on the occasion of her80th birthday
DYNAMICS ON HIGHER DIMENSIONAL REAL OR COMPLEX FRACTALS
EUGEN MIH ˘AILESCU
In this paper we review several notions and some older as well as recent results about the fractal structure of invariant sets in higher dimension. For this purpose, the tools of classical complex analysis like analytic sets, currents, plurisubhar- monic functions must be supplemented with tools coming from ergodic theory and thermodynamical formalism, like entropy, pressure, inverse pressure, Lyapunov coefficients, equilibrium measures in order to better describe intrinsic invariants (Hausdorff dimension, box dimension, invariant measures supported on the set, etc.). In fact some of these properties can only be attacked with ergodic methods.
We also give some new results about transversal families, as well as the asymptotic distributions of preimages for a class of hyperbolic, not necessarily expanding en- domorphisms; in particular this includes hyperbolic toral endomorphisms which are conformal on stable manifolds.
AMS 2000 Subject Classification: Primary 37D35, 37A05; Secondary 37D20.
Key words: hyperbolic non-invertible map, equilibrium measure, Hausdorff di- mension, Julia set.
We want to apply the methods of ergodic theory and thermodynamical formalism to dimension estimates and the problem of the existence of invariant, dynamically generated, measures on basic sets. Let us take first a simple example, the holomorphic mapf(z) :=z2+c,c6= 0 and|c|small. We look for the largestf-invariant set on whichf has chaotic behaviour. If c= 0 this set is the circle S1, but ifc6= 0, then it is a complicated fractal set denoted byJc
and called theJulia set off. The Julia set of a rational function represents the complement of the largest open set where the iterates form a normal family.
In general, the Hausdorff dimension of Jc is not equal to 1, ifc6= 0.
The Hausdorff dimension is one of the most important invariants of a fractal set and the main way to deal with this, in general, is through tools from thermodynamical formalism.
REV. ROUMAINE MATH. PURES APPL.,54(2009),5–6, 513–524
Bowen [2] and Ruelle [15] were the first to notice that thermodynamical formalism, namely topological pressure (and its particular case, the topolog- ical entropy) can be applied to estimates of the Hausdorff dimension. Let us also mention that Hausdorff dimension is much harder to estimate than other invariants (like the entropy, Lyapunov coefficients, etc.), especially for fractals which are not obtained by an infinite repetition of a simple procedure (like the uniform Cantor sets, self-similar sets and others).
Definition 1. Let f : X → X be a continuous function on a compact metric space X and for a positive integer n define the metric dn(x, y) :=
sup
0≤i≤n
d(fix, fiy), x, y ∈X. A set F ⊂X is called an (n, ε)-spanning forX if the union of balls of radius ε(in the metricdn), centered at the points from F is equal to the entire spaceX.
For a continuous functionφ:X →R, let us define Sn(φ)(x) :=φ(x) +. . .+φ(fnx), x∈X.
Definition 2. Theε-pressure ofφis P(φ, ε) := lim
n→∞
1 nlog inf
( X
x∈Fn
eSnφ(x), Fn(n, ε)-spanning forX )
. Then the topological pressureof φis defined as P(φ) := lim
ε→0P(φ, ε).
Theorem 1 ([15]). Let us consider the function fc(z) := z2+c, z ∈C and its Julia set Jc. Then iffc is hyperbolic onJc (in particular if |c| is close enough to 0), then the Hausdorff dimension of Jc is equal to the unique zero of the pressure functional
t→PJc(tΦc), where Φc(z) :=−log|Df(z)|, z∈Jc.
This is an exact formula for the Hausdorff dimension but it does not necessarily give exact numerical values, only estimates. In general, except for some few cases, it is not possible to give exact numerical values for the Hausdorff dimension of fractal sets which are not constructed by a self-similar finitely generated procedure.
Definition 3. The Mandelbrot set isM:={c∈C,(fcn(0))nis bounded}, where fc(z) =z2+c.
We have also the following classical theorems (for example in [21]):
Theorem 2 (Bowen, Ruelle). If d(c) denotes the Hausdorff dimension of the Julia set Jc, then the real-valued function c → d(c) is real-analytic in the main cardioid C:={λ/2−λ2/4,|λ|<1} of the Mandelbrot set.
Theorem3 (Ruelle). In a neighbourhood of0, we haved(c) = 1+4 log 2|c|2 + o(|c|2). The minimum value ofd(c)in that neighbourhood is attained forc= 0 when J0 =S1 and d(0) = 1.
The question is to extend these results to higher dimensional holomorphic or conformal cases. The dynamics for non-invertible maps in higher dimensions presents many differences from the diffeomorphism case, and also from the one complex variable case ([1], [4], [11], [17], etc.) First, let us define two important concepts.
Definition 4. For a differentiable function f :U → Rm (U an open set in Rm) which is hyperbolic on a basic set Λ ⊂U, the stable dimension at a point x ∈ Λ is HD(Wrs(x)∩Λ), where r > 0 is a small positive number for which the local stable manifolds makes sense. Similarly, the unstable dimension atx∈Λ is HD(Wru(x)∩Λ).
The first such case we consider is that of automorphisms ofC2, in parti- cular that of Henon maps; these maps are defined as g(z, w) = (w, p(w)−az), for pa polynomial of degree d≥2,a6= 0.
For Henon mapsg which are hyperbolic on their Julia setJ, Verjovsky and Wu [18] proved that δs(x) :=HD(J∩Wrs(x)) is equal to the unique zero of the pressure function t → P(tΦs), Φs(y) = log|Dgs(y)|, y ∈ J; similarly, δu(x) := HD(J ∩Wru(x)) is equal to the unique zero of the function t → P(tΦu), where Φu(y) := −log|Dgu(y)|, y ∈ J. Hence both δs, δu do not depend on the pointx∈J; also they depend real-analytically on the function.
In this case the Julia set is locally bi-Lipschitz the product (Wrs(x)∩J)× (Wru(x)∩J), soHD(J) =δs+δu.
However, in the case of non-invertible maps, the situation becomes more difficult and such a formula as the one for Henon maps, is no longer valid; also a priori the stable dimension may depend on the point.
Examples of non-invertible maps are abundant in the literature for both the real and the holomorphic case, for example [1], [4], [11], [13], [14], [17], [21].
The behaviour of non-invertible maps (called also endomorphisms) is in many instances very different than the one for diffeomorphisms. For example we showed in [11] that the polynomial map fε(z, w) = (z2+aεz+bεw+c+ dεzw +eεw2, w2), (z, w) ∈ C2 (which is clearly not invertible on C2) has a basic set Λε contained in the non-wandering set, and the stable dimension at x ∈ Λε is greater than some positive constant α which does not depend on ε. In this case we also showed that Λε is not a Jordan curve. In higher dimensions the situation is complicated by the fact that there may exist both stable and unstable directions.
One way to deal with dimension estimates in higher dimensions for non- invertible maps has been provided in [12], where we studied further applica- tions of the inverse pressure. For the concept of inverse pressure P−, one works with sets Λ(C, ε) which are modeled after a finite prehistory C with elements from Λ. Indeed, for a function f :M → M differentiable on a man- ifold M, and having a basic set Λ, we define Λ(C, ε) := {z ∈ M, ∃ a finite prehistory ofz,(z, z−1, . . . , z−n),such thatd(z−j, x−j)< ε, 0≤j≤n},where C = (x, x−1, . . . , x−n) is a prehistory of length n of x with elements from Λ.
Then the inverse pressure is defined by a dimension type formula using the sets Λ(C, ε) in order to cover Λ. We proved
Theorem 4 ([12]). a) The stable dimension of a c-hyperbolic basic set of f is bounded above by the unique zero ts of the functiont→P−(tΦs).
b) In the same setting as in a), we have|δs(x)−δs(y)| ≤ (dimBΛ) logχu
logχ−1s , where χu := sup
z∈Λ
|Dfu(z)|, χs:= sup
z∈Λ
|Dfs(z)|.
The estimate in Theorem 4 is better than the one which uses the zero of the usual pressure, since P− is modelled after the preimages rather than the forward iterates; however, in general, the stable dimension is not neces- sarily constant on the basic set, but we can estimate the maximum oscilla- tion as in b).
One major difficuly with non-invertible maps is that we do not know in general how many of the f-preimages of a point from a basic set Λ really remain in Λ. This affects the dimension estimates for the respective stable intersections. The following result has been proved in [7].
Theorem5 ([7], [8]). Letf be a holomorphic map on a complex compact manifold. Assume that f is hyperbolic on a basic set Λ and that its critical set does not intersect Λ. Let d(x) denote the number of preimages of x ∈ Λ belonging to Λ.
i) Assume that the function d(·) is locally constant onΛ. Then δs(x)≥ td(·), ∀x ∈ Λ, where td(·) denotes the unique zero of the pressure function t→P(tΦs−logd(·)).
ii) In the same setting, if there exists a continuous function ω(·) on Λ such thatd(x)≤ω(x),x∈Λ (or forxin a dense subset ofΛ), thenδs(x)≥tω, x∈Λ.
iii)If the functiond(·) is locally constant onΛ, thenδs(x)≤td(·),x∈Λ, where td(·) denotes the unique zero of the pressure t→P(tΦs−logd(·)).
iv)If there exists a continuous functionω(·)onΛsuch thatd(x)≥ω(x), x∈Λ, then δs(x)≤tω, x∈Λ.
Theorem 5 has the following
Corollary 1 ([7]). In the same setting as in Theorem 5, let d be the maximum value attained by the function d(·) onΛ. Assume that there exists a point x∈Λ with δs(x) =td, where td denotes the unique zero of the function t→P(tΦs−logd). Then d(y) =d, ∀y ∈Λ.
Theproofof Theorem 5 is difficult, due to the fact thatf is not invertible.
Thus one cannot simply apply the results from the diffeomorphism case to the inversef−1; in the hyperbolic non-invertible case onedoes not haveMarkov partitions on Λ (so, the possibility of coding is severely reduced), and there exist no local unstable laminations. On the other hand, if one uses the inverse limit construction, the differentiability properties are lost.
What is interesting is that from these considerations of ergodic theory and thermodynamic formalism, one can obtain geometric results about fractal sets. Indeed, let us consider a perturbationf of the mapf0(z, w) = (z2+c, w2);
then f will have a basic set Λf close to {pc} ×S1, where pc denotes the unique fixed attracting point for z → z2+c (for |c| small). It can be shown (see [16]) that f(Λf) = Λf, f is transitive on Λf, and f has a hyperbolic structure on Λf. In general,f|Λf is not conjugate topologically tof0|{pc}×S1, another difference from the case of diffeomeorphisms. Instead, we only have a topological conjugacy between natural extensions of these maps (i.e., spaces of prehistories), but then we loose all the differentiability and hyperbolicity properties. In this setting, we proved [8] the following “flattening” result.
Theorem 6 ([8]). In the above setting, if there exists a point z ∈ Λf with δs(z) = 0, then f is 2-to-1 on Λf and moreover Λf is contained in the global unstable set of a point.
Thus, dimension has deep implications upon the global geometry of the fractal set.
The stable dimension also appears in connection to the problem of the global unstable set of a basic set Λ. We formulate it, more generally, for c- hyperbolic maps (i.e., maps which are hyperbolic on Λ and conformal on the local stable manifolds, so that Λ does not intersect the critical set off), but it applies also to holomorphic maps, for which it is possible to prove that local repellors cannot appear ([12], [9]).
Theorem7 ([9]). Let M be a smooth compact Riemannian manifold of real dimension 4 and f : M → M be a c-hyperbolic map on a basic set of saddle type Λ which is not a local repellor. Then for any pointx∈Λ, we have δs(x) < 2. In particular, this also holds in the case of a holomorphic map f :P2 →P2 which is s-hyperbolic on a basic set of saddle type Λ.
In the holomorphic case onPk, the most interesting dynamics occurs not on the whole Julia set, like in one dimension, but instead on the basic sets of saddle type and on the invariant support of the measure of maximal entropy.
In the non-invertible case, there may pass many (even uncountably many) local unstable manifolds through the same point. Thus the product structure from the diffeomorphism case, between the stable and unstable intersections, fails here and we need new tools and results to deal with this situation. The global unstable set of an endomorphism does not have in general a foliation structure. However Theorem 7 can be applied to show that the Hausdorff di- mension of the global unstable set is strictly less than 4 in the dissipative case:
Theorem 8 ([9]). In the same setting as in Theorem 7, and assuming that sup
bx∈Λb
|Dfs(x)| · |Dfu(x)|b <1, we have HD(Wu(Λ))b <4.
As can be noticed, the conditions in Theorem 7 are stable under holo- morphic perturbations.
Another strategy to deal with the loss of invertibility is to embed the map into a parametrized family and to prove dimension estimates for Lebesgue almost all parameters. This was achieved in [13], where we employed a no- tion of transversalityfor a parametrized family of hyperbolic skew-products Fλ(x, y) = (f(x),Φλ,x(y)), (x, y) ∈X×V, with the parameter λ∈W ⊂Rd, where f : X → X is expanding on the compact metric space X and Φλ,x are contracting on V (for V open set in Rq), for x ∈ X, λ ∈ W. This con- dition has been used in a different form (and setting) by Peres, Solomyak, etc. We adapted it to the case of non-invertible maps and used it to prove Bowen type formulas almost everywhere. The invariant fibers of Fλ are the sets Yλ,x := T
n≥0
S
z∈f−n(x)
Φnλ,z(V), x ∈ X. We see that to any prehistory xb = (x, x−1, . . .) of x ∈ X (with respect to f) there corresponds a unique point in V obtained by the intersection of the decreasing sequence of sets Φnλ,x−n(V) (we denote below the metric space of all prehistories in X by X,b understanding that these are prehistories with respect tof).
Definition 5. We say that the parametrized family (Fλ)λ∈W istransver- salif for allx∈X and λ0 ∈W there existδ(x, λ0)>0 and a constantC1 >0 such that for all prehistories x,b ybofx withx−16=y−1 and allr >0 we have
ld({λ∈B(λ0, δ(x, λ0)),|πλ(x)b −πλ(y)| ≤b r})≤C1rq, where πλ :Xb →V and πλ(bx) is defined as the intersection T
n
Φnλ,x−n(V).
We also denote by hλ the unique zero of the pressure function on the natural extension Λ,b t→P(tlog|DΦλ,x|(π(bx))).
Theorem9 ([13]). Suppose that{Fλ}λ∈W is a transversal family of hy- perbolic fiberwise conformal skew product endomorphisms. Then the function λ→hλis continuous onW, and for allx∈Xthere exists a Borel setWx⊂W
such that ld(W \Wx) = 0 and HD(Yλ,x) = min{hλ, q}, ∀λ∈Wx. Moreover, ld({λ∈W, hλ > q and ld(Yλ,x)>0}) =ld({λ∈W, hλ > q}).
The ideea of the proof is to introduce conditional measures on the fibers Yλ,x obtained from equilibrium measures of certain Holder continuous poten- tials.
We also introduced a stronger condition, that of uniform transversality, in which we require that the condition from Definition 5 is satisfied with the same constantC2 for all points (instead of C1). This stronger condition guarantees a Bowen type formula for Yλ,x for λ in a set fW ⊂ W of full measure in W, which does not depend on x∈X.
We applied these results (see [13]) to estimates of the stable dimension for parametrized families of holomorphic skew products like
Fλ(z, w) = (z2, z2+λ1z+λ2zw2), W =
(λ1, λ2)∈C2,|λ1|< 1 50, 1
10 <|λ2|< 1 8
, V =
w∈C, 1
2 <|w|< 3 2
.
We notice that these maps can be extended to rational maps on P2 and thus their dynamics can be also studied from the point of view of their indetermi- nacy sets (see [4]).
The results can be also applied to the familyFλ :X×[0,1]→X×[0,1]
with Fλ(x, y) = (f(x), λi + Φi(x, y, λ)), where f : X → X is a topologically exact, distance expanding map for which there are mclosed mutually disjoint sets X1, . . . , Xm such that X =X1∪. . .∪Xm, and f(Xi) = X, i= 1, . . . , m and where Φi are contractions in they variable. This family can be obtained as a perturbation of an IFS (Iterated Function System) with finitely many gen- erators. It was proved in [13] that under certain conditions on the derivatives of Φi, the family is uniformly transversal, thus Theorem 9 holds.
Another important problem is that of the asymptotic distributions of preimages for endomorphisms f. In the diffeomorphic case the asymptotic distribution of preimages is given in a neighbourhood of a repellor by a SRB measure for the inverse f−1. We give below the theorem of existence of SRB measures for attractors ([20]). It collects various results of Sinai, Ruelle, Bowen and other authors:
Theorem 10 (SRB measures for attractors of diffeomorphisms). Let f be a C2 diffeomorphism on a manifold M having an Axiom A attractor Λ.
Then there exists a unique f-invariant Borel probability measure µ+ on Λ, which is characterized by each of the following equivalent conditions:
i)there exists a setV contained in a neighbourhoodU of Λ, such that V has full Lebesgue measure in U (i.e., the Lebesgue measure of U \V is equal to zero), such that for every pointx∈V we have δx+...+δn f nx →µ+ as n→ ∞ (convergence is the weak convergence of measures);
ii) µ+ has absolutely continuous conditional measures on unstable mani- folds;
iii) µ+ is the unique equilibrium measure of the potential Φu(y) :=
−log|detD f|Eu
y|, y∈Λ;
iv) hµ+(f) = R
log|detDf|Eu
x|dµ+(x), where hµ(f) denotes in general the metric entropy of an arbitrary f-invariant probability measure µ.
In the case of Anosov diffeomorphisms, the conclusions are the same and the measure µ+ is supported on the entire manifold M.
Corollary 2. If f is a C2 diffeomorphism on a manifold M having an Axiom A repellor, then Theorem 10 holds for the inverse f−1.
Let us mention that SRB measures have been also studied for families of non-invertible maps (endomorphisms). For example Tsuji [17] proved that they display strange behaviour, namely, for some parameters of a family of skew product maps, the associated SRB measures are absolutely continuous with respect to the Lebesgue measure, and for other parameters they are totally singular.
But, as we said above, one is also interested in the distribution of preima- ges, given hopefully by an “inverse SBR measure”. If f is just an endomor- phism, hyperbolic over a repellor Λ, then first of all it is not clear how to average over prehistories, given that a point may have several preimages. Let us notice also that f is not assumed to be expanding on Λ, but instead it can have also contracting directions. In the expanding case, the analysis is facilitated by the fact that local inverse iterates contract small balls.
There are several ways to average over prehistories. Let us consider in the sequel maps which ared-to-1 over Λ. One such procedure (detailed in [6]) is to consider the measures:
(1) µxn:= 1
n
n
X
k=1
P
y∈Ek(x)
δy dk ,
where Ek(x) := f−k(x)∩Λ, k ≥ 0. Due to the fact that f|Λ is d-to-1, the measure µxn is a probability Borel measure. If Λ is a repellor, i.e., there exists a neighbourhoodU of Λ such thatf(U)⊃U, then sincef isd-to-1 on Λ, it is easy to prove (eventually by shrinkingU) that every point fromU has exactly d f-preimages in U. So, if Λ is a repellor, we can form the measures µxn for points x ∈ U, too. It can be shown that any limit measure of the sequence (µxn)nis a Borel probability f-invariant measure supported on Λ.
A second procedure is as follows: let Λ be a basic set forf(not necessarily a repellor). Let also x ∈ Wu(bΛ) be a point in the global unstable set of a prehistory from Λ. Then if x ∈ Wu(y), then there exists a prehistoryb bx of x which is r-shadowed by yb (for r > 0 small) starting from some level p = p(r, x,by), i.e., d(x−i, y−i) < r, i > p. Then we can consider the Borel probability measures
(2) νnxb:= 1
n
n
X
i=1
δx−i.
Again, it can be shown (see [6]) that any limit measure of (νnx)n is a Borel probability f-invariant measure supported on Λ. There appears the question to describe these limit measures; Theorem 10 cannot be applied since the map f is not necessarily invertible on Λ. Also, the Birkhoff Ergodic Theorem can be applied only to one prehistory at a time, but not for all prehistories simultaneously.
In [6] we proved that there are good prehistories along which the measures νnbx (for appropriate prehistories bx) converge to the equilibrium measure of a Holder continuous potential.
Theorem 11 ([6]). Let Λ be a repellor for the d-to-1 C2 map f on a compact Riemannian manifold M. Assume also that f is c-hyperbolic on Λ and that the unstable manifolds over Λ only depend on their base points, not on the whole prehistories. Then there exists a set T(Λ) ⊂ Wu(Λ) with λ(Wu(Λ)\ T(Λ)) = 0 such that for any x ∈ T(Λ) and any continuous g ∈ C(M,R) we have
inf
x−n∈fΛ−n(x)
g(x) +. . .+g(x−n)
n −
Z gdµs
−→n 0,
whereµsis the equilibrium measure ofΦs−logd, withΦs(y) := log|detDfs(y)|, y ∈Λ.
Then we proved results about the whole collection of n-preimages of a point, and the measuresµxn,n≥1. Again, the mapf is not necessarily expan- ding on Λ, so the local inverse images of balls may grow in the stable direction.
Theorem 12 ([6]). Let us consider a smooth map f : M → M on a smooth (C2) Riemannian manifold M, and a repellor Λ for f such that f is d-to-1 on Λ and f is c-hyperbolic on Λ. Assume also that the local unstable manifolds over Λ only depend on their base points, i.e., Wru(z) =b Wru(bx) for any prehistories z,b bx∈Λb of z (if x=z). Then there exists a small ε >0 such
that for any continuous function g∈ C(M,R) we have Z
Wεu(Λ)
|µzn(g)−µs(g)|dλ(z) −→
n→∞0.
In particular, there exist a measurable setA⊂Wεu(Λ)withλ(A) =λ(Wεu(Λ)), and an infinite subsequence (pk)k such that for any point z ∈ A we have the weak convergence of measures µzpk −→
k→∞µs.
In particular, Theorem 12 gives information about the asymptotic dis- tributions of preimages for hyperbolic (not necessarily expanding) endomor- phisms of tori. These maps are constant-to-one on the entire torus, namely, if the endomorphism f onTm is given by an integer-valuedm×m matrixA, then any point inTm has exactly|detA|f-preimages inTm. So we proved
Corollary 3 ([6]). Let f : Tm → Tm be a hyperbolic toral endomor- phism (group homomorphism in particular) such that f is conformal on stable manifolds. Denote by λ the normalized Lebesgue measure (Haar measure) on Tm. Then R
Tm|µzn(g)−µs(g)|dλ(z)→
n 0 for any continuous real function g on Tm. Also, there exist a Borel set A⊂Tm withλ(A) =λ(Tm) and an infinite sequence (nk)k such that we haveµznk→
k λ for anyz∈A.
In particular, the conclusions above hold for any hyperbolic toral endo- morphism on T2.
For an Anosov endomorphism f we denote by µ+ the usual SRB mea- sure and by µ− the inverse SRB measure whose existence results from Theo- rem 12. Then we obtained [6] conditions on periodic points, guaranteeing that an Anosov endomorphism has the same measure giving both the asymptotics of forward iterates and the asymptotics of prehistories; these conditions are similar to those from the diffeomorphism case (see [3]), but the proof involves natural extensions and lifts of measures. Periodic points prove thus to be very important for the dynamics of non-invertible maps as well (see also [3], [4], [5]).
Corollary4. Assume we are in the same setting as in Theorem 12but this time f is a transitive Anosov endomorphism, soΛ =M. Then there exist both the SRB measure µ+ and the inverse SRB measure µ−. In this case we haveµ+ =µ− if and only if for everyx∈M withfm(x) =x,m≥1, we have
|detDfm(x)|=dm.
In [10] we studied the weighted distributions of preimages for endomor- phisms, obtaining equilibrium measures in the (weak) limit. This represents a step forward in the understanding of the nature of equilibrium measures, both as weak limits of distributions of periodic points as well as weak limits of distributions of consecutive preimage sets. For such limits involving all
n-preimages in Λ, one cannot simply employ Birkhoff Ergodic Theorem. In- stead, new methods and combinatorial arguments were devised.
Theorem 13 ([10]). Let f : M → M be a smooth map (say C2) on a smooth Riemannian manifold M. Assumef is hyperbolic and finite-to-one on a basic setΛ; assume also that the critical setCf of f does not intersectΛ. Let also φ be a Holder continous potential on Λ and µφ the equilibrium measure of φon Λ. Then
Z
Λ
1 n
X
y∈f−nx
eSnφ(y) P
z∈f−nx
eSnφ(z) ·
n−1
X
i=0
δfiy−µφ, g
dµφ(x) −→
n→∞0, ∀g∈ C(Λ,R).
Acknowledgements.This work was partially supported by the Romanian Ministry of Education, Research and Innovation (PN II Program, Contract No. 472/2009, CNCSIS code: 1185).
REFERENCES
[1] H.G. Bothe,Shift spaces and attractors in noninvertible horseshoes. Fundamenta Math.
152(1997),3, 267–289.
[2] R. Bowen, Hausdorff dimension for quasi-circles. Publ. Math. IHES No. 50 (1979), 11–25.
[3] R. Bowen,Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lec- ture Notes in Math.470. Springer-Verlag, Berlin–New York, 1975.
[4] J. E. Fornaess,Dynamics in Several Complex Variables. CBMS Conf. Series in Math.
87. Amer. Math. Soc., Providence, RI, 1996.
[5] E. Mih˘ailescu,Periodic points for actions of tori in Stein manifolds. Math. Ann.314 (1999),1, 39–52.
[6] E. Mih˘ailescu,Asymptotic distributions of preimages for endomorphisms. Submitted for publication 2008.
[7] E. Mih˘ailescu and M. Urba´nski,Relations between stable dimension and the preimage counting function on basic sets with overlaps. Accepted Bull. London Math. Soc., 2009.
[8] E. Mih˘ailescu,Stable dimension rigidity and geometric consequences of some thermody- namical formalism results on basic sets. Submitted for publication 2009.
[9] E. Mih˘ailescu,Metric properties of some fractal sets and applications of inverse pressure.
Accepted Math. Proc. Camdrige, 2009, d.s.i.: 10.1017/S030500 4109990326.
[10] E. Mih˘ailescu,Weighted distributions for hyperbolic endomorphisms. Submitted for pub- lication 2009.
[11] E. Mih˘ailescu and M. Urba´nski, Estimates for the stable dimension for holomorphic maps. Houston J. Math.31(2005),2, 367–389.
[12] E. Mih˘ailescu and M. Urba´nski, Inverse pressure estimates and the independence of stable dimension for non-invertible maps. Canad. J. Math.60(2008),3, 658–684.
[13] E. Mih˘ailescu and M. Urba´nski, Transversal families of hyperbolic skew-products. Dis- crete Contin. Dyn. Syst.21(2008),3, 907–928.
[14] M. Qian and S. Zhu, SRB measures and Pesin’s entropy formula for endomorphisms.
Trans. Amer. Math. Soc.354(2002),4, 1453–1471.
[15] D. Ruelle,Repellers for real analytic maps. Ergodic Theory Dynam. Systems2(1982), 99–107.
[16] D. Ruelle, Elements of Differentiable Dynamics and Bifurcation Theory. Academic Press, New York, 1989.
[17] M. Tsuji,Fat solenoidal attractors. Nonlinearity14(2001), 1011–1027.
[18] A. Verjovsky and H. Wu,Hausdorff dimension of Julia sets of complex Henon mappings.
Ergodic Theory Dynam. Systems16(1996), 849–861.
[19] P. Walters,An Introduction to Ergodic Theory. Springer-Verlag, New York–Berlin, 1982.
[20] L.S. Young,What are SRB measures and which dynamical systems have them. J. Statist.
Phys.108(2002), 733–754.
[21] M. Zinsmeister, Formalisme thermodynamique et syst`emes dynamiques holomorphes.
Panoramas et Synth`eses4. Soc. Math. France, Paris, 1996.
Received 20 January 2009 Romanian Academy
“Simion Stoilow” Institute of Mathematics P.O. Box 1-764
014700 Bucharest, Romania Eugen.Mihailescu@imar.ro
www.imar.ro/∼mihailes
