Invariant measures for interval maps
with critical points and singularities
We prove that, under a mild summability condition on the growth of the derivative on critical orbits any piecewise monotone interval map possibly containing discontinuities and singularities with infinite derivative (cusp map) admits an ergodic invariant probability measures which is absolutely continuous with respect to Lebesgue measure.
1. Introduction and statement of results
The existence of absolutely continuous invariant probability measures (acip’s) for dynamical systems is a problem with a history going back more than 70 years, see for example pioneering papers by Hopf  and Ulam and von Neumann . Notwithstanding an extensive amount of research in this direction in the last two or three decades, the problem is still not completely solved even in the one-dimensional setting which is the focus of this paper. Quite general conditions are known which guarantee the existence of acip’s for uniformly expanding maps in the smooth case or possibly admitting singularities, i.e. discontinuities with possibly unbounded derivatives (see  for additional remarks and references), and for smooth maps with a finite number of critical points (see  for first and strongest results including decay of correlations, and  for the most recent and possibly the most general conditions for the existence of absolutely continuous invariant measures in this setting) and even for smooth maps with a countable number of critical points . We are interested here in a general class of maps which contain critical points and singularities.
A natural family of maps belonging to this class was introduced in [11, 12] and motivated by the study of the return map of the Lorenz equations near classical parameter values, see Figure 1. It is clear from the arguments in these papers, that the presence of both critical points and singularities and their interaction can give rise to significant technical as well as fundamental issues. In particular, as we shall see in the present setting, it is not enough to have just some expansivity conditions in order to obtain the existence of an acip, as expansivity might occur due to the regions of unbounded derivative even when the deeper dynamical structure of the map is very pathological. Moreover, it is possible that the interaction of critical points and singularities could give rise to new phenomena which are still unexplored.
1.1.1. Exponential growth and subexponential recurrence
Some general results for the existence of acip’s and their properties in maps with critical points and singularities were obtained in  under the assumption that Lebesgue almost every point satisfy some exponential derivative growth and subexponential recurrence conditions. These conditions provide an interesting conceptual picture but may be hard to verify in practice. On the other hand, it was proved in   that with positive probability in the parameter space of Lorenz-like families, the orbits of the critical points satisfy such exponential derivative growth and subexponential recurrence conditions. In  it was shown, within a more general setting of maps with multiple critical points and singularities, that these conditions are in fact sufficient to guarantee the existence of an ergodic acip (from which it can in fact be proved that Lebesgue almost every point also satisfies such conditions).
1.1.2. Summability conditions
Our aim in this paper is to obtain the same conclusion but relax as much as possible the conditions on the orbit of the critical points, to include in particular cases in which the derivative growth may be subexponential and/or the recurrence of the critical points exponential. A crucial observation concerning the difference between the smooth case and the case with singularities discussed here is that in the smooth case, for which in particular the derivative is bounded, any condition on the growth of the derivative is also implicitly a condition on the recurrence to the critical set. Indeed sufficiently strong recurrence to the critical set will always kill off any required derivative growth. On the other hand, this is not the case in our setting. Derivative growth may be exponential but arise as a consequence of very strong recurrence to the singularities even if we have at the same time very strong recurrence to the critical set. Strong recurrence to either the singular or the critical set brings its own deep structural problems and can be an intrinsic obstruction to the existence of an acip. We shall formulate below a condition which simultaneously keeps track of the growth of the derivative along critical orbits and of the recurrence of such orbits to the critical set within a single summability condition. This optimizes the result to include a larger class of maps than would be possible by having to independent conditions both of which need to be satisfied. We conjecture that it is not possible to obtain a general result on the existence of acip’s in the presence of both critical points and singularities by assuming only conditions on the derivative growth of critical points.
1.2. Statement of results
Every map admits a finite number of absolutely continuous invariant (physical) probability measures whose basins cover up to a set of measure 0.
1.2.1. Nondegenerate critical/singular set
Let be an interval and be a piecewise map: By this we mean that there exists a finite set such that is and monotone on each connected component of and admits a continuous extension to the boundary so that exists. We denote by the set of all “one-sided critical points” and and define corresponding one-sided neighbourhoods
for each . For simplicity, from now on we use to represent the generic element of and write for . We assume that each has a well-defined (one-sided) critical order in the sense that
for all in some . Note that we say that if the ratio is bounded above and below uniformly in the stated domain. If we say that is a singular point as this implies unbounded derivative near ; if we say that is a critical point as this implies that the derivative tends to near . We shall assume also that for every as this would be a degenerate case which is not hard to deal with but would require having to introduce special notation and special cases, whereas the other cases can all be dealt with in a unified formalism.
For future reference we point out that this immediately implies
for all , where denotes the distance of the point to the critical/singular set (indeed this is the actual property of which we will make use).
1.2.2. Uniform expansion outside the critical neighbourhood
We suppose that is “uniformly expanding away from the critical points”, meaning that the following two conditions are satisfied: there exists a constant , independent of , such that for every point and every integer such that for all and we have
and, for every there exist constants and such that
for every and such that for all .
We remark that both these conditions are quite natural and are often satisfied for smooth maps without discontinuities. More specifically, the first one is satisfied if is , has negative Schwarzian derivative and satisfies the property that the the derivative along all critical orbits tends to infinity, see Theorem 1.3 in . The second is satisfied in even greater generality, namely when is and all periodic points are repelling .
1.2.3. Summability condition along the critical orbit
For each we write
to denote the derivative along the orbit of and the distance of from the critical set respectively. We then assume that for every critical point with we have
This condition plays off the derivative against the recurrence in such a way as to optimize to some extent the class of maps to which it applies. As mentioned in Section 1.1.2 above, we cannot expect to obtain the conclusions of our main theorem in this setting using a condition which only takes into account the growth of the derivative. Notice that condition () is satisfied if the derivative is growing exponentially fast and the recurrence is not faster than exponential in the sense that
Here and in the rest of the paper, the symbol means that the inequality holds up to some multiplicative constant, i.e. there exists a constant independent of or any other constants, such that and .
2. The main technical theorem
Our strategy for the proof is to construct a countable partition of (mod 0) into open intervals , define an inducing time function which is constant on elements of , and let denote the induced map defined by
This induced map is uniformly expanding on each element of but does not have many desirable properties such as uniformly bounded distortion or long branches. Nevertheless it has the two key properties we shall require which are summable inducing times and summable variation. We recall that the variation of a function over a subinterval of is defined by
where the supremum is taken over all and all choices of points . For each we define the function by
Our main technical result in this paper is the following
There exists a countable partition of (mod 0) and an inducing time function , constant on elements of , such that the induced map is uniformly expanding and satisfies the the following properties.
(Summable inducing times)
Theorem 1 implies the Main Theorem by known arguments. Indeed, by a result of Rychlik the summable variation property together with uniform expansion implies that admits a finite number of ergodic absolutely continuous invariant measure whose basins cover up to a set of measure zero [14, 16, 3]. By standard arguments the summable inducing time property implies that these measures can be pulled back to a absolutely continuous invariant probability measure for the original map satisfying the same properties .
The arguments used in [4, 1, 8] also involve the construction of an induced map with summable return times, but in those papers the induced map has some very strong properties such as uniformly bounded distortion and the Gibbs-Markov property (the image of each partition element maps diffeomorphically to the entire domain of definition of the induced map). To achieve these properties a quite complicated construction is required, involving the inductive definition of an infinite number of finer and finer partitions together with a combinatorial and probabilistic argument showing that the procedure eventually converges. Besides the fact that we deal here with a significantly larger class of systems, a major difference is the construction of an induced map satisfying a different set of conditions as formalized in the summable variation property stated in the theorem. These induced maps do not necessarily have bounded distortion and there is no uniform lower bound for the size of the images. For this reason the construction of these induced maps is *much* simpler, and in fact can be fully achieved in less than two pages of text in the following section. The rest of the paper is just devoted to checking the required properties.
2.2. Definition of the induced map
The induced map can in fact be defined in complete generality with essentially no assumptions on the map . We will only require our assumptions to show that this induced map has the desired properties.
For a point in the neighbourhood of one of the critical points , we let
For an arbitrary interval we let denote the length of and denote it’s distance to the critical set , i.e. the minimum distance of all point in to . For each critical point with , and every integer we let
It follows immediately from the summability condition () that
Given ,we define the binding period of a point as follows. If we just define the binding period as . Otherwise we define the binding period as the smallest such that
For each and , define to be the interval of points such that . Observe that from the definition of binding it follows immediately that
monotonically when . Notice also that the interval may be empty and indeed that is the case, for instance, for all .
Using the monotonicity of we can fix at this moment and for the rest of the paper sufficiently small so that
the critical neighbourhood of size of all critical/singular points are disjoint and the images of the critical/singular neighbourhoods are also disjoint from the critical/singular neighbourhoods themselves;
for all ;
for all . The symbol here means that must be larger than some constant factor of for a constant which depends only on the map itself and which is determined in the course of the proof but which could in principle we specified explicitly at this point.
We now fix an integer sufficiently large so that
Notice that the constants and come from the expansion outside the critical neighbourhoods given in Section 1.2.2. The choice of is motivated by the fact that any finite piece of orbit longer than iterations staying outside a neighbourhood of the critical points has an accumulated derivative of at least 2.
2.2.5. The inducing time
so that denotes the set of points of which remain outside for the first iterations, and denotes those which enter at some time before . For let
so that is the first time the orbit of enters and denotes the binding period corresponding to the point . Then we define the inducing time by
2.2.6. The induced map
We define the induced map as
and let denote the partition of into the maximal intervals restricted to which the induced map is smooth, and write and . This completes the definitions of the induced map.
3. Variation, Distortion and Expansion
In this section we prove a general formula relating the variation, the distortion and the expansion. First of all we define the notion of generalized distortion. This is a very natural notion which is no more difficult to compute than standard distortion and which appears in variation calculations. Strangely it does not seem to us to have been defined before in the literature. For any interval and integer we let for and define the (generalized) distortion
We remark here that we are taking the supremum over all choices of sequences . If these sequences are chosen so that for some then we recover the more standard notion of distortion. In particular, by choosing the sequence arbitrary and the sequence as the actual orbit of a point, we can compare the two products and,in this case, the definition given above of generalized distortion immediately implies
for any . For future reference we remark also that by the mean value theorem, there exists some such that
Therefore we have
We are now ready to state the main result of this section.
For any interval and integer such that is a diffeomorphism, we have
Before starting the proof we recall a few elementary properties of functions with bounded variation which will be used here and later on. Proofs can be found, for instance, in  or . For any interval , , and ,
if is a homeomorphism;
if is of class then .
for any interval , any bounded variation function , and any probability on ,
In particular, this holds when normalized Lebesgue measure on .
We start by writing
Thus, from (V3) we have
Since the supremum of the product is clearly less than or equal to the product of the supremums this gives
Thus, multiplying and dividing through by both the first and last term of the right hand side of this expression, we get
We have used here the simplified notation to denote . Using this bound recursively we get
Continuing in this way and and then using (V4) we arrive at
From the definition of generalized distortion, in particular (7), this gives
Finally from (V5) and (2) we get
4.1. Distortion during binding periods
For any , , the critical point closest to , , and any we have
In particular there exists independent of such that for all we have
and for all we have
The definition of binding period is designed to guarantee that the length of the interval is small compared to its distance to the critical set. Indeed, from the definition we have and therefore, for every we have
In particular this also implies, from the order of the critical points, that and and therefore
The right hand side is uniformly bounded by the summability of the ’s. Indeed, taking logs and using the inequality for all we get This proves the uniform bound on the distortion . The fact that then follows directly from the definition of and the fact that it is uniformly bounded. Finally notice that and therefore the required bound follows from (12). ∎
4.2. The binding period partition
The partition is defined quite abstractly and we do not have direct information about the sizes of the partition elements and in particular the relation between their sizes and their distances to the critical set. However, using the distortion bounds obtained above, we can prove the following
Let with and in the neighbourhood of a critical point with order . Then
In particular, letting denote the order of the critical/singular point closest to we have
We remark that the distortion not uniformly bounded in implying that the induced map does not have uniformly bounded distortion. Notice also that for some values of it may happen that ; in this case the corresponding interval would necessarily be empty, i.e. there is no with binding period .
From Lemma 4.1 and the definition of binding period we have, for any ,
By taking a sufficiently small we can assume that is sufficiently large so that and therefore, from the definition of the sequence we get
Thus, substituting into the expressions above gives
This gives the first set of inequalities. As a consequence we immediately get
This gives the first inequality in (14). To get the second inequality we simply use the fact that . To get the last inequality we simply integrate over the interval to get
and then argue as above. ∎
4.3. Expansion during binding periods
For all , and , we have
In particular we can choose small enough so that
5.1. Expansion of the induced map
For every we have
5.2. Distortion of the induced map
We now study the distortion of the induced map on each of its branches.
There exists a constant such that
for all (in which case ) and (in which case ) respectively, where is the order of the critical point associated to . Also, we have
respectively for and .
For we have standard distortion estimates for uniformly expanding maps which give a uniform distortion bound depending on the size of . For on the other hand we write
The first term consists of iterates for which lies always outside and therefore is bounded above by the same constant as above. The second and third term have already been estimated above in Lemmas 4.1 and 4.2. Combining these estimates we complete the first set of estimates.
For , using the uniform expansion outside we have and therefore
For we again split the sum into three parts corresponding to the initial iterates outside , the first iterate in , and the following binding period. The fist part of the sum is bounded by the same constant as above. The second and third have already been estimated above. Thus, from Lemmas 4.1 and 4.2 and in particular (15) we get the statement.
We are now ready to prove the summable variation and the summable inducing time properties.
6.1. Summable variation
From the definition of that we have
For the supremum we have, from Lemma 5.1,