Ergodic Maximizing Measures of Non-Generic, Yet Dense Continuous Functions
Ergodic optimization aims to single out dynamically invariant Borel probability measures which maximize the integral of a given “performance” function. For a continuous self-map of a compact metric space and a dense set of continuous performance functions, we show that the existence of uncountably many ergodic maximizing measures. We also show that, for a topologically mixing subshift of finite type and a dense set of continuous functions there exist uncountably many ergodic maximizing measures which are fully supported and have positive entropy.
2010 Mathematics Subject Classification:28D05, 37D20, 37D35
Ergodic optimization aims to single out dynamically invariant Borel probability measures which maximize the integral of a given “performance” function. It originates in variational methods in mechanical systems [Man, Mat] and several applications can be considered, for instance, in the thermodynamic formalism [BLL], in chaos control [HO, YH]. (See bibliographical notes in [J] for more details).
Consider a continuous self-map of a compact metric space and a continuous performance function . A -invariant Borel probability measure is called a -maximizing measure if
where denotes the space of -invariant Borel probability measures endowed with the weak*-topology. We investigate properties of -maximizing measures by considering the set of all -maximizing measures. A performance function is uniquely maximized if is a singleton. Jenkinson shows that a generic continuous function is uniquely maximized [J]. If has the specification property, the unique maximizing measure of a generic continuous function is fully supported and has zero entropy [BJ, Br, J, Mo]. However, it is difficult to tell whether or not a given performance function is generic. Indeed, no concrete example of a continuous function is known which is uniquely maximized by a fully supported measure [J, Problems 3.9 and 4.3]. Hence it is natural to investigate properties which are non-generic, yet hold for a reasonably large set of functions.
In this paper we pay attention to non-generic properties of maximizing measures. Denote by the space of continuous functions endowed with the supremum norm and by the set of ergodic elements of . Recall that is the set of extrema points of . Let us say that is arcwise-connected if for every and there exists a homeomorphism on its image such that and . Note that if is arcwise-connected then is not a singleton. We prove that the set of uncountably maximized continuous functions is dense in , provided is arcwise-connected.
Let be a continuous self-map of a compact metric space . Suppose is arcwise-connected. There exists a dense subset of such that for every in the set contains uncountably many ergodic elements.
Examples to which Theorem A applies include topologically mixing subshifts of finite type and Axiom A diffeomorphisms. The space of invariant measures of these systems is the Poulsen simplex [Si1]: the infinite simplex for which the set of its extremal points is dense. The denseness of the set of ergodic measures implies its arcwies-connectedness because the Poulsen simplex and the set of its extremal points are homeomorphic to the Hilbert cube and its interior respectively [GK]. Sigmund shows that the specification, which the above examples actually have, implies the denseness of the set of ergodic elements [Si2]. Several extensions of Sigmund’s result have been considered under some generalized versions of the specification (see [GK]). In one-dimensional case, Blokh shows that continuous topologically mixing interval maps have the specification [Bl] and a discontinuous version is studied in [Bu].
The arcwise-connectedness of the set of ergodic measures is strictly weaker than the denseness of it. For example the set of ergodic measures of the Dyck shift [Kr] is not dense but arcwise-connected. The connectedness of the set of ergodic measures for some partially hyperbolic systems is studied in [GP]. On the other hand, there do exist systems for which is not arcwise-connected: Cortez and Rivera-Letelier show that for the restriction of some logistic maps to the omega limit set of the critical points, the sets of ergodic measures become totally disconnected [CR].
An idea of our proof of Theorem A is to perturb a given continuous function to create another so that the function defined on an arc of ergodic measures has a “flat” part (see FIGURE 1). The Bishop Phelps theorem allows us to construct such a perturbation. In order to use the Bishop Phelps theorem, we use the fact that maximizing measures are characterized as “tangent measures” to the convex functional
It is worthwhile to remark that our perturbation does not work in the Lipschitz topoligy. In the course of the proof of Theorem A we show the following statement.
Let and . For any neighborhood of and any open neighborhood of there exists in such that contains uncountably many ergodic elements.
On the other hand, in the space of Lipschitz continuous functions a phenomenon called a “lock up on periodic orbits” occurs: for a Lipschitz continuous functions which is uniquely maximized by a periodic measure one cannot realize a perturbation breaking the uniqueness of maximizing measure [YH] (See also [BLL]).
For the subshift of finite type, one can choose the arc of ergodic measures used in the proof of Theorem A from fully supported measures with positive entropy. Hence, slightly modifying the proof of Theorem A we obtain the next theorem.
Let be a topologically mixing subshift of finite type. There exists a dense subset of such that for every in the set contains uncountably many ergodic elements which are fully supported and positive entropy.
2.1. Bishop Phelps Theorem
First we see the Bishop Phelps theorem, which is concerned with a convex functional on a Banach space. We begin with definitions of basic notions.
A functional on a Banach space is convex if
for all and .
Let be a convex and continuous functional on a Banach space . A bounded linear functional is tangent to at if
for all .
A bounded linear functional is bounded by if for all .
For a bounded linear functional on a Banach space define
This becomes a norm of the set of all bounded linear functionals on . Note that becomes a Banach space with this norm. The Bishop Phelps theorem states that -bounded functionals can be approximated by -tangent ones with respect to this norm.
[I, Theorem V.1.1.] Let be a convex and continuous functional on a Banach space . For every bounded linear functional bounded by , and , there exist a bounded linear functional and such that is tangent to at and
2.2. The Space of Borel Probability Measures
Denote by the set of all Borel probability measures on a compact metric space . In our setting, is compact and metrizable in the weak*-topology. Then we will also consider in the following section. Denote by the Banach space of all continuous functions on with the supremum norm. By the Riesz representation theorem, the set
can be identified with . With this identification, the norm and the notion of being tangent and being bounded in Definition 1 carry over to elements of . For we denote by the integral of a continuous function by .
2.3. Ergodic Decomposition
For a -invariant measure there exists a unique Borel probability measure on such that and
for all . We call the ergodic decomposition of . Note that is a nonempty compact convex set and coincides with the set of its extremal points. Since the ergodic decomposition of a -invariant measure is unique, is a Choquet simplex. From the Theory of a Choquet simplex, for the ergodic decompositions of , we have . See [R, Appendix A.5] and the references therein.
3. Proofs of the theorems
3.1. On the proof of Theorem A
Define a functional by
Note that is continuous and convex. Maximizing measures are characterized by tangency to .
[Br, Lemma 2.3] Let be a continuous self-map of a compact metric space and . Then is tangent to at if and only if .
First we consider the ergodic decomposition of a -maximizing measure. The following proposition states that invariant measures in the support of the ergodic decomposition of a -maximizing measure are also -maximizing. The support of an ergodic decomposition is defined by where the intersection is taken over all closed subsets of with . Note that , since has a countable basis.
Let be a continuous self-map of a compact metric space and . Let and let be the ergodic decomposition of . Then is contained in .
Let . Suppose . Then
This is a contradiction and then we have . Since and is closed, we have . ∎
Second we construct a non-atomic Borel probability measure on supported in for a given . The point of the construction is this gives positive weight to the set of ergodic measures for which the integrals of are -close to the maximum value . The arcwise-connectedness of is essential for the following construction.
Let be a continuous self-map of a compact metric space . Suppose is arcwise-connected. Then for every there is a non-atomic Borel probability measure on such that and for all ,
Pick and let be a -maximizing measure. Pick . By the assumption, there exists an arc from to . Let denote the Lebesgue measure on and . Since is a homeomorphism, the inverse image of a point in is a singleton. Hence is non-atomic. Since is continuous, the set has nonempty interior. Since the image of the set by is contained in , is supported in and satisfies the desired inequality. ∎
We now prove Theorem A.
Proof of Theorem A.
Note that every Borel probability measure is bounded by . Pick and . Let be a non-atomic Borel probability measure for which the conclusion of Proposition 5 holds with . Put and
Denote by the conditinal measure of on , namely
for every Borel subsets of . Since by Proposition 5, is well-defined. Note that is also supported in .
Let . By Theorem 2 applied to , and , there exist and such that is tangent to at , and
Then is -close to and by Proposition 3 is a -maximizing measure.
Next we show the existence of uncountably many ergodic -maximizing measures. Let be the ergodic decomposition of and we have
Let . Since is a Borel probability measure and is a closed set, there is an open set such that and . Since is a metric space, there is a continuous function which vanishes on and is identically on . Hence we have
The inequality in (1) implies
for all with . Hence we have
Since is non-atomic and supported in , contains uncountably many ergodic elements. By Proposition 4 we have , and the proof is complete. ∎
3.2. On the proof of Theorem B
The following result by Sigmund is essential for our proof of Theorem B. For a continuous function which satisfies and is called a path from to .
[Si3] Let be a topologically mixing subshift of finite type. Then for every there exists a path from to with the following properties: (i) for every measure , is a countable set; (ii) every measure except for countably many ones is fully supported and has positive entropy.
Proof of Theorem B.
Pick and . We obtain a non-atomic Borel probability measure on by modifying the proof of Proposition 5. Let be a -maximizing measure and pick . Let be a path from to for which the conclusion of Theorem 6 holds. Since the inverse image of any point is countable, becomes a non-atomic Borel probability measure supported in . Then for the inequality in Proposition 5 holds with and .
Following the proof of Theorem A, we define to be the restriction of to the set and obtain and a Borel probability measure such that and .
By Theorem 6, contains uncountably many ergodic elements which are fully supported and have positive entropy. The definition of implies
and still contains uncountably many ergodic elements which are fully supported and have positive entropy. By (2) in the proof of Theorem A we have
Since is non-atomic and supported in , this implies contains uncountably many ergodic elements which are fully supported and have positive entropy. By Proposition 4 we have
and the proof is complete. ∎
This research is partially supported by the JSPS Core-to-Core Program “Foundation of a Global Research Cooperation Center in Mathematics focused on Number Theory and Geometry”.
- A. Baraviera, R. Leplaideur and A. Lopes, Ergodic optimization, zero temperature limits and the max-plus algebra, IMPA Mathematical Publications 29th Brazilian Mathematics Colloquium Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, 2013.
- A. Blokh, Decomposition of dynamical systems on an interval, Russian Mathematical Surveys, 38, 1983, pp.133–134.
- T. Bousch and O. Jenkinson, Cohomology classes of dynamically non-negative functions, Inventiones Mathematicae, 148, 2002, pp.207–217.
- J. Brémont, Entropy and maximizing measures of generic continuous functions, Comptes Rendus Mathématique. Académie des Sciences. Paris, 346, 2008, pp.199–201.
- J. Buzzi, Specification on the interval, Transactions of the American Mathematical Society, 349, 1997, pp.2737–2754.
- G. Contreras, Ground states are generically a periodic orbit, Inventiones Mathematicae, 205, 2016, pp.383–412.
- M. Cortez and J. Rivera-Letelier, Invariant measures of minimal post-critical sets of logistic maps, Israel Journal of Mathematics, 176, 2010, pp.157–193.
- K. Gelfert and D. Kwietniak, On density of ergodic measures and generic points, Ergodic Theory and Dynamical Systems, 2016.
- A. Gorodetski and Y. Pesin, Path connectedness and entropy density of the space of ergodic hyperbolic measures, arXiv:1505.02216, 2015.
- B. Hunt and E. Ott, Optimal periodic orbits of chaotic systems, Physical review letters, 1996, 76, 2254.
- R. Israel, Convexity in the theory of lattice gases, Princeton Series in Physics, Princeton University Press, Princeton, 1979.
- O. Jenkinson, Ergodic optimization, Discrete and Continuous Dynamical Systems. Series A, 15, 2006, pp.197–224.
- W. Krieger, On the uniqueness of the equilibrium state, Mathematical Systems Theory. An International Journal on Mathematical Computing Theory, 8, 1974/75, pp.97–104.
- R. Mañé, Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity, 9, 1996, pp.273–310.
- J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems, Mathematische Zeitschrift, 207, 1991, pp.169–207.
- I. Morris, Ergodic optimization for generic continuous functions, Discrete and Continuous Dynamical Systems. Series A, 27, 2010, pp.383–388.
- F. Przytycki and M. Urbański, Conformal fractals: ergodic theory methods, London Mathematical Society Lecture Note Series, 371, Cambridge University Press, Cambridge, 2010.
- D. Ruelle, Thermodynamic formalism, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004.
- K. Sigmund, Generic properties of invariant measures for Axiom diffeomorphisms, Inventiones Mathematicae, 11, 1970, pp.99–109.
- K. Sigmund, On dynamical systems with the specification property, Transactions of the American Mathematical Society, 190, 1974, pp. 285–299.
- K. Sigmund, On the connectedness of ergodic systems, Manuscripta Mathematica, 22, 1977, pp.27–32.
- G. Yuan and B. Hunt, Optimal orbits of hyperbolic systems, Nonlinearity, 12, 1999, pp.1207–1224.