Removal of phase transition of the Chebyshev quadratic
and thermodynamics of Hénon-like maps
near the first bifurcation
We treat a problem at the interface of dynamical systems and equilibrium statistical physics. It is well-known that the geometric pressure function
of the Chebyshev quadratic map is not differentiable at . We show that this phase transition can be “removed”, by an arbitrarily small singular perturbation of the map into Hénon-like diffeomorphisms. A proof of this result relies on an elaboration of the well-known inducing techniques adapted to Hénon-like dynamics near the first bifurcation.
2010 Mathematics Subject Classification:37D25, 37D35, 37G25, 82C26
The thermodynamic formalism, i.e., the formalism of equilibrium statistical physics developed by G. W. Gibbs and others, has been successfully brought into the ergodic theory of chaotic dynamical systems (see e.g., [3, 16] and the references therein). In the classical setting, it deals with a continuous map of a compact metric space and a continuous function on , and looks for equilibrium measures which maximize (the minus of) the free energy among all -invariant Borel probability measures on . A relevant problem is to study the regularity of the pressure function , where .
The existence and uniqueness of equilibrium measures depends upon details of the system and the potential. For transitive uniformly hyperbolic systems and Hölder continuous potentials, the existence and uniqueness of equilibrium measures as well as the analyticity of the pressure function has been established in the pioneering works of Bowen, Ruelle and Sinai [3, 16, 19]. The latter property is interpreted as the lack of phase transition.
One important problem in dynamics is to understand structurally unstable, or nonhyperbolic systems . The main problem which equilibrium statistical physics tries to clarify is that of phase transitions . Hence, it is natural to study how phase transitions are affected by small perturbations of dynamics.
A natural candidate for a potential is the so-called geometric potential , where denotes the unstable direction at which reflect the chaotic behavior of . For nonhyperbolic systems, is often merely measurable, and may even be unbounded as in the case of one-dimensional maps with critical points. These defects sometimes lead to the occurrence of phase transitions, e.g., the loss of analyticity or differentiability of the pressure function. Typically, at phase transitions, there exist multiple equilibrium measures.
As an emblematic example, consider the family of quadratic maps and the associated family of geometric pressure functions given by
Here, denotes the Kolmogorov-Sinai entropy of and the supremum is taken over all -invariant Borel probability measures.
For , the Julia set does not contain the critical point , and so the dynamics is uniformly hyperbolic and structurally stable. According to the classical theory, for any there exists a unique equilibrium measure for the potential , and the geometric pressure function is real analytic. At the first bifurcation parameter the Julia set contains the critical point, and so the dynamics is nonhyperbolic and structurally unstable. The Lyapunov exponent of any ergodic measure is either or , and it is only for the Dirac measure, denoted by , at the orientation-preserving fixed point. Equilibrium measures for the potential are: (i) if ; (ii) and if ; (iii) if , where denotes the absolutely continuous invariant probability measure. Correspondingly, the pressure function is not real analytic at :
We say displays the freezing phase transition in negative spectrum, to be defined below (see the paragraph just before the Main Theorem).
This phase transition is due to the fact that the measure is anomalous: it has the maximal Lyapunov exponent, and this value is isolated in the set of Lyapunov exponents of all ergodic measures. For all the Dirac measure at the orientation preserving fixed point continues to be anomalous, and therefore all the quadratic maps continue to display the freezing phase transition [8, Proposition 4]. The freezing phase transition in negative spectrum is often caused by anomalous periodic points. For example, see  for results on certain two-dimensional real polynomial endomorphism, and  for a complete characterization on rational maps of degree on the Riemannian sphere.
An elementary observation is that
any nonhyperbolic one-dimensional map sufficiently close to in the -topology
displays the freezing phase transition in negative spectrum.
This raises the following question: is it possible to remove the phase transition of
by an arbitrarily small singular perturbation to higher dimensional nonhyperbolic maps?
More precisely we ask:
(Removability problem) Is it possible to “approximate” by higher dimensional nonhyperbolic maps
which do not display the freezing phase transition in negative spectrum?
The aim of this paper is to show that the phase transition of can be removed, by an arbitrarily small singular perturbation along the first bifurcation curve of a family of Hénon-like diffeomorphisms
where is near , is bounded continuous in and in . The parameter controls the nonlinearity, and the controls the dissipation of the map. Note that, with the family degenerates into the family of quadratic maps.
We proceed to recall some known facts on the first bifurcation of the family of Hénon-like diffeomorphisms. If there is no fear of confusion, we suppress from notation and write for , and so on. For near let , denote the fixed saddles of near and respectively. The stable and unstable manifolds of are respectively defined as follows:
if , then the non wandering set is a uniformly hyperbolic horseshoe;
if , then there is a single orbit of homoclinic or heteroclinic tangency involving (one of) the two fixed saddles (see FIGURE 1). In the case (orientation preserving), meets tangentially. In the case (orientation reversing), meets tangentially. The tangency is quadratic, and the one-parameter family unfolds the tangency at generically. An incredibly rich array of dynamical complexities is unleashed in the unfolding of this tangency (see e.g.,  and the references therein);
The curve is a nonhyperbolic path to the quadratic map , consisting of parameters corresponding to nonhyperbolic dynamics. The main theorem claims that does not display the freezing phase transition in negative spectrum.
To give a precise statement of result we need a preliminary discussion. We first make explicit the range of the parameter to consider. Assume . Let denote the compact curve in containing such that has two connected components of length . Let denote the isometric embedding such that and . Let
Note that , , as , and that at , is tangent to quadratically. Since the family unfolds the tangency at generically, . In this paper we assume .
Let denote the non wandering set of , which is a compact -invariant set. For nonhyperbolic dynamics beyond the parameter , the notion of “unstable direction” is not clear. In the next paragraph, we circumvent this point with the Pesin theory (See e.g., ), by introducing a Borel set on which an unstable direction makes sense.
Given , for each integer define to be the set of points for which there is a one-dimensional subspace of such that for every integers , and for all vectors ,
Since expands area, the subspace with this property is unique when it exists, and characterized by the following backward contraction property
Here, denotes the norm induced from the Euclidean metric on . Note that is a closed set, and is continuous. Moreover, if then . Therefore, the Borel set
is -invariant: . Then the Borel set
is -invariant as well, and the map is Borel measurable with the invariance property . The one-parameter family of potentials we are concerned with is
where . Since is compact and is a diffeomorphism, is bounded from above and bounded away from zero. We shall only take into consideration measures which give full weight to .
The chaotic behavior of is produced by the non-uniform expansion along the unstable direction , and thus a good deal of information will be obtained by studying the associated geometric pressure function defined by
and denotes the entropy of , and
which we call an unstable Lyapunov exponent of . Let us call any measure in which attains the supremum in an equilibrium measure for the potential .
We suggest the reader to compare (1) and (3). One important difference is that the function in (1) is unbounded, while the function in (3) is bounded. Another important difference is that the class of measures taken into consideration is reduced in (3).
It is natural to ask in which case . This is the case for because from the result in . In fact, still holds for “most” parameters immediately right after the first bifurcation at . See Sect.4.2 for more details.
The potential and the associated pressure function deserve to be called geometric, primarily because Bowen’s formula holds at [18, Theorem B]: the equation has the unique solution which coincides with the (unstable) Hausdorff dimension of . We expect that the same formula holds for the above “most” parameters.
We are in position to state our main result. Let
and define freezing points , by
This denomination is because equilibrium measures do not change any more for or . Indeed it is elementary to show the following:
if , then any equilibrium measure for (if it exists) has positive entropy;
if , then . If , then .
Let us say that displays the freezing phase transition in negative (resp. positive) spectrum if (resp. ) is finite.
Let be a family of Hénon-like diffeomorphisms. If is sufficiently small and , then does not display the freezing phase transition in negative spectrum. If , then as .
The main theorem states that the graph of the pressure function does not touch the line . At we have more information: this line is the asymptote of the graph of as (see FIGURE 4).
The main theorem reveals a difference between the bifurcation structure of quadratic maps and that of Hénon-like maps from the thermodynamic point of view. As mentioned earlier, the quadratic maps display the freezing phase transition in negative spectrum for all parameters beyond the bifurcation, while this is not the case for Hénon-like maps.
The freezing phase transition for negative spectrum does occur for some parameters . It is well-known that there exists a parameter set of positive Lebesgue measure corresponding to non-uniformly hyperbolic strange attractors [2, 13, 24]. For these parameters, the non-wandering set is the disjoint union of the strange attractor and the fixed saddle near [4, 5]. For these parameters it is possible to show that the Dirac measure at the saddle is anomalous.
Regarding freezing phase transitions in positive spectrum of Hénon-like maps, the known result is very much limited. Let denote the Dirac measure at . It was proved in [23, Proposition 3.5(b)] that if and , then does not display the freezing phase transition in positive spectrum. However, since and as , it is not easy to prove or disprove this equality.
For a proof of the main theorem we first show that is the unique measure which maximizes the unstable Lyapunov exponent (see Lemma 2.4). Then it suffices to show that for any there exists a measure such that To see the subtlety of showing this, note that from the variational principle must satisfy
where denotes the topological entropy of . As becomes large, the unstable Lyapunov exponent becomes more important and we must have as . A naive application of the Poincaré-Birkhoff-Smale theorem  to a transverse homoclinic point of indeed yields a measure whose unstable Lyapunov exponent is approximately that of , but it is not clear if the entropy is sufficiently large for the first inequality in (4) to hold.
Our approach is based on the well-known inducing techniques adapted to the Hénon-like maps, inspired by Makarov Smirnov  (see also Leplaideur ). The idea is to carefully choose for each a hyperbolic subset of such that the first return map to it is topologically conjugate to the full shift on a finite number of symbols. We then spread out the maximal entropy measure of the first return map to produce a measure with the desired properties. As becomes large, more symbols are needed in order to fulfill the first inequality in (4).
The hyperbolic set is chosen in such a way that any orbit contained in it spends a very large proportion of time near the saddle , during which the unstable directions are roughly parallel to . More precisely, for any point with the first return time to , the fraction
is nearly . A standard bounded distortion argument then allows us to copy the unstable Lyapunov exponent of . Note that, if the unstable direction is not continuous (which is indeed the case at  and considered to be the case for most ), then the closeness of base points does not guarantee the closeness of the corresponding unstable directions .
In order to let points stay near the saddle for a very long period of time, one must allow them to enter deeply into the critical zone. As a price to pay, the directions of along the orbits get switched due to the folding behavior near the critical zone. In order to restore the horizontality of the direction and establish the closeness to , we develop the binding argument relative to dynamically critical points, inspired by Benedicks Carleson . The point is that one can choose the hyperbolic set so that the effect of the folding is not significant, and the restoration can be done in a uniformly bounded time. This argument works at the first bifurcation parameter , and even for all parameters in because only those parts in the phase space not being destroyed by the homoclinic bifurcation are involved.
The rest of this paper consists of three sections. In Sect.2 we introduce the key concept of critical points, and develop estimates related to them. In Sect.3 we use the results in Sect.2 to construct the above-mentioned hyperbolic set. In Sect.4 we finish the proof of the main theorem and provide more details, on the abundance of parameters satisfying .
2. Local analysis near critical orbits
For the rest of this paper, we assume and . In this section we develop a local analysis near the orbits of critical points. The main result is Proposition 2.7 which controls the norms of the derivatives in the unstable direction, along orbits which pass through critical points.
For the rest of this paper we are concerned with the following positive small constants: , , chosen in this order, the purposes of which are as follows:
is used to exclusively in the proof of Proposition 2.7;
determines the size of a critical region (See Sect.2.5);
determines the magnitude of the reminder term in (1).
We shall write with or without indices to denote any constant which is independent of , , . For we write if both and are bounded from above by constants independent of , , . If and the constants can be made arbitrarily close to by appropriately choosing , , , then we write .
For a nonzero tangent vector at a point , define if , and if . Similarly, for the one-dimensional subspace of spanned by , define . Given a curve in , the length is denoted by . The tangent space of at is denoted by . The Euclidean distance between two points of is denoted by . The angle between two tangent vectors , is denoted by . The interior of a subset of is denoted by .
2.1. The non wandering set
Recall that the map has exactly two fixed points, which are saddles: is the one near and is the other one near . The orbit of tangency at the first bifurcation parameter intersects a small neighborhood of the origin exactly at one point, denoted by . If then . If then (See FIGURE 1).
By a rectangle we mean any compact domain bordered by two compact curves in and two in . By an unstable side of a rectangle we mean any of the two boundary curves in . A stable side is defined similarly.
In the case (resp. ) let denote the rectangle which is bordered by two compact curves in (resp. ) and two in , and contains . The rectangle with these properties is unique, and is located near the segment . One of the stable sides of contains , which is denoted by . The other stable side of is denoted by . We have . At , one of the unstable sides of contains the point of tangency near (See FIGURE 1).
2.2. Non critical behavior
and call it a critical region.
By a -curve we mean a compact, nearly horizontal curve in such that the slopes of tangent vectors to it are and the curvature is everywhere .
Let be a -curve in . Then is a -curve.
Follows from Lemma 2.2 and the lemma below. ∎
Let and be an integer such that . Then for any nonzero vector at with ,
If moreover , then
Follows from the fact that outside of and that is very small. ∎
([20, Lemma 2.3]) Let be a curve in and . For each integer let denote the curvature of at . Then for any nonzero vector tangent to at ,
2.3. Lyapunov maximizing measure
Recall that is the set of -invariant Borel probability measures which gives total mass to the set . The next lemma states that is the unique measure which maximizes the unstable Lyapunov exponent among measures in .
For any , .
From the linearity of the unstable Lyapunov exponent as a function of measures, it suffices to consider the case where is ergodic. Let
Reducing if necessary, one can show that holds for any ergodic with . In the case we have . From the Ergodic Theorem, it is possible to take a point such that
Define a sequence of nonnegative integers inductively as follows. Start with . Let and be such that . If , then define
If , then define
Since , .
The form of our map (1) gives for every . This implies
If then . Hence
2.4. -closeness due to disjointness
Corollary 2.6 below states that the pointwise convergence of pairwise disjoint -curves implies the -convergence. This fact was already used in the precious works for Hénon-like maps, e.g., [17, 18]. We include precise statements and proofs for the reader’s convenience.
Let and let , be two disjoint -curves parametrized by arc length such that:
, are defined for ;
Then the following holds:
for all .
Write . By the mean value theorem, for any in between and there exists in between and such that , where the dot denotes the -derivative. Integrating this equality gives
We argue by contradiction assuming . The assumption , (ii) and give
A comparison of (6) with (7) shows that the sign of coincides with that of . The same argument shows that the sign of coincides with that of . From the intermediate value theorem it follows that for some , namely intersects , a contradiction. Hence and (a) holds. (b) follows from (6), (ii), (a) and . ∎
Let be a sequence of pairwise disjoint -curves which as a sequence of functions converges pointwise to a function as . Then the graph of is a -curve and the slopes of its tangent directions are everywhere .
2.5. Critical points
From the hyperbolicity of the fixed saddle , there exist mutually disjoint connected open sets , independent of such that , , and a foliation of by one-dimensional vertical leaves such that:
, the leaf of containing , contains ;
if , then ;
let denote the unit vector in whose second component is positive. Then is , and ;
if , then .
Let be a -curve in . We say is a critical point on if and is tangent to . If is a critical point on a -curve , then we say admits . For simplicity, we sometimes refer to as a critical point without referring to .
Let be a critical point. Note that , and the forward orbit of spends a long time in . Hence it inherits the exponential growth of derivatives near the fixed saddle . For an integer write , and define
More precisely, from the bounded distortion near the fixed saddle ,
2.6. Binding to critical points
In order to deal with the effect of returns to , we now establish a binding argument in the spirit of Benedicks Carleson  which allows one to bind generic orbits which fall inside to suitable critical points, to let it copy the exponential growth along the piece of the critical orbit.
Let be a critical point and let . We define a bound period in the following manner. Consider the leaf of the stable foliation through . This leaf is expressed as a graph of a function: there exists an open interval containing and independent of , and a function on such that
Choose a small number such that any closed ball of radius about a point in is contained in . For each integer define
Write . If , then define