Geometric pressure for multimodal maps of the interval
This paper is an interval dynamics counterpart of three theories founded earlier by the authors, S. Smirnov and others in the setting of the iteration of rational maps on the Riemann sphere: the equivalence of several notions of non-uniform hyperbolicity, Geometric Pressure, and Nice Inducing Schemes methods leading to results in thermodynamical formalism. We work in a setting of generalized multimodal maps, that is smooth maps of a finite union of compact intervals in into with non-flat critical points, such that on its maximal forward invariant set the map is topologically transitive and has positive topological entropy. We prove that several notions of non-uniform hyperbolicity of are equivalent (including uniform hyperbolicity on periodic orbits, TCE & all periodic orbits in hyperbolic repelling, Lyapunov hyperbolicity, and exponential shrinking of pull-backs). We prove that several definitions of geometric pressure , that is pressure for the map and the potential , give the same value (including pressure on periodic orbits, ”tree” pressure, variational pressures and conformal pressure). Finally we prove that, provided all periodic orbits in are hyperbolic repelling, the function is real analytic for between the ”condensation” and ”freezing” parameters and that for each such there exists unique equilibrium (and conformal) measure satisfying strong statistical properties.
2000 Mathematics Subject Classification:37E05, 37D25, 37D35.
1 Introduction. The main results
- 1.1 Generalized multimodal maps, maximal invariant sets and related notions
- 1.2 Periodic orbits and basins of attraction. Bounded distortion property
- 1.3 Statement of Theorem A: Analytic dependence of geometric pressure on temperature, equilibria
- 1.4 Characterizations of geometric pressure
- 1.5 Non-uniformly hyperbolic interval maps
- 1.6 Complementary Remarks
- 1.7 Acknowledgements
- 2 Preliminaries
- 3 Non-uniformly hyperbolic interval maps
- 4 Equivalence of the definitions of Geometric Pressure
- 5 Pressure on periodic orbits
- 6 Nice inducing schemes
- 7 Analytic dependence of Geometric Pressure on temperature. Equilibria.
- 8 Proof of Key Lemma. Induced pressure
- A More on generalized multimodal maps
- B Uniqueness of equilibrium via inducing
- C Conformal pressures
1. Introduction. The main results
This paper is devoted to extending to interval maps results for iteration of rational functions on the Riemann sphere concerning statistical and ‘thermodynamical’ properties obtained mainly in [PR-LS2], [PR-L1] and [PR-L2]. We work with a class of generalized multimodal maps, that is smooth maps of a finite union of compact intervals in into with non-flat critical points, and investigate statistical and ‘thermodynamical’ properties of restricted to the compact set , maximal forward invariant in , such that is topologically transitive and has positive topological entropy.
This includes all sufficiently regular multimodal maps, and other maps that are naturally defined on a union of intervals, like fixed points of generalized renormalization operators, see e.g. [LS] and references therein, Examples 1.9, 1.10, 2.12, and the ones mentioned in Subsection 1.6.
Several strategies are similar to the complex case, but they often need to be adapted to the interval case in a non-trivial way.
The paper concerns three topics, closely related to each other.
1. Extending the results for unimodal maps in [NS], and multimodal maps in [NP] and [R-L] to generalized multimodal maps considered here, we prove the equivalence of several notions of non-uniform hyperbolicity, including uniform hyperbolicity on periodic orbits, Topological Collet-Eckmann condition (abbr. TCE) & all periodic orbits hyperbolic repelling, Lyapunov hyperbolicity, and exponential shrinking of pull-backs. For the complex setting see [PR-LS1] and the references therein.
2. We prove that several definitions of geometric pressure , that is pressure for the map and the potential , give the same value (including pressure on periodic orbits, ”tree” pressure and variational pressures). For the complex setting, see [PR-LS2] and the references therein.
3. We prove that, provided all periodic orbits in are hyperbolic repelling, the geometric pressure function is real analytic for (inverse of temperature) between the ”condensation” and ”freezing” parameters, and , and for each such there exists unique equilibrium (and conformal) measure satisfying strong statistical properties. All this is contained in Theorem A, the main result of the paper. For the complex setting, see [PR-L1] and [PR-L2].
Our results extend [BT1], which proved existence and uniqueness of equilibria and the analyticity of pressure only on a small interval of parameters and assumed additionally a growth of absolute values of the derivatives of the iterates at critical values.
Our results extend also [PS, Theorem 8.2], where the real analyticity of the geometric pressure function on a neighborhood of was proved, also assumed some growth of absolute values of the derivatives of the iterates at critical values and additionally assumed a slow recurrence condition.
Our main results are stronger in that growth assumptions are not used and the domain of real analyticity of is the whole , i.e. the maximal possible domain. Precisely in this domain it holds , where supremum is taken over all -invariant ergodic probability measures on , and (it is therefore clear that analyticity cannot hold at neither nor , as is affine to the left of and to the right of ).
Let us mention also the paper [IT] where, under the restriction that has no preperiodic critical points, the existence of equilibria was proved for all . The authors proved that is of class and that their method does not allow them to obtain statistical properties of the equilibria.
1.1. Generalized multimodal maps, maximal invariant sets and related notions
We shall assume throughout the paper that intervals are non-degenerate (i.e. containing more than one point). For an integer and a finite union of intervals , a function is of class , if it is the restriction of a function of class defined on an open neighborhood of .
Let be a finite union of intervals, and a map of class . A critical point of is a point such that . An isolated critical point of is inflection (resp. turning) if is locally (resp. is not locally) injective at . Furthermore, is non-flat if for some real and a diffeomorphism in a neighbourhood of with , see [dMvS, Chapter IV]. The set of critical points will be denoted by turning critical points by and inflection critical points by .
Let be the union of a finite family of pairwise disjoint compact intervals in the real line . Moreover, consider a map of class with only non-flat critical points. Let be the maximal forward invariant set for , more precisely the set of all points in whose forward trajectories stay in . We call the maximal repeller of . The map is said to be a generalized multimodal map if is infinite and if is topologically transitive (that is for all open in it, non-empty, there exists such that ).
A generalized multimodal map is said to be reduced if and if
We will show that and is either a finite union of compact intervals or a Cantor set, see Lemma 2.1.
Note that we do not assume neither nor .
We shall usually assume that is dynamically sufficiently rich, namely that the topological entropy is positive.
In examples of generalized multimodal maps, topological transitivity and positive entropy of must be verified, which is sometimes not easy.
Every generalized multimodal map determines in a canonical way a subset of such that the restriction is a reduced generalized multimodal map with as follows:
Remove from a finite number of open intervals , each containing a critical point of . Next, using for the same notation , consider for each the set being the convex hull of . Finally define . Then is a reduced generalized multimodal map. In fact, the maximality of in obviously follows, since . Since has no isolated points, it also follows that is non-degenerate.
Thus we shall consider only reduced generalized multimodal maps. We shall usually skip the word ‘reduced’.
All assertions of our theorems concern the action of on . So it is natural to organize definitions in the opposite order as follows.
Let be an infinite compact subset of the real line. Let be a continuous topologically transitive mapping. Assume there exists a covering of by a finite family of pairwise disjoint closed intervals with end points in such that extends to a generalized multimodal map on their union and is the maximal forward invariant set in it. Reducing if necessary, assume all the critical points of in are in . Then is a reduced generalized multimodal map.
Let be a generalized multimodal map. Consider a extension of to an open neighbourhood of each . We consider ’s small enough so that they are pairwise disjoint. Moreover we assume that all critical points in are in . Thus together with (1.1) we assume that the union of satisfies
We call the quadruple a (reduced) generalized multimodal quadruple. In fact it is always sufficient to start with a triple , because this already uniquely defines .
As we do not assume , when we iterate , i.e. consider for positive integers , we consider them on their domains, which can be smaller than for ,
Note that we do not assume to be maximal forward invariant in . We assume maximality only in .
See Proposition A.2 for an alternative approach, replacing maximality of in by so-called Darboux property.
1. Let us stress that the properties of the extension of beyond are used only to specify assumptions imposed on the action of on , in particular the way it is embedded in . So we will sometimes talk about generalized multimodal pairs , understanding that this notion involves and as above, sometimes about the triples , sometimes about the triples .
2. Denote the space of all (reduced) generalized multimodal quadruples (pairs, triples) discussed above by . For and (change of coordinates as in the definition of non-flatness) of class we write , so . If is assumed we write or .
1.2. Periodic orbits and basins of attraction. Bounded distortion property
As usual we call a point periodic if there exists such that . We denote by its periodic orbit.
Define the basin of attraction of the periodic orbit by
The orbit is called attracting if . Notice that this happens if , when we call the orbit hyperbolic attracting, and it can happen also if . The orbit is called repelling if for , where is a small neighbourhood of in , we have and as . If then is repelling and we call the orbit hyperbolic repelling.
When is neither attracting nor repelling, we call indifferent. The union of the set of indifferent periodic orbits will be denoted by . If is indifferent, then it is said to be one-sided attracting if its basin of attraction contains a one-sided neighbourhood of each point of the orbit. Finally is said to be one-sided repelling if it is not repelling and if for a local inverse, the orbit of every point in a one-sided neighborhood of converges to .
We say also that a periodic point is (hyperbolic) attracting, (hyperbolic) repelling, one-sided attracting, one-sided repelling or indifferent if is (hyperbolic) attracting, (hyperbolic) repelling, one-sided attracting, one-sided repelling, or indifferent, respectively.
When we discuss a specific (left or right) one-side neighbourhood of a point in in the above definitions, we sometimes say attracting on one side or repelling on one side.
If an indifferent periodic point of period is neither one-sided attracting nor one-sided repelling on the same side, then obviously it must be an accumulation point of periodic points of period from that side. Notice that indifferent one-sided attracting (repelling) implies preserves the orientation at , equivalently: ; otherwise, if , the point would be attracting or repelling.
For it follows by the maximality of that there are no periodic orbits in . By the topological transitivity of on , there are no attracting periodic orbits in . Moreover each point in an indifferent periodic orbit in is one-sided repelling on the side from which it is accumulated by ; then it is not accumulated by from the other side. Also by the topological transitivity and by smoothness of the number of periodic orbits in that are not hyperbolic repelling is finite. The proof uses [dMvS, Ch. IV, Theorem B]. Therefore, the number of periodic orbits in that are not hyperbolic repelling is finite.
For further details see Corollary A.3 in Appendix A and remarks following it.
By changing on if necessary, one can assume that the only periodic orbits in are hyperbolic repelling. See Appendix A, Lemma A.4, for details.
For an attracting or a one-sided attracting periodic point , the immediate basin is the union of the components of whose closures intersect . The unique component of whose closure contains will be denoted by .
Notice that if is attracting then . If is one-sided attracting then is a boundary point of and vice versa. Finally notice that for each component of there exists such that . (See an argument in Proof of Proposition A.2 in Appendix A). We need to add above in case is indifferent and some contains a turning critical point whose forward trajectory hits . (Compare also Example 1.9 below, where Julia set need not be backward invariant.)
If for a generalized multimodal map its domain is just one interval and , then we have a classical case of an interval multimodal map. However the set of non-escaping points is the whole in this case, usually too big (not satisfying topological transitivity, and not even mapped by onto itself). So one considers smaller -invariant sets, see below.
Define Julia set by
compare [dMvS, Ch.4], and its core Julia set by
Notice that the sets and are compact and forward invariant. They need not be backward invariant, even for , namely a critical preimage of a indifferent point can be in ), whereas .
The definition of is compatible with the definition as a complement of the domain of normality of all the forward iterates of as in the complex case.
Notice however that without assuming that is topologically transitive on the comparison to Julia set is not justified. For example for mapping into itself defined by where (to exclude an attracting or indifferent fixed point or an escape from ), we have where is not topologically transitive. However, if is not renormalizable, if we restrict to then is topologically transitive on for (independent of ), where , and is the critical point. Notice that by and the non-renormalizability there can be no basin of attraction. Our and are both equal to
Notice that since belongs to the interior of the set is not maximal invariant in being a neighbourhood of in whatever small we take.
Multimodal maps considered in the previous example, restricted to still need not be topologically transitive. Then, instead, examples of generalized multimodal pairs in are provided by , where is an arbitrary maximal topologically transitive set in ’s in the spectral decomposition of the set of non-wandering points for , as in [dMvS, Theorem III.4.2, item 4.], so-called basic set, for which . It is easy to verify that basic sets are weakly isolated.
In regularity lower than there can exist a wandering interval, namely an open interval such that all intervals are pairwise disjoint and not in . In the multimodal case wandering intervals cannot exist, see [dMvS, Ch.IV, theorem A]. We shall use this fact many times in this paper.
Following [dMvS] we say that for and an interval , an interval is an -scaled neighbourhood of if has two components, call them left and right, and , such that .
We say that a (or ) map for an open subset of (in particular a generalized multimodal triple for ) satisfies bounded distortion, abbr. BD, condition if there exists such that for every there exists such that the following holds: For every pair of intervals , such that and for every , if maps diffeomorphically an interval containing onto an interval being an -scaled neighbourhood of and , then for every we have for
(Equivalently, for every , .)
Notice that BD easily implies that for every there is such that if in the above notation is an -scaled neighbourhood of then contains an -scaled neighbourhood of .
We denote the space of or satisfying BD, by or , respectively. Sometimes we omit and use notation or for generalized multimodal pairs.
Notice that for all repelling periodic orbits in are hyperbolic repelling.
Furthermore, since every indifferent periodic orbit in K is one-sided repelling by Remark 1.7, it follows from BD that every indifferent periodic orbit in K is also one-sided attracting.
The finiteness of the set of indifferent periodic orbits in was in fact noted already in Remark 1.7 without assuming BD.
The finiteness of the set of non-repelling periodic orbits in (where we treat an interval of periodic points as one point) follows from the standard fact that by BD for every (one-sided) attracting , the immediate basin must contain a critical point or the boundary of the basin must contain a point belonging to and that there is only a finite number of such points (in fact critical points cannot be in since we have assumed ). This implies also that the only indifferent periodic points that are points of accumulation of periodic points of the same or doubled period are those belonging to intervals of periodic points. See Proposition A.5 in Appendix A for more details. In particular, by shrinking , one can assume that every periodic orbit in is hyperbolic repelling.
1.3. Statement of Theorem A: Analytic dependence of geometric pressure on temperature, equilibria
Fix . Let be the space of all probability measures supported on that are invariant by . For each , denote by the measure theoretic entropy of , and by the Lyapunov exponent of .
If is supported on a periodic orbit we use the notation .
Given a real number we define the pressure of for the potential by,
A measure is called an equilibrium state of for the potential , if the supremum in (1.3) is attained for this measure.
As in [PR-L2] define the numbers,
Later on we write sometimes and or just . By Ergodic Decomposition Theorem we can assume that all in the definition of and above are ergodic. In Section 8 we use an equivalent definition, see Proposition 4.7 in Section 4.
the condensation point and the freezing point of , respectively. As in the complex case the condensation (resp. freezing) point can take the value (resp. ).
Similarly to [PR-L2] we have the following properties:
for all we have ;
for all we have .
Let be a map of class with only non-flat critical points, and let be a compact subset of that is forward invariant by . Moreover, let be a Borel function. We call a finite Borel measure on for a -conformal measure if it is forward quasi-invariant, i.e. , compare [PU, Section 5.2], and for every Borel set on which is injective
Let be in . We say that for intervals in intersecting , the map is a -diffeomorphism if it is a diffeomorphism, in particular it is well defined, i.e. for each , and , compare Definition A.1 and Lemma 2.2.
Since , is -diffeomorphism if and only if is -diffeomorphism for all .
Let be in . Denote by the “conical limit part of ” for and , defined as the set of all those points for which there exists and an arbitrarily large positive integer , such that on , the component of the -preimage of the interval containing , is a -diffeomorphism onto .
Notice that .
The main theorem in this paper corresponding to [PR-L2, Main Theorem] is
Let , in particular let be topologically transitive on and have positive entropy, and assume all -periodic orbits in are hyperbolic repelling. Then
- 1. Analyticity of the pressure function:
The pressure function is real analytic on the open interval , and linear with slope (resp. ) on (resp. ).
- 2. Conformal measure:
For each , the least value for which there exists an -conformal probability measure on is . The measure is unique among all -conformal probability measures. Moreover is non-atomic, positive on all open sets in , ergodic, and it is supported on .
- 3. Equilibrium states:
For each , for the potential , there is a unique equilibrium measure of . It is ergodic and absolutely continuous with respect to with the density bounded from below by a positive constant almost everywhere. If furthermore is topologically exact on , then this measure is mixing, has exponential decay of correlations and it satisfies the Central Limit Theorem for Lipschitz gauge functions.
It is easy to see that the assertion about the analyticity of can be false without the topological transitivity assumption. See also [Dobbs] for an example where analyticity fails at an infinite set of values of .
For a generalized multimodal map having only hyperbolic repelling periodic points in , as in Theorem A, the assumption can be replaced by a bounded distortion hypothesis that is more restrictive than BD, see Definition 2.13, which is in fact what we use in our proof. See Remark 2.14 and Lemma A.4.
Let us now comment on the properties of topological transitivity and positive entropy assumed in Theorem A, and also notions of exactness.
Let be a continuous mapping of a compact metric space .
We call topologically exact if for every non-empty open there is such that (sometimes the name locally eventually onto is used).
We call weakly topologically exact (or just weakly exact) if there exists such that for every non-empty open , there exists such that
This property clearly implies that maps onto , compare Lemma 2.1.
Notice that the equality (1.5) implies automatically the similar one with replaced by any . To see this apply to both sides of the equation and use .
In the sequel we shall usually use these definitions for setting , . For some immediate technical consequences of the property of weak exactness see Remark 2.6 below.
Clearly topological exactness implies weak topologically exactness which in turn implies topological transitivity. In Appendix A,
Lemma A.7, we provide a proof of the converse fact for , saying that topological transitivity and positive topological entropy of imply weak topological exactness. This allows in our theorems to assume only topological
transitivity and to use in proofs the formally stronger weak exactness.
This fact seems to be folklore. Most of the proof in Appendix A was told to us by Michał Misiurewicz. We are also grateful to Peter Raith for explaining us how this fact follows from [Hof]. See also [Buzzi, Appendix B].
In fact for the properties: weak topological exactness and positive topological entropy are equivalent for , see Proposition A.8.
Now we provide the notion of exceptional and related ones, substantial in the paper, though not explicitly present in the statements of the main theorems. They are important in the study of various variants of conformal measures and pressures, see Appendix C, and in explaining the meaning if , and substantial in the study of e.g. Lyapunov spectrum, see [MS], [GPRR] and [GPR2].
1. End points. Let . We say that is an end point if . We shall use also the notion of the singular set of in , defined by .
Of more importance will be the notion of the restricted singular set
where is the set of points where is not an open map, i.e. points such that there is an arbitrarily small neighbourhood of in whose -image is not open in . Notice that , see Lemma 2.2.
In the proof of Theorem A we shall use inducing, that is return maps to nice domains, as commented already at the very beginning, as in [PR-L2], but we cover by the components (open intervals) of the nice sets the set rather than only .
Analogously, replacing by , we define -exceptional subsets of .
Another useful variant of this definition is weakly-exceptional where we do not assume is forward invariant. For example each unimodal map of interval, i.e. with just one turning critical point , has the one-point set being a weakly -exceptional set.
Notice that it easily follows from this definition for or that if belongs to a weakly exceptional set, then the set
is weakly exceptional. So it makes sense to say that the point is weakly exceptional.
It is not hard to see that if weak topological exactness of on is assumed, then each -exceptional set is finite, moreover with number of elements bounded by a constant, see Proposition 2.7. Therefore the union of -exceptional sets is -exceptional and there exists a maximal -exceptional set which is finite. It can be empty. If it is non-empty we say that is -exceptional, or that is -exceptional. Analogous terminology is used and facts hold for weakly exceptional sets and for in place of .
More generally the above facts hold for an arbitrary finite in place of or , where is called then -exceptional or weakly -exceptional, depending as we assume it is forward invariant or not, if , see Proposition 2.7.
3. No singular connection condition. To simplify notation we shall sometimes assume that no critical point is in the forward orbit of a critical point. This is a convention similar to the complex case. Moreover we shall sometimes assume that no point belonging to is in the forward orbit of a point in , calling it no singular connection condition.
These assumptions are justified since no critical point, neither a point belonging to , can be periodic, see Lemma 2.2, hence each trajectory in can intersect in at most number of times, hence with difference of the moments between the first and last intersection bounded by a constant. In consequence several proofs hold in fact without these assumptions.
1.4. Characterizations of geometric pressure
For , all the definitions of pressure introduced in the rational functions case, see [PR-LS2], make sense for . In particular
Definition 1.20 (Hyperbolic pressure).
supremum taken over all compact -invariant (that is ) isolated hyperbolic subsets of .
Isolated (or forward locally maximal or repeller), means that there is a neighbourhood of in such that for all implies . Hyperbolic or expanding means that there is a constant such that for all large enough and all we have . We call such isolated expanding sets expanding repellers following Ruelle.
We shall prove that the space of such sets is non-empty.
Notice that by our definitions is a maximal invariant set in its neighbourhood in , whereas the whole need not be maximal invariant in any of its neighbourhoods in , see Definition 1.4 and Example 1.9.
From this definition it immediately follows (compare
[PU, Corollary 12.5.12] in the complex case) the following
(Generalized Bowen’s formula) The first zero of is equal to the hyperbolic dimension of , defined by
supremum taken over all compact forward -invariant isolated hyperbolic subsets of .
Sometimes we shall assume the following property
We call an -invariant compact set weakly isolated if there exists an open neighbourhood of in the domain of such that for every -periodic orbit , if it is in , then it is in . We abbreviate this property by (wi).
In the case of a reduced generalized multimodal quadruple it is sufficient to consider in this property a neighbourhood of . Indeed, by the maximality property if is not contained in it is not contained in . For an example of a topologically exact generalized multimodal pair which does is not satisfy (wi) see Example 2.12.
Definition 1.23 (Tree pressure).
For every and define
Under suitable conditions for (safe and expanding, as defined below) limsup can be replaced by liminf, i.e. limit exists in in this definition, see Proof of Theorem B, more precisely Lemma 4.4 and the Remark following it.
To discuss the (in)dependence of tree pressure on we need the following notions.
Definition 1.24 (safe).
See [PU, Definition 12.5.7]. We call safe if and for every and all large enough .
Notice that by this definition all points except at most a set of Hausdorff dimension 0, are safe.
Sometimes it is sufficient in applications to replace here by or , and write appropriately -safe and -safe.
Definition 1.25 (expanding or hyperbolic).
See [PU, Definition 12.5.9]. We call expanding or hyperbolic if there exist and such that for all large enough maps 1-to-1 the interval
to and . Here and further on means the component containing .
Sometimes we shall use also the following technical condition
Definition 1.26 (safe forward).
A point is called safe forward if there exists such that for all .
For every , there exists which is safe, safe forward and expanding. The pressure does not depend on such .
For such we shall just use the notation or and use the name tree pressure.
For rational function it is enough to assume is safe, i.e. does not depend on , except in a thin set (of Hausdorff dimension 0). In the interval case we do not know how to get rid of the assumption is expanding.
One defines periodic orbits pressure hyperbolic and variational pressure for , analogously to [PR-LS2], by
Definition 1.28 (Periodic orbits pressure).
Let be the set of all -periodic points in of period (not necessarily minimal period). Define
Definition 1.29 (Hyperbolic variational pressure).
Notice that compared to variational pressure in (1.3) we restrict here to hyperbolic measures, i.e. measures with positive Lyapunov exponent. The space of hyperbolic measures is non-empty, since . Indeed, then there exists , an -invariant measure on of entropy arbitrarily close to by Variational Principle, hence positive. Hence, by Ruelle’s inequality, .
For every weakly isolated, all pressures defined above coincide for all . Namely
For the assumption (wi) can be skipped.
Denote any of these pressures by or just and call geometric pressure.
The first equality holds for complex rational maps, under an additional assumption H, see [PR-LS2], and we do not know whether this assumption can be omitted there. In the interval case, we prove that this assumption (and even H*, a stronger one, see Section 5) holds automatically.
The proof of uses Katok-Pesin’s theory, similarly to the complex case, but in the setting, allowing flat critical points, see Theorem 4.1.
The proofs of the equalities in the interval case must be slightly modified since we cannot use the tool of short chains of discs joining two points, omitting critical values and their images, see e.g. [PR-LS1, Geometric Lemma]. Instead, we just use strong transitivity property, Definition 2.3. This difficulty is also the reason that we assume is expanding when proving that is independent of .
The definition of conformal pressure is also the same as in the complex case:
Definition 1.30 (conformal pressure).
see Definition 1.14
The proof of Theorem A yields the extension of Theorem B to the conformal pressure for , for maps having only hyperbolic repelling periodic orbits. We obtain
For every (or ), all whose periodic orbits in are hyperbolic repelling for every
Our proof will use inducing and will accompany Proof of Theorem A, see Subsection 7.2.
1.5. Non-uniformly hyperbolic interval maps
The main result of Section 3 is the equivalence of several notions of non-uniform hyperbolicity conditions, which are closely related to the Collet-Eckmann condition: For every critical point c in K, whose forward trajectory does not contain any other critical point,
Namely the following is an extension to generalized multimodal maps of the results for multimodal maps in [NP] and [R-L, Corollaries A and C], see also [NS] for the case of unimodal maps and [PR-LS1] for the case of complex rational maps. See Section 3 for definitions.
For every or , weakly isolated, the following properties are equivalent.
TCE (Topological Collet-Eckmann) and all periodic orbits in hyperbolic repelling,
ExpShrink (exponential shrinking of components),
CE2* (backward Collet-Eckmann condition at some for preimages close to ),
UHP. (Uniform Hyperbolicity on periodic orbits in .)
UHPR. (Uniform Hyperbolicity on repelling periodic orbits in .)
Lyapunov hyperbolicity (Lyapunov exponents of invariant measures are bounded away from 0),
Negative Pressure ( for large enough).
The only place we substantially use (wi) is that UHP on periodic orbits in is the same as on periodic orbits in a sufficiently small neighbourhood of , since both sets of periodic orbits are the same by the definition of (wi). We will use (wi) also to reduce the proof to the case.
A novelty compared to the complex rational case is the proof of the implication CE2* ExpShrink, done by the second author for multimodal maps in [R-L]. Here we present a slight generalization, following the same strategy, with a part of the proof being different.
1. For complex rational maps, the TCE condition is invariant under topological conjugacy. However, since a topological conjugacy between interval maps need not preserve inflection critical points, the TCE condition is not, by itself, invariant under topological conjugacy.
2. As opposed to the case of complex rational maps, for interval maps the TCE condition does not exclude existence of periodic orbits that are not hyperbolic repelling. So, the TCE condition is not, by itself, equivalent to the other notions of non-uniform hyperbolicity considered above.
1.6. Complementary Remarks
Then the function for , called dimension spectrum for Lyapunov exponents, is equal to , where
(a Legendre-like transform of geometric pressure ), provided is non-exceptional and satisfies the weak isolation condition, see Definition 1.22.
So Theorem A item 1 yields the real analyticity of for , where .
The same in the complex case was noted in [PR-L2, Appendix B].
2. As mentioned at the beginning, if has no preperiodic critical point in , or is not exceptional, then (under additional mild assumptions), see [IT] or [Zhang, Theorem B]. It may happen that even if is Collet-Eckmann, in particular for for a real quadratic polynomial with a non-recurrent critical point, see [CR-L].
The property: TCE and all periodic orbits in hyperbolic repelling, is equivalent to which is hyperbolic dimension of and the first zero of , see Subsection 1.4.
In addition, if a generalized multimodal map is TCE with all periodic orbits in hyperbolic repelling and if (a union of closed intervals) then and Theorem A yields immediately the existence of an absolutely continuous invariant probability measure (acip); this is the equilibrium measure whose existence is asserted there. For multimodal maps this was first shown in [R-LS, Corollary 2.19].
If is not TCE or if has a periodic point that is not hyperbolic repelling, then and the existence (or non-existence) of acip is an object of an extensive theory, see e.g. [BRSS].
3. There are several interesting examples of generalized multimodal maps in the literature. In [R-L2] generalized multimodal maps were considered each being the restriction to its domain of a degree 4 polynomial, with being a Cantor set with one turning critical point in . In these examples is Collet-Eckmann, but the expansion rates in UHP and ExpShrink occur different (), what cannot happen in the complex case.
A bit similar example also belonging to was considered in [P-span], for which the periodic specification property, uniform perfectness of (in the plane) and some other standard properties, fail.
Both examples satisfy the assumptions of Theorems A,B,C.
4. It has been proven recently in [P-span] that does not depend on safe, in particular it is constant except in a thin set, as in the rational case, thus answering the question asked after Proposition 1.27 (for , assumed all periodic orbits are hyperbolic repelling and is weakly isolated).
5. Given a generalized multimodal map , one can consider its factor by contracting components of , and their iterated preimages to points, see Appendix A: Remark A.6 and Step 2 in the proof of Lemma A.7. This is piecewise monotone, piecewise continuous map of an interval .
Notice that it is continuous iff for each bounded component of , is one point or there is a component of such that .
Clearly if is a classical multimodal map of an interval into itself and, in the notation in Example 1.9, is its core Julia set, if is topologically transitive and , then the factor defined above is continuous.
We thank Weixiao Shen for pointing out to us subtleties in the construction of conformal measures for interval maps. We thank also Michał Misiurewicz and Peter Raith, see Remark 1.18.
2.1. Basic properties of generalized multimodal pairs
If is a continuous map for a compact metric space and it is topologically transitive, then maps onto and if is infinite, it has no isolated points and is uncountable.
If we assume additionally that and moreover that for being an open neighbourhood of