∈ X n it holds |(f n ) (y )|/|(f n ) (y)| ≤ C. This, after passing to limits, proves P spanning (t) ≥ P sep (f | X , φ t | X ) − ξ. Hence letting ξ → 0 and choosing appropriate X we obtain P spanning (t) ≥ P hyp (t).
We considered here all (n, ε)-spanning sets, so it is natural to call this Part II of the proof the ALL part. Notice that this proof corresponds to the left hand side inequality in (1.4).
To end this section let us provide the lemma we have already referred to Lemma 3.5. For every t > 0 and ξ > 0 there exists δ 0 > 0 and n 0 ≥ 0 such that for every 0 < δ ≤ δ 0 and n ≥ n 0 and every z 1 , To prove this lemma we use the fact being a part of [12, Lemma 3.1] (see also [5] and [14, Geometric Lemma]) Lemma 3.6. There exists C > 0 such that for every set W of m > 0 points in C and 0 < r < 1/2 such that m ≥ log 1/r, for every z 1 , z 2 ∈ C \ W there exists a sequence of discs in the Riemann sphere metric B 1 = B(q 1 , r 1 ), ..., B k = B(q k , r k ) such that for every j = 1, ..., k each 2B j := In other words the quasi-hyperbolic distance between z 1 and z 2 in C \ W is bounded by Const √ m log 1/r.
Proof of Lemma 3.5. Given n set W = j=1,...,n f j (Crit(f )) and m = n# Crit(f ). Using Lemma 3.6 and Koebe distortion lemma we obtain for a sequence s j ∈ B j ∩ B j+1 for j = 1, ..., k − 1 and s 0 = z 1 , s k+1 = z 2 , and for a distortion constant C Dist > 0 Q n (s j , t) Q n (s j+1 , t) Dist . Hence, for r = exp −nδ, due to for δ small enough and n satisfying (2.2) and large enough to satisfy the latter inequality.

Weak backward Lyapunov stability and further corollaries in the complex case
For every x ∈ C and a rational mapping f : C → C define the lower Lyapunov exponent by Let us start with the following Proof. Take arbitrary ε, δ > 0 and x ∈ J, and consider large n. Consider B := B(x, exp −nδ) and an arbitrary y ∈ f −n (x). For every 0 < j ≤ n consider U j = B(x, a j exp −nδ), where a j = j s=1 (1 − s −2 ). Let V j be the pull-back of U j for f j containing f n−j (y). Let j = j 1 > 0 be the least non-negative integer for which V j+1 contains a critical point c. Then c ∈ J(f ) if n is large enough. Indeed, the only other possibility would be a critical point c / ∈ J(f ) attracted to a parabolic periodic orbit. Then however the convergence of f n (c) to this orbit, and moreover to J(f ) would be subexponential, so f s (c) / ∈ B(x, exp −nδ) for s = 1, 2, ..., n if n is large enough.
(In fact we can omit this part of the proof, since the proof below does not use c ∈ J(f ). It uses only χ(f (c)) ≥ 0, true if f n (c) converges to a parabolic periodic orbit.) Then, for diameters and derivatives in the spherical metric, This method of controlling distortion was introduced in [11, Definition 2.3] and developed and called in [4] shrinking neighbourhoods. j = j 1 is called the first essential critical time.
In (4.1) ξ > 0 is arbitrary and C 2 is an appropriate constant. The latter inequality follows from the assumption that χ(f (c) := lim inf n→∞ 1 n log |(f n ) (f (c))| ≥ 0. In the sequel we shall assume that ξ δ.
Consider now B 0 = B(x, κ exp −nδ), for 0 < κ 1 small enough that B 0 is deeply in B(x, ∞ s=1 (1 − s −2 ) exp −nδ) so that for the associated pullbacks W t of B 0 , for t = 1, 2, ..., j we have diam W t ≤ ε. This is possible due to bounded distortion before the capture of c. Now notice that for j = j 1 ≥ N for a constant N = N (δ, ξ) by (4.1) Hence, denoting by τ = τ (c) the multiplicity of f at the critical point c, If j 1 < N then we obtain (4.3) automatically if we replace B 0 by a disc centered at x of diameter a exp(−nδ) with a small enough.
Denote n 1 = n − j 1 − 1. Apply the shrinking neighbourhood procedure starting from B(f n 1 (y), exp −n 1 δ/τ ). Let 0 < j 2 ≤ n 1 be the first essential critical time, if it exists. Denote the captured critical point by c (it can be different from the former c) Denote Denote the consecutive pull-backs of B 1 by W 1,j . Repeating (4.1) we obtain, analogously to (4.2), using an analogon of (2.5), Here we have j 2 > N (δ, ξ) automatically, provided c = c and n 1 large enough, since otherwise c is periodic attracting hence not in J(f ).
Denote n 2 = n 1 − j 2 − 1 and continue, choosing j 3 , j 4 , ..., until an essential critical time j k does not exist; then the last pull-back is just the pull-back of B k−1 f j k (y) for f j k , containing y, j k ≥ 0. By the 'telescoping' construction and isolating annuli of moduli log Const κ, all the pull-backs W s of B 0 , s = 1, ..., n have diameters not exceeding ε.
If there is more than one critical point in J(f ) then the proof should be modified in a standard way. It relies on the observation that for n large enough the pull-backs under consideration have small diameters so j s is small only if f js+1 ( c) = c which can happen consecutively only #(Crit(f ) ∩ J(f )) number of times, otherwise a critical point in J(f ) is periodic.

From Proposition 4.1 and Theorem 3.2 it follows
Theorem 4.2. For every rational mapping f : C → C of degree at least 2 such that for every critical point c ∈ J(f ) the lower Lyapunov exponent χ(f (c)) is non-negative, and for every t > 0, the equality P spanning (t) = P tree (t) holds.

Now let us invoke the following part of [7, Theorem 5.1]
Theorem 4.3. For every rational mapping f : C → C of degree at least 2, such that there is exactly one critical point c whose forward orbit has an accumulation point in J(f ) (i.e. c ∈ J(f ) or the forward trajectory of c being attracted to a parabolic periodic orbit), we have χ(f (c)) ≥ 0.
This and Theorem 4.2 yield Corollary 4.4. Let f : C → C be a rational mapping of the Riemann sphere of degree at least 2, such that there is at most one critical point whose forward trajectory has an accumulation point in J(f ), then P spanning (t) = P tree (t).
Without the assumption of weak backward stability, i.e. in the full generality, we can prove only the following in place of Theorem 3.2  Continuation of Proof of Theorem 4.5. We can now write, using (4.5), for a constant C ≥ 0 depending on C f and the multiplicities of the critical points. The factor exp(tC∆M n)L tM n takes care of (at most) M integers k omitted in . For these k we use ρ(f k (y), Crit(f )) ≥ L −(n−k) exp(−n∆), where L = sup |f |, true since otherwise dist(f n (y), f n−k (Crit(f ))) < exp −n∆ which contradicts the definition of r in B above.
Hence, collecting our estimates,

Geometric pressure via spanning sets. The real case
We start from a notion refining the definition of safe, see Definition2.1 Definition 5.1. For (f, K) ∈ A a point z ∈ K is called safe from outer folds if for every η > 0 and all n large enough, for every pull-back W n of W = B(z, exp(−ηn)) for f n , intersecting K, containing a turning critical point for f n , there is a point z n ∈ ∂W n such that f j (z n ) ∈ I for all j = 0, 1, ..., n.
Theorem 5.2. For every (f, K) ∈ A BD + (or A 3 + ) without parabolic periodic orbits, weakly isolated, for every t > 0 and every safe z ∈ K it holds P spanning (t) ≥ P tree (z, t).
If every periodic z ∈ ∂ I is safe from outer folds, then the equality of the pressures holds. In particular it holds, provided K = I = I, namely it is a single interval Proof. I. The CONSTRUCTION inequality.
We mostly repeat parts of the proof of Theorem 2.6 Fix an arbitrary safe z ∈ K and δ is chosen to ε as in the Definition of backward Lyapunov stability.
We shall prove that the set f −n (A) ∩ K itself happens to be an (ε, n)spanning set (roughly, for w with f n (w) as in case (i) below) under an assumption as in Theorem 5.2. Then immediately P spanning (t) ≤ P tree (z, t).
Indeed, if w = f n (w) ∈ W = [z 0 , z 0 ] with its endpoints belonging to A whose distance is at most δ then for its pull-back W n = [z n , z n ] containing w we have for all j = 0, ..., n, |f j (z n ) − f j (w)| < ε (and the same for z n ). The proof that z n or z n is in K is the same as in Proof of Theorem 2.6 and uses the weak isolation assumption.
A trouble is with w such that w = f n (w) is not in any W as above. Then, as in Proof of Theorem 2.6 there is a large gap (a component in R \ K) of length at least δ/4 within the distance at most δ of w .
Then we have two cases.
(i) For some m bounded by a constant depending only on (f, K) and δ the point f m (w ) belongs to some W with endpoints z 0 , z 0 ∈ A(z, δ ) for δ satisfying (2.6). Then ρ n (w, z n+m ) ≤ ρ n+m (w, z n+m ) < ε for an appropriate z n+m in the boundary of the pull-back of W for f n+m containing w.
(ii) For some n + m the point w = f n+m (w) is close to a periodic point p in the boundary of a large gap G.
Notice that in fact p ∈ ∂ I, see [13][Lemma 2.9, Case 2]. Indeed, if p and all other points of its periodic orbit belong to the interior of I, then also f j (G) ⊂ int I. Otherwise, if j 0 is the least integer such that f j 0 (G) intersects ∂ I at a point y, then y j 0 = (f j 0 | G ) −1 (y) ∈ G and it belongs to K since y ∈ K and all f j (y j 0 ) belong to I hence to K by the maximality of K. This contradicts G ∩ K = ∅. If all f j (G) are in I, then by the maximality G ⊂ K, a contradiction.
If w / ∈ B , then for some k ≤ κχn the point v = f k (w ) is far from the periodic orbit of p but f k is still invertible on B(p, |p − w |). In particular there exist z 0 , z 0 ∈ A such that |z 0 − z 0 | < δ and v ∈ [z 0 , z 0 ]. Hence w ∈ [z k , z k ], the pull-back. Hence, as before, w ∈ [z n+m+k , z n+m+k ] where one of the ends say z n+m+k is in K and ρ n (w, z n+m+k ) < ε.
If w ∈ B , then by the assumption that p is safe from the outer fold for the constant η for n large enough, for [z(w), z (w)] being the pull-back of B for f n+m containing w, all f j (z(w)), j = 0, ..., n + m belong to I (or the same for z (w)). In particular u := f n+m (z(w)) is the point of ∂B in I.
By our definitions, z is between w and u. Since w ∈ K, f j (w) ∈ I for all j ≥ 0. Hence there exists z n+m ∈ [w, z(w)] ∩ f −n−m ( z). such that f j ( z n+m ) ∈ I as belonging to f j ([w, z(w)] being intervals shorter than ε with ends in I. These ends may be of the form f j (w), f j (z(w)) or f i (c) for a turning critical point c ∈ K hence in K ⊂ I.
Hence z n+m ∈ K and ρ n (w, z n+m ) ≤ ε.
So, given ε > 0 and safe z ∈ K, for all κ > 0, for all n large enough, the set SP(z, n) := is (n, ε)-spanning. Const(ε) depends on N and m above which depend on ε.
For n → ∞ and κ → 0 this holds for ξ arbitrarily small and finally P spanning (t) ≤ P tree (z, t).
Notice that unlike in Proof of Theorem 2.6 we have not needed here to compare the derivatives |(f n ) (w)| and the shadowing |(f n ) (z n )|. In particular we consider all w, rather than having f n (w) safe.
Notice finally that if K = I = I is a single interval, then every z ∈ I is safe from outer folds. Otherwise both ends z n , z n of W n are outside I, since if, say, z n ∈ I then all f j (z n ) ∈ I by the forward invariance of K = I here. So z n and z n are on the different sides of I. This is not possible since W n is short by backward Lyapunov stability of f .
II. The ALL inequality. The proof is the same as in the complex case, via P spanning (t) ≥ P hyp (t).
Example 5.3. We show that the assumption on the safety from outer folds is really needed in Theorem 5.2 above.
• Consider quadratic polynomials f a (x) = ax(1 − x) for 0 < a < 4 large enough that the entropy of f a is positive. For each a let p a denote the unique fixed point in the open interval (0, 1). It is repelling; let us make a small perturbation of f a close to p a so that p a becomes attracting and a repelling orbit Q a of period 2, being the boundary of B 0 (p) ⊂ (1/2, 1) which is the immediate basin of attraction to p a , is created.
One can do it in such a way that Schwarzian derivative Sg of the new map g = g a is negative except in B 0 (p). Write Q a = {q a , q a } with q a < q a . Omit the subscript a. Define otherwise.
One can choose ε > 0 arbitrarily small and b > 0 so that the above function is continuous. Then b is also small hence by Sf < 0 Schwarzian of g stays negative except in [p − ε, p + ε].
• Let I a = (I n ) n=1,2,...,N denote the kneading sequence for g a , that is the sequence of letters L, R, C depending whether c n = g n a (1/2) lies to the left of the critical point 1/2, to the right of 1/2, or at 1/2. We put N the least integer n for which I n = 1/2. If no such integer exist we put N = ∞. See [1] for these definitions. Let where N = ∞, all n j are finite positive, even, their sequence is increasing and n j → ∞ exponentially fast as j → ∞.
I is a maximal sequence for every sequence (n j ) satisfying above conditions, hence there exists a such that g = g a has this kneading sequence, see [1][Theorem III.1.1].
For I as above for c n left of 1/2 we have c n+1 right of 1/2 and close to q a , left of it (remember Q a = {q a , q a } with q a < q a ). Next the trajectory c n+2 , c n+3 , ... follows Q a outside of [q a , q a ], in the interval (1/2, 1) until c n+n k +1 occurs to the left of 1/2 moreover to the left and close to the point symmetric to q a with respect to 1/2.
• Now consider I = [c 2 , q a ] ∪ [q a , c 1 ], g restricted to a neighbourhood U of I and K the maximal forward invariant subset of I. Clearly 1/2 ∈ K since otherwise g n (1/2) → p so I would consist solely of R's for n large enough. K = I \ B(p a ), where B(p a ) is the basin of attraction by g to p a . Due to Sg < 0 on a neighbourhood U of K we obtain (g, K, I, U) ∈ A BD , provided we prove • Claim: g is topologically transitive on K.
Let a be so that the kneading sequence for f = f a is the same as for g a , that is I. Due to the lack of attracting and parabolic periodic orbits for f a (otherwise I would be eventually periodic), there is a monotone increasing continuous semiconjugacy h : h is defined first in a standard way on T (g) := n≥0 g −n (1/2) to the corresponding T (f ) := n≥0 f −n (1/2), increasing since the orders in the interval [c 2 , c 1 ] of points in these sets are (combinatorially) the same, due to the same kneading sequences.
The mapping h can be continuously extended to the closures, and notice that cl T (f ) = [c 2 , c 1 ] due to the absence of wandering intervals for f . This h collapses B 0 (p) and its g n -preimages to points, provided we extend h to these gaps by constant functions. In other words h identifies the pairs of points being ends of gaps B(p) being components in the basin of p. There are no other gaps in [c 2 , c 1 ] \ cl T (g) since there are no wandering intervals (see [8]) and no attracting or parabolic periodic orbits other than p. This in turn holds since the Schwarzian Sg is negative outside B(p) so the basin of such an orbit would contain a critical point that is 1/2 which is not possible since I is not eventually periodic. Therefore h is injective on K except the abovementioned pairs of points.
Notice that our I is not a *-product, see [1][Section II.2] for the definition. Hence there is no interval T ⊂ I f = [f 2 (1/2), f (1/2)] such that f k (T ) ⊂ T for some k > 1 containing 1/2 with f k unimodal on it (i.e. with one turning point), i.e. there is no renormalization interval. (In other words f is not renormalizable). This follows from [1][Corollary II.7.14].
Consider now any interval T ⊂ I f and V = j≥0 f j (T ). By definition V is forward invariant. Let W be a connected component of V . Then there are integers 0 ≤ k 1 < k 2 such that f k 1 (W ) ∩ f k 2 (W ) = ∅ since W is non-wandering, see [8][Chapter IV, Theorem A] for the non-existence of wandering intervals. Hence, for k = k 2 − k 1 , and W = f k 1 (W ), f k (W ) ⊂ W . We consider k the smallest such integer. We can assume that 1/2 ∈ W (or some f j (W )), since otherwise W would be attracted to a periodic orbit and we have assumed such orbits do not exist. No f (W ), 0 < < k contains 1/2 by its disjointness from W . So f k is unimodal on W . So k = 1, since otherwise f would be renormalizable. So f (1/2) and f 2 (1/2), the end points of I f , belong to W . Hence V = I f , hence f is topologically transitive on I f . This due to our semiconjugacy and the fact that K has no isolated points, implies the topological transitivity of g on K. The Claim is proved.
The property we proved in particular, (∀ open W ) (∃k), f k (W ) = domain(f ) is called topological exactness or leo -"locally eventually onto". This is stronger than topological transitivity. See [13][Lemma A7] for a discussion of a general case. Since the topological entropy h top (g|K) > 0 we can write (g, K, I, U) ∈ A BD + • Notice that K is weakly isolated for g on U, see Definition 2.5. This is so because if a periodic trajectory P in U has a point z / ∈ K then z belongs to the basin of attraction to p, i.e. g n (z) → p. In other words the trajectory g| n U (z) leaves U. Hence P ⊂ K. Note that above argument proves the weak isolation property in general situations, namely if K is Julia set in the sense of [8][Chapter IV, Lemma] i.e. the domain being an interval with the basins of attracting or parabolic periodic orbits removed (provided there is a finite number of them).
• Notice that q a is not safe from outer folds, see Definition 5.1. Indeed. Denote 2 + j=1,...k n j + k + 1 by m k . The summands n j correspond to the blocks of R's, the first summand 2 corresponds to the starting RL and the final 1 to the first R in the k + 1'th block of R's. We obtain |c m k − q a | ≤ Const(a) exp −n k+1 χ(q a ), where χ(q a ) = 1 2 log |(g 2 ) (q a )|. Consider the pullback W m k of W = B(q a , exp(−ηm k )) for g m k containing 1 2 . The critical point 1/2 is not recurrent since c m k −1 corresponding to L approach to the point symmetric to q a since n j grow, so they are in the distance from c 2 bounded away from 0. Hence the only points we need to care about, c m k −3 are in the distance from 1/2 also bounded away from 0.
Hence for all n = 1, 2, ...m k − 1 the map g is injective on g n (W m k ) but g has a turning critical point 1/2 in W m k . Using the assumption that all n j are even we conclude that each g m k has a minimum at 1/2, hence if n k+1 χ(q a ) m k η the boundary points of W m k are mapped by g m k into the gap (basin B 0 (p)). In other words ∂W m k ⊂ B m k ∪ B m k the latter being the union of a symmetric pair of pull-backs of B 0 (p) for g m k on both sides of 1/2.
Hence for ε small enough and an integer N such that all g j (1/2), j ≥ N are close to ω(1/2), if |g j (1/2) − g j (x)| ≤ ε for all j : N ≤ j ≤ n, then for all N ≤ j ≤ n, g j (x) ∈ g j (W n ), where W n is the pull-back of W as above, but for W = B(q a , ε) (unlike above). Then this holds also automatically also for 0 ≤ j < N , maybe on the cost of taking a smaller ε.
We conclude that P spanning (t) = ∞. Remark 5.4. 1. In the example above K is not uniformly perfect (considered in the plane), unlike in the complex case where the uniformly perfect property of Julia set allowed us to prove Theorem 4.
2. In this example the so-called Bowen's periodic specification property does not hold. This property is defined for any continuous map f : X → X of a compact X as follows: For every ε > 0 there exists an integer N such that for every x ∈ X and every integer n ≥ 0 there exists y ∈ X of period k : n ≤ k ≤ n + N such that for every 0 ≤ j ≤ n, dist(f j (x), f j (y)) ≤ ε.
Even a weaker periodic specification does not hold, where N = N (ε) is replaced by N (n, ε) for ε small enough (see the survey [6]). Namely for every function N (n, ε) there exists a such that for g a with an appropriate kneading sequence I the specification with N (ε, n) does not hold. Consider blocks of the g-trajectories 1/2, c 1 , c 2 , ...c m j with n k growing fast enough. Then for every y being (m j , ε)-close to 1/2, y is in fact ξ-close to 1/2 for ξ > 0 arbitrarily small, depending on n j+1 . Then the period of y must be long since otherwise y would be an attracting periodic point.
3. One can have an additional insight in the topological dynamics of g a or f = f a if one uses the existence of a semiconjugacy of f to a tent map τ (of slopes ±h top (f ), see [9][Theorem 7.4], which must be a conjugacy since f has no renormalization or wandering interval [8][Chapter IV, Theorem A].