THERMODYNAMIC FORMALISM METHODS IN THE THEORY OF ITERATION OF MAPPINGS IN DIMENSION ONE, REAL AND COMPLEX

Received: 10.09.2020.Accepted: 08.11.2020. (2020) Mathematics Subject Classification: 37D35, 37E05, 37F10, 37F35, 31A20.


Introduction
In equilibrium statistical physics, originated by Boltzmann (1877) and Gibbs (1902), the Ising model of ferromagnetism is considered. Let Ω be the configuration space of functions Z n → A on the integer lattice Z n with interacting values in A over its sites, e.g. "spin" values + or -, assigning the resulting energy (potential) for each configuration. One considers probability distributions on Ω, invariant under translation, called equilibrium states depending of this potential functions on Ω and on "temperature".
In 1960/70 Yakov Sinai, David Ruelle and Rufus Bowen applied this theory to investigate invariant sets in dynamics distributing measures on them, see [29], [28] and [1].
Let us start with the following important

Introduction: corresponding dynamics notions
Let f : X → X be a continuous map for a compact metric space (X, ρ) and φ : X → R be a continuous function (potential).  supremum over all Y ⊂ X such that for distinct x, y ∈ Y , ρ n (x, y) := max{ρ(f i (x), f i (y)), 0 ≤ i ≤ n} ≥ ε. For this and related theory see e.g. [30] or [25]. In view of Theorem 1.4 we can omit subscripts and write P (f, φ).
Call f : X → X distance expanding if there exist λ > 1, C > 0 such that for all x, y ∈ X, sufficiently close to each other, then ρ(f n (x), f n (y)) ≥ Cλ n ρ(x, y) for all n ∈ N.
Sometimes we use the word hyperbolic.
Lemma 1.1 becomes in the infinite (dynamical, expanding) setting: Theorem 1.5 (Gibbs measure -uniform case). Let f : X → X be a distance expanding, topologically transitive continuous open map of a compact metric space X and φ : X → R be a Hölder continuous potential. Then, there exists exactly one µ φ ∈ M(f, X), called a Gibbs measure, such that for constants C, r 0 > 0, all x ∈ X and all n ∈ N called the Gibbs property, where f −n x is the local branch of f −n mapping f n (x) to x and S n φ(x) := n−1 j=0 φ(f j (x)).
• µ φ is the unique equilibrium state for φ, and is ergodic. It is equivalent to the unique exp −(φ − P )-conformal measure m φ , that is an f -quasi-invariant measure with Jacobian exp −(φ − P ) for a constant P . • P = P (f, φ) := lim n→∞ 1 n log x∈f −n (x 0 ) exp S n φ(x). This normalizing limit exists and is equal P(f, φ) for every x ∈ X.

Introduction to dimension one
Thermodynamic formalism is useful for studying properties of the underlying space X. In dimension one, for f real of class C 1+ε or f holomorphic (conformal) for an expanding repeller X, considering φ = φ t := −t log |f | for t ∈ R, the Gibbs property gives, The latter follows from a comparison of the diameter with the inverse of the absolute value of the derivative of f n at x, due to bounded distortion. All this is not literally true if f has critical points in X, i.e. points where the derivative f is zero. Then the "escalator" f n to large scale deforms shapes when passing close to critical points. Also φ is not Hölder at these points. Some correctness of Theorem 1.5 depends then on recurrence of critical points and on t where 1/t mimics temperature for t > 0.
When t = t 0 is a zero of the function t → P (φ t ), this gives (for expanding (f, X)) for all small balls B, hence HD(X) = t 0 . Moreover, the Hausdorff measure H t 0 of X in this dimension is finite and nonzero. The potentials −t log |f |, their pressure and equilibria are called geometric since they provide a tool for a local geometrical insight in the space.
preserve length on S 1 and are ergodic. Hence h preserves so it is a rotation, identity for appropriate R 1 , R 2 . Hence R 1 and R 2 glue together to a holomorphic automorphism R of the Riemann sphere, a homography. (Compare Mostov rigidity theorem.) Therefore R −1 • f c • R(z) = λz 2 for λ with |λ| = 1 and in consequence c = 0.
-Complex case In the complex case we consider f a rational mapping of degree at least 2 of the Riemann sphere C. We consider f acting on its Julia set K = J(f ) (generalizing the z 2 + c model). Formally the Julia set is the complement in the sphere of the Fatou set which is the set where the family of the iterates f n is locally equicontinuous. J(f ) is compact completely invariant and f on it acts in a "chaotic" way.

Complex case
In the complex case we consider f a rational mapping of degree at least 2 of the Riemann sphere C. We consider f acting on its Julia set K = J(f ) (generalizing the z 2 + c model), see Fig. 2. Formally the Julia set is the complement in the sphere of the Fatou set which is the set where the family of the iterates f n is locally equicontinuous. J(f ) is compact completely invariant and f on it acts in a "chaotic" way.

Real case
Definition 2.2 (Real case, [20]). f ∈ C 2 is called a generalized multimodal map if it is defined on a neighbourhood of a compact invariant set K, critical points are not infinitely flat, bounded distortion (BD) property for iterates holds, is topologically transitive, and has positive topological entropy on K.
with J(f ) being the boundary between white (basilica) and black (rabbit), Sierpiński-Julia carpet f (z) = z 2 − 1/16z 2 i.e. boundaries of Fatou set components do not touch each other (the corona-like shapes are lines of the same speed of escape to ∞).
Also K is a maximal forward invariant subset of a finite unionÎ of pairwise disjoint closed intervals, whose endpoints are in K.
This maximality corresponds to the Darboux property. We write (f, K) ∈ A BD + , where + marks positive entropy. In place of BD one can assume C 3 (and write (f, K) ∈ A 3 + ) and assume that all periodic orbits in K are hyperbolic repelling. Then changing f outside K allows to get (f, K) ∈ A BD + .
The CLT follows from sufficiently fast convergence of iteration of transfer operator (spectral gap). The LIL is proved via LIL for a return map (inducing) to a nice domain related to µ φ (Mañé, Denker, Urbański) providing a Markov structure (Infinite Iterated Function System) avoiding critical points, satisfying BD.

Non-uniform hyperbolicity
Define the following conditions, both for real and complex (rational) cases: for all critical points c ∈ K whose forward orbit is disjoint from the set Crit(f ) of all critical points of f . Moreover there are no indifferent periodic orbits in K. (b) Backward Collet-Eckmann condition at z 0 ∈ K (CE2(z 0 )). There exist λ > 1 and C > 0 such that for every n ≥ 1 and every w ∈ f −n (z 0 ) (in a neighbourhood of K in the real case) (c) Topological Collet-Eckmann condition (TCE), [23]. There exist M ≥ 0, P ≥ 1, r > 0 such that for every x ∈ K there exist increasing n j , j = 1, 2, . . . , such that n j ≤ P · j and for each j and discs B(·) below, understood in C or R, (d) Exponential shrinking of components (ExpShrink). There exist λ > 1 and r > 0 such that for every x ∈ K, every n > 0 and every connected (e) Lyapunov hyperbolicity (LyapHyp). There is λ > 1 such that the Lyapunov exponent χ(µ) := K log |f | dµ of any ergodic measure µ ∈ M(f, K) satisfies χ(µ) ≥ log λ. (f) Uniform hyperbolicity on periodic orbits (UHP). There exists λ > 1 such that every periodic point p ∈ K of period k ≥ 1 satisfies Note that whereas in the complex case for a ball B = B(f (x), τ ) and in the real case it may be false, because of "folds". Therefore in the real case additional difficulties in this theory appear, in particular in TCE it is not equivalent to write that degrees of all f n j on See e.g. [21]. For polynomials (b)-(f) are equivalent for K = J(f ) = Fr Ω ∞ (f ), to Ω ∞ the basin of ∞, being Hölder (Graczyk, Smirnov). Note that for rational maps f satisfying TCE, if J(f ) = C, then it is mean porous hence HD(J(f )) < 2, see [23].
An order of proving the equivalences in Theorem 4.1 is, for z 0 safe (defined below), Separately one proves ExpShrink ⇔ TCE using for ⇒ the following − log |f j (x) − c| ≤ Qn for a constant Q > 0 every c ∈ Crit(f ), every x ∈ K and every integer n > 0. Σ means that we omit in the sum an index j of smallest distance |f j (x) − c|.
) and for every > 0 and all n large enough B(z, exp(− n)) ∩ n j=1 (f j (Crit(f ))) = ∅. Notice that this definition implies that all points except at most a set of Hausdorff dimension 0, are safe. Hyperbolic points (see below) are e.g. all points in invariant hyperbolic (expanding) subsets of K. Such sets are abundant.
Definition 4.4 (hyperbolic). We call z ∈ K hyperbolic if there exist λ > 1, r > 0, C > 0 such that for all n ∈ N the map f n is injective on Comp x (f −n (B(f n (x), r))) and |f n (x)| ≥ Cλ n .

Geometric variational pressure and equilibrium states
For φ = φ t := −t log |f |, the variational definition of pressure, here still makes sense by the integrability of log |f |, [P:93]. Moreover K log |f | dµ = χ(µ) ≥ 0 for all ergodic µ even in presence of critical points where φ = ±∞. t → P (t) is convex, monotone decreasing. We usually assume t > 0 later on.   Notice that this definition implies that all points except at most a set of Hausdorff dimension 0, are safe. Hyperbolic points (see below) are e.g. all points in invariant hyperbolic (expanding) subsets of K. Such sets are abundant.
Definition 4.4 (Hyperbolic point). We call z ∈ K hyperbolic if there exist λ > 1, r > 0, C > 0 such that for all n ∈ N the map f n is injective on Comp x (f −n (B(f n (x), r))) and |(f n ) (x)| ≥ Cλ n .

Geometric variational pressure and equilibrium states
For φ = φ t := −t log |f |, the variational definition of pressure, here still makes sense by the integrability of log |f |, [13]. Moreover K log |f | dµ = χ(µ) ≥ 0 for all ergodic µ even in presence of critical points where φ = ±∞. t → P (t) is convex, monotone decreasing. We usually assume t > 0 later on. Here t + is the phase transition "freezing" parameter, where t → P (t) is not analytic. P (t) is equal to several other quantities, in the complex case see [15] and [22], in real [20]. E.g.
where H (f, K) is defined as the space of all compact forward invariant, i.e. f (X) ⊂ X, expanding subsets of K, repellers.  HD(X).
Recall that for expanding f : X → X, t 0 (X) = HD(X), see (2.1). Passing to sup we obtain: Proposition 5.3 (Generalized Bowen's formula). The first zero t 0 of t → P hyp (K, t) is equal to HD hyp (K).
• If furthermore f is topologically exact on K (that is for every V an open subset of K there exists n ≥ 0 such that f n (V ) = K), then this measure is mixing, has exponential decay of correrations and satisfies CLT for Lipschitz observables.
Theorem 5.5 (Przytycki, Rivera-Letelier, the complex case, [19]). The assertion is the same. One assumes a very weak expansion: the existence of arbitrarily small nice, or pleasant, couples and hyperbolicity away from critical points.
Remark. For real f satisfying LyapHyp and K =Î, we have the unique zero of pressure t 0 = 1 and for − log |f | we conclude that a unique equilibrium state exists which is absolutely continuous with respect to Lebesgue measure (probability), acip. In general for K = I it holds assumed only e.g. |(f n ) (f (c))| → ∞ for all c ∈ Crit(f ), see [2]. For t > t + for f LyapHyp, equilibria do not exist, see [9].
Proofs use inducing (and Lai-Sang Young towers), compare Theorem 3.1 though here we find nice sets (pairs) geometrically, independently of equilibria. For a different proof, the real case, see a recent [5].

Lasota-Yorke Theorem
Sometimes to find an absolutely continuous invariant measure (probability) it is sufficient to find a function u : I → R invariant for the transfer operator (Perron-Frobenius) directly for f rather than for a return map via inducing as above. Then u · Leb will be acip. This is so in the classical Lasota-Yorke Theorem.
Proof. We find u := lim 1 n n−1 k=0 P k (φ), where φ is an arbitrary function of class C 1 , may be 1 1. The convergence follows from the conditional weak compactness of the family P N k (φ), where the Perron-Frobenius operator P is defined by P (φ)(x) := f (y)=x φ(y)/|f (y)|.
The weak compactness follows for φ with bounded variation, for α > 0, β < 1, k = 0, 1, ..., and some N , from This estimate with the use of two (semi)norms allows even to prove an exponential convergence to u (Ionescu-Tulcea, Marinescu). Proof of ≥: Given α consider t where inf is attained. The tangent to P (t) at t is parallel to −αt and for µ t the equilibrium, it is h µ t (f ) − tχ(µ t ). So the infimum is h µ t (f ), see Fig. (By the variational definition, P (t) and h µ are mutual Legendre type transforms.) Dividing by α gives ≥ using Mañé's equality (Notice that this equality is related to (2.1).) The proof of ≤ uses conformal measures. Using of the Legendre transform of P (t), see Fig. 5 allows us to also give formulae for Hausdorff dimension of (irregular) sets of points with given lower and upper Lyapunov exponent for β > 0, see [6] and [7].

More on Lyapunov exponents
In analogy to χ(µ) ≥ 0 one has: Theorem 5.7 (Levin, Przytycki, Shen, [11]). If for a rational function f : C → C there is only one critical point c in J(f ) and no parabolic periodic orbits, then χ(f (c)) ≥ 0.
For S-unimodal maps of interval this was proved much earlier by T. Nowicki and D. Sands.

Other definitions of geometric pressure
Definition 6.1 (Tree pressure). For every z ∈ K and t ∈ R define Theorem 6.2. P tree (z, t) does not depend on z for z safe.
• In the complex case to prove P tree (z 1 , t) = P tree (z 2 , t) one joins z 1 to z 2 with a curve not fast accumulated by critical trajectories, see [15] and [22]. • In the real case there is no room for such curves. Instead, one relies on the topological transitivity. See [16] and [20].
• For φ = −t log |f | pressure via separated sets does not make sense. Indeed, in presence of critical points for f , for t > 0, it is equal to +∞. So it is replaced by P tree . • One can consider however spanning geometric pressure P span (t) using (n, ε)spanning sets (in place of separated) and infimum. Assumed weak backward Lyapunov stability, wbls (see the definition below) it is indeed equal to P (t) in the complex case, see [16]. This is however not so in the real case, where wbls always holds if all periodic orbits are hyperbolic repelling. It happens that P span (t) = ∞ for t > 0, if some x with big |(f n ) (x)| −1 is well isolated in the metrics ρ n in Definition 1.3. See Fig. 6.  Figure 7. A Riemann map and its radial limit Consider harmonic measure ω = R * (l), where l is normalized length measure on ∂D and R is radial limit, defined l-a.e. Since 0 is a fixed point for g, l is g-invariant, hence ω is f -invariant. Denote by H 1 Hausdorff measure in dimension 1.
HD(ω) = 1 was proved in 1985 by Makarov without assuming existence of f .
This theorem applies also e.g. to snowflake-type Ω's.
Proofs. To prove HD(Ω hyp ) > 1 in Theorem 7.1, we can find X with HD(X) ≥ HD(ω) − by A. Katok's method and using HD = h/χ, see (5.1). This is not enough. However we can do better: σ 2 > 0 yields by CLT large fluctuations of the sums n−1 j=0 ψ • ς j from 0, allowing to find expanding X with HD(X) > HD(ω). One builds an iterated function system, for which X is the limit set. A special care is needed to get X ⊂ Fr Ω.
Substituting in LIL n ∼ (log 1/r n )/χ(ω) for r n = |(f n ) (x)| −n , comparing log |(g n ) | − log |(f n ) | • R with 2σ 2 n log log n for a sequence of n's, we get The above theorems hold for every connected, simply connected open Ω ⊂ C, different from C, without existence of f . Of course one should add ess sup over ζ ∈ ∂D and over z ∈ Fr Ω in Refined Volume Lemma and reformulate the case i). There is a universal Makarov's upper bound C M < ∞ for all c(Ω), C M ≤ 1.2326 (Hedenmalm, Kayumov, 2007, [8]). In 1989 I gave a weaker estimate.
Above theorems hold in an abstract setting of a geometric coding tree T in f (U ) for f : U → C, f (U ) ⊃ U proper. We obtain a coding from the left shift space, see Introduction, π : Σ d → Λ of the limit set Λ (in place of R : ∂D → Fr Ω). If f extends holomorphically beyond cl Λ we call Λ a quasirepeller.
The vertices are defined as the ends of γ n (α), denoted then z n (α) and z n−1 (α). For every α ∈ Σ d the subgraph composed of z, z n (α) and γ n (α) for all n ≥ 0 is called an infinite geometric branch and denoted by b(α). It is called convergent if the sequence γ n (α) is convergent to a point in cl U . This convergence holds for all a except a thin set, see [24]. Λ is defined as the set of limits of all convergent infinite branches.
For a constant potential, µ = µ max is a measure of maximal entropy on Julia set J(f ) for f : C → C rational. Then 1) If σ 2 > 0 then HD hyp (J(f )) > HD(µ max ).
2) If σ 2 = 0 then for each x, y ∈ J(f ) not postcritical, if z = f n (x) = f m (y) for some positive integers n, m, the orders of criticality of f n at x and f m at y coincide. In particular all critical points in J(f ) are pre-periodic, f is postcritically finite with parabolic orbifold, in particular z d , Chebyshev or some Lattès maps, (Zdunik, 1990, [31]). In the Ω version it is sufficient to assume f is defined only in a neighbourhood of ∂Ω repelling on the side of Ω, called RB-domain.