Bifurcations in families of polynomial skew products

Bifurcations in families of polynomial skew products

Matthieu Astorg Université d’Orléans, MAPMO
UMR CNRS 7349
45067 Orléans Cedex 2
France
matthieu.astorguniv-orleans.fr
 and  Fabrizio Bianchi Imperial College
South Kensington Campus
London SW7 2AZ
UK
f.bianchiimperial.ac.uk
.

We initiate a parametric study of families of polynomial skew products, i.e., polynomial endomorphisms of of the form that extend to endomorphisms of . Our aim is to study and give a precise characterization of the bifurcation current and the bifurcation locus of such a family. As an application, we precisely describe the geometry of the bifurcation current near infinity, and give a classification of the hyperbolic components. This is the first study of a bifurcation locus and current for an explicit and somehow general family in dimension larger than 1.

This research was partially supported by the ANR project LAMBDA, ANR-13-BS01-0002, and by a PEPS grant Jeune-s Chercheur-e-s awarded to the first author
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1. Introduction

A polynomial skew product in two complex variables is a polynomial endomorphism of of the form that extends to an endomorphism of . The dynamics of these maps was studied in detail in [jonsson1999dynamics]. Despite (and actually because of) their specific form, they have already provided examples of dynamical phenomena not displayed by one-dimensional polynomials, such as wandering domains [astorg2014two], or Siegel disks in the Julia set [b_tesi]. They have also been used to construct examples of non dynamical Green currents [dujardin2016nonlaminar] or stable manifolds dense in [taflin_blender]. In this paper we address the question of understanding the dynamical stability of such maps. In order to do this, let us first introduce the framework for our work and previous known results.

A holomorphic family of endomorphisms of is a holomorphic map of the form . The complex manifold is the parameter space and we require that all have the same degree. In dimension , the study of stability and bifurcation within such families was initiated by Mané-Sad-Sullivan [mane1983dynamics] and Lyubich [lyubich1983some] in the 80s. They proved that many possible definitions of stability are actually equivalent, allowing one to decompose the parameter space of any family of rational maps into a stability locus and a bifurcation locus. In 2000, by means of the Lyapunov function, DeMarco [demarco2001dynamics, demarco2003dynamics] constructed a natural bifurcation current precisely supported on the bifurcation locus. This allowed for the start of a pluripotential study of the bifurcations of rational maps.

The simplest and most fundamental example we can consider is the quadratic family. We have a parameter space and, for every , we consider the quadratic polynomial . In this case, it is possible to prove that the bifurcation locus is precisely the boundary of the Mandelbrot set (notice that, in this context, the equality between the bifurcation measure and the harmonic measure for the Mandelbrot set is due to N. Sibony, see [sibonyorsay]). The study of this particular family is of fundamental importance for all the theory, since, by a result of McMullen [mcmullen2000mandelbrot] (see also [lei1990similarity]), copies of the Mandelbrot set are dense in the bifurcation locus of any family of rational maps. Understanding the geometry of the Mandelbrot set is still today a major research area.

The theory by Mané-Sad-Sullivan, Lyubich and De Marco was recently extended to any dimension by Berteloot, Dupont, and the second author [bbd2015, b_misiurewicz], see Theorems 2.6 and 2.8 below (the second, for an adapted version of the main result in our context). Despite the quite precise understanding of the relation between the various phenomena related to stability and bifurcation (motion of the repelling cycles, Lyapunov function, Misiurewicz parameters), apart from specific examples ([bt_desboves]) or near special parameters ([bb_hausdorff, dujardin2016non, taflin_blender]), we still miss a concrete and somehow general family whose bifurcations can be explicitly exhibited and studied. This paper aims at providing a starting point for this study for any family of polynomial skew products. More precisely, we will establish equidistribution properties towards the bifurcation current (Section 3), study the possible hyperbolic components (Sections 7 and 8) and give a precise description of the accumulation of the bifurcation locus at infinity (Section 6). This will be achieved by means of precise formulas for the Lyapunov exponents, which will in turn allow for a precise decomposition of the bifurcation locus (Section 4).

Les us now be more specific, and enter more into the details of our results and techniques. While many of the results apply to more general families, we will mainly focus on the family of quadratic skew products. It is not difficult to see (Lemma 5.41) that the dynamical study of this family can be reduced to that of the family

It is also quite clear (see also Section 4) that the bifurcation of this family consists of two parts: the bifurcation locus associated to the polynomial family , and a part corresponding to “vertical” bifurcation in the fibres. We thus fix the base polynomial and consider the bifurcations associated with the other three parameters. Our first result is the following basic decomposition of both the bifurcation locus and current. This is essentially based on formulas for the Lyapunov exponents of a polynomial skew product due to Jonsson [jonsson1999dynamics] (see Theorem 2.2). Our result is actually more general than the situation described above, and holds in any family of polynomial skew products, see Section 4.

Theorem A.

Let be a family of polynomial skew products of degree . Then

(1)

and

(2)

Here, we denote by (respectively ) the bifurcation locus (respectively current) associated to the non autonomous iteration of the polynomials . These are defined respectively as the non-normality locus for the iteration of some critical point in the fibre at , and the Laplacian of the corresponding sum of the Green function evaluated at the critical points (see Section 4.1). In the case of a periodic point , we are actually considering iterations of the return maps to the fibre. The result follows combining the above mentioned result by Jonsson, the equidistribution of the periodic points with respect to the equilibrium measure, and the characterization of the bifurcation locus as the closure of the Misiurewicz parameters ([bbd2015], and Theorem 2.8 below).

Another classical way to approximate the bifurcation current is by seeing this as a limit of currents detecting dynamically interesting parameters. A first example of this is a theorem by Levin [levin1982bifurcation] (see also [levin1990theory]) stating that the centres of the hyperbolic components of the Mandelbrot set equidistribute the bifurcation current, which is supported on its boundary. This means that the bifurcation current detects the asymptotic distribution in the parameter space of the parameters possessing a periodic critical point of period . This result was later generalized in order to cover any family of polynomials [bassanelli2011distribution, okuyama2014equidistribution], the distribution of parameters with a cycle of any given multiplier [bassanelli2011distribution, buff2015quadratic, gauthier2016equidistribution] and also the distribution of preperiodic critical point [dujardin2008distribution, favre2015distribution]. See also [gauthier2017hyperbolic] for the most recent account and results in this direction for families of rational maps.

In our situation, we can prove the following equidistribution result, giving the convergence for the parameters admitting a periodic point with vertical multiplier (the study of “horizontal” multipliers naturally gives the bifurcation current for the base ). We also have a more general statement (see Theorem 3.15 and Corollary 3.16) valid for any family of endomorphism of . This gives also a first parametric equidistribution result for holomorphic dynamical systems in several complex variables.

Theorem B.

Let be a holomorphic family of polynomial skew-products of degree over a fixed base , and parametrized by . For all outside of a polar subset, we have

where

The equidistribution result stated above plays a crucial role in the next step of our investigation: the description of the accumulation of the bifurcation locus at infinity. Notice that the study of degenerating dynamical systems, and in particular of their Lyapunov exponents, is an active current area of research, see for instance [favre2016degeneration]. In the following result not only we describe this accumulation locus, but we quantify this accumulation by means of the equilibrium measure of the base polynomials. An analogous result for quadratic rational map is proved in [berteloot2015geometry].

Theorem C.

Let . Then the bifurcation current extends to a current on and

where and is the map that associates to the point such that are the roots of .

A similar result also applies for subfamilies of . In particular, for the subfamily given by , we have the following description. Notice that this is not only a by-product of the previous theorem, but actually a main step in the proof. Indeed, the current in Theorem C will be constructed by slicing it with respect to lines corresponding to special subfamilies. Tools from the theory of horizontal currents as developed by Dinh and Sibony also play a crucial role in our proof.

Theorem C’.

Let . Then the bifurcation current extends to a current on and

where is given by .

All the results presented until now are related to the description of the bifurcation locus. Our last result concerns stable components, and in particular unbounded hyperbolic components. It follows from the results above that the stability of a polynomial skew product is determined by the behaviour of the critical points of the form , with . We prove that there exists a natural decomposition of the parameter space: a (compact) region , where all these critical points have bounded orbit, a region where all the critical points escape under iteration, and a region where maps admit critical points with either behaviour.

In particular, components in can be thought of as the analogous in this situation of the complement of the Mandelbrot set. By the description above, the accumulation of the bifurcation locus at infinity is given by the image under of the points in with at least one component in the Julia set of . Thus, in particular consists of components accumulating on the image of by the map of Theorem C (here denotes the Fatou set of ). Let be the set of these components. Our last result gives a complete classification of these components, essentially stating that distinct couples of Fatou components are associated to distinct hyperbolic components.

Theorem D.

Let such that the Julia set of is locally connected. Let be the set of parameters such that all critical points escape to infinity, and the set of the components in accumulating on some point of . There is a natural bijection between and the (non ordered) pairs of connected components of the Fatou set of .

As above, we state also the analogous result for the simpler case of the family .

Theorem D’.

Let such that the Julia set of is locally connected. Let be the set of parameters such that all critical points escape to infinity, and the set of the components in accumulating on some point of . There is a natural bijection between and the connected components of the Fatou set of .

Theorem D is proved by exhibiting distinct topologies for the Julia sets of maps in components associated to distinct Fatou components (see Theorem 7.73). In Section 8 we also study the problem of the existence of unbounded hyperbolic components in . By adapting an example by Jonsson, we can prove that such a component can actually exist (see Propositions 8.78 and 8.81).

The paper is organized as follows. After the presentation of the necessary preliminaries on polynomial skew products and bifurcations in several variables, the exposition is divided in two parts. In the first we prove Theorems A and B, giving the approximations for the bifurcation current that we need in the sequel. In this part, we do not need to restrict to quadratic skew products, and actually, as mentioned above, we can prove an equidistribution formula for any family of endomorphisms of . Then, in the second part, we focus on quadratic skew products, and in particular on the study of their parameter space near infinity (proving Theorems C and D).

2. Preliminaries: polynomial skew products and bifurcations

2.1. Polynomial skew products

In this section we collect notations, definitions, and results concerning the dynamics of polynomial skew products that we will need through all the paper. Unless otherwise stated, all the results are due to Jonsson [jonsson1999dynamics].

Definitions and notations

We consider here a polynomial skew product of the form

of algebraic degree . Recall that we require to be extendible to . This means that has degree , and that the coefficient of in the second coordinate is non zero. We will assume that this is 1. The second coordinate will be also written as , to emphasize the variable and see the map as a family of polynomials depending on . We shall denote by the orbit of a point under the base polynomial . In this way, we can write

We shall denote by the second coordinate of .

We will be mainly interested in the recurrent part of the dynamics. In particular, we restrict in the following to the points of the form , with (we denote by and the Julia set and the filled Julia set of , respectively). The family gives rise to a (non autonomous) iteration on the fibre . Since , the orbit of is bounded if and only if the orbit of under the non autonomous iteration of is bounded. We then define as the points in (to be thought of as the fibre over ) with bounded orbit under this iteration. The set is the boundary of . Notice that and can be defined for any point , but only detect the boundedness of the second coordinate of the orbit.

Green functions and Julia set

A standard way to detect the boundedness of an orbit (or its escape rate) is by means of a dynamical Green function. In our setting, given , we can consider three possible Green functions:

  1. the Green function of , defined as ;

  2. the horizontal Green function , where is the Green function of the base polynomial , and

  3. for , the vertical Green function .

The last function detects the escape rate of the sequence . The set is then the zero level of the function . In the case of points , the Green function reduces to the vertical Green function. By means of the Green functions, it is immediate to deduce that the maps and are respectively upper and lower semicontinuous with respect to the Hausdorff topology.

Another useful feature of the Green function is that it allows to construct the equilibrium measure for the map . The construction is now classical and proceeds as follows (see for instance [fs_higher, ds_cime]). First of all, we consider the Green current (on ). This is positive, closed (1,1)-current. Its support is precisely the non normality locus of the sequence of iterates of . Since the potential of is continuous (actually Holder continuous) it is possible to consider the wedge product . This is a positive measure, which turns out to be invariant, of constant Jacobian, ergodic, mixing. It detects the distribution of periodic points and preimages of generic points. Its support is the Julia set of the map . In the case of polynomial skew products, we have the following structure result for .

Theorem 2.1.

Let be a polynomial skew product. Then

Moreover, coincides with the closure of the repelling points.

Notice that the last assertion in this result is known not to hold for general endomorphisms of ([hubbard1991superattractive, fs_examplesP2]).

Lyapunov exponents

Since the equilibrium is ergodic, by Oseledec Theorem we can associate two Lyapunov exponents to it. The idea, again now classical, is the following. There exists an invariant splitting of the tangent space at (-almost all) points , depending measurably on . The differential of acts on these sub bundles , and there exists constants such that . More precisely, for -almost every and every we have

The constants are called the Lyapunov exponents of the ergodic system . In the case of polynomial skew products, we have the following explicit formulas for the Lyapunov exponents (for more general formulas, valid for any regular polynomial, see [bedford2000dynamics]).

Theorem 2.2.

Let be a polynomial skew product. Then the equilibrium measure of admits the two Lyapunov exponents

where and are the critical set and the equilibrium measure of and is the critical set of .

Notice, in particular, that the Lyapunov exponent coincides with the Lyapunov exponent of the system . We use the notation and (vertical) because we will not assume any ordering between these two quantities.

Hyperbolicity and vertical expansion

We conclude this section by introducing an adapted notion of hyperbolicity, particularly useful in the study of polynomial skew products. Recall that an endomorphism of is hyperbolic or uniformly expanding on the Julia set if there exist constants such that, for every and , we have (with respect for instance to the standard norm on ). In the case of polynomial skew products, this condition in particular forces the base polynomial to be hyperbolic. Since we will be mainly concerned with the vertical dynamics over the Julia set of , the following definition gives an analogous notion of hyperbolicity, more suitable to our purposes.

Given an invariant set for (we shall primarily use ) we set

  • for the critical set over ,

  • for the postcritical set over , and

  • for the Julia set over .

When dropping the index , we mean that we are considering . We then have the following definition.

Definition 2.3.

Let be a polynomial skew product and such that . We say that is vertically expanding over if there exist constants and such that for every , and .

For polynomials on , hyperbolicity is equivalent to the fact that the closure of the postcritical set is disjoint from the Julia set. In our situation, we have the following analogous characterization.

Theorem 2.4.

Let be a polynomial skew product. Then the following are equivalent:

  1. is vertically expanding on ;

  2. .

The previous results allows one to give a similar characterization of hyperbolicity for polynomial skew products. Notice that the same result is not known for general endomorphisms of .

Theorem 2.5.

Let be a polynomial skew product. Then the following are equivalent:

  1. is hyperbolic;

  2. ;

  3. is hyperbolic, and is vertically expanding over .

2.2. Stability, bifurcations, and hyperbolicity

In this paper we will be concerned with stability and bifurcation of polynomial skew product of (extendible to ). The definition and study of these for endomorphisms of projective spaces of any dimension is given in [bbd2015, b_misiurewicz]. For a general presentation of this and related results, see also [bb_warsaw].

Theorem 2.6 ([bbd2015, b_misiurewicz]).

Let be a holomorphic family of endomorphisms of of degree . Then the following are equivalent:

  1. asymptotically, the repelling cycles move holomorphically;

  2. there exists an equilibrium lamination for the Julia sets;

  3. ;

  4. there are no Misiurewicz parameters.

The holomorphic motion of the repelling cycles is defined as in dimension 1 (see e.g. [berteloot2013bifurcation, dujardin2011bifurcation], or [bbd2015, Definition 1.2] in this context). The asymptotically essentially means that we can follow out of the repelling points, see [b_misiurewicz, Definition 1.3]. This condition can be improved to the motion of all repelling cycles contained in the Julia set if , or if the family is an open set in the family of all endomorphisms of a given degree. See also [berteloot2017cycles] for another description of the asymptotic bifurcations of the repelling cycles. denotes the sum of the Lyapunov exponents, which is a psh function on the parameter space. Thus, is a positive closed (1,1) current on the parameter space. Finally, we just mention that the equilibrium lamination is a weaker notion of holomorphic motion, that provides an actual holomorphic motion for a full measure (for the equilibrium measure) subset of the Julia set (see [bbd2015, Definition 1.4]). Since it will be used in the sequel, we give the precise definition of the last concept in the theorem above. This is a generalization to any dimension of the notion of non-persistently preperiodic (to a repelling cycle) critical point.

Definition 2.7.

Let be a holomorphic family of endomorphisms of and let be the critical set of the map . A point of the parameter space is called a Misiurewicz parameter if there exist a neighbourhood of and a holomorphic map such that:

  1. for every , is a repelling periodic point;

  2. is in the Julia set of ;

  3. there exists an such that belongs to some component of ;

  4. is not contained in a component of satisfying 3.

In view of Theorem 2.6, it makes sense to define the bifurcation locus as the support of the bifurcation current . If any (and thus all of the) conditions in Theorem 2.6 hold, we say that the family is stable.

Since we will be mainly concerned with families of polynomial skew products in dimension 2, we cite an adapted version of the result above in our setting.

Theorem 2.8 ([bbd2015]).

Let be a holomorphic family of polynomial skew products of degree . Then the following are equivalent:

  1. the repelling cycles move holomorphically;

  2. there exists an equilibrium lamination for the Julia sets:

  3. ;

  4. there are no Misiurewicz parameters.

As mentioned above, since the dimension is , we can promote the first condition in Theorem 2.6 to the motion of all the repelling cycles contained in the Julia set. By the result of Jonsson (Theorem 2.1), we then know that polynomial skew products cannot have repelling points outside the Julia set.

Let us now consider a hyperbolic parameter for a family of endomorphisms of . A natural question is to ask whether all the corresponding stability component consists of hyperbolic parameters. While this is not known in general, by Theorem 2.5 it follows that this is true for polynomial skew products.

Lemma 2.9.

Let be a stable family of polynomial skew products. If there exists such that is hyperbolic, then is hyperbolic for every in the parameter space.

Proof.

Let be any parameter. Assume that is not hyperbolic. By Theorem 2.5, this implies that the postcritical hypersurfaces have an accumulation point on the Julia set of .

Since the family is stable, by Theorem 2.8 the repelling points move holomorphically. Denote by the collection of the holomorphic graphs in the product space . Since repelling points are dense in the Julia set, there exists a sequence of points with converging to . By compactness, there exist a graph over which is an accumulation point for the sequence of the holomorphic motions of the repelling points . Since the family is stable there are no Misiurewicz parameters. Thus, all these motions do not create transverse intersections with the postcritical set. This, by Hurwitz theorem, implies that the graph is contained in the closure of the postcritical set. This gives the desired contradiction, since this cannot be true at the hyperbolic parameter in . ∎

Remark 2.10.

By the same argument (and using Theorem 2.4) we can prove that if a parameter in a stable component is vertically expanding, the same is true for all the parameter is the component. An improvement of this fact will be given in Lemma 7.63

For families of polynomial skew products, it thus makes sense to talk about hyperbolic components (respectively vertically expanding components), i.e., stable components whose elements are (all) hyperbolic (respectively, vertically expanding). We will characterize and study some components of this kind in Sections 7 and 8. We conclude this section with a result relating the existence of non hyperbolic component for families of polynomial skew products to the same question for the (possibly non autonomous) iteration of polynomial maps in dimension 1.

Proposition 2.11.

If a polynomial family of skew products has a non hyperbolic stable component, then some one-dimensional (possibly non autonomous) polynomial family has a non hyperbolic stable component.

The proof of this proposition exploits the characterization of the stability of a skew product family with respect to the stability of the dynamics on the periodic fibres, and will be given in Section 4.3.

Part I Approximations for the bifurcation current

3. Equidistribution results in the parameter space (Theorem B)

Our aim in this section is to prove Theorem 3.23. We will actually prove a more general result (see Theorem 3.15 and Corollary 3.16), valid for any family of endomorphisms of . We shall later specialise to our setting of polynomial skew products (see Section 3.2).

3.1. A general equidistribution result for endomorphisms of

Let us begin with a rather general equidistribution result that holds for families of endomorphisms of , . Let be a complex manifold, and let

be a holomorphic map, defining a holomorphic family of endomorphisms of . Assume that for all there exists at least one parameter such that for all periodic points of exact period for ,

Let

where we denote by the determinant of the Jacobian matrix, and let be the closure of in . As we will see below, is an analytic hypersurface in .

Proposition 3.12.

There exists a sequence of holomorphic maps such that:

  1. For all , is a monic polynomial of degree

  2. if and only .

Moreover, if , there exists and dividing such that , , and is an eigenvalue of .

Proof.

Set

It is an analytic hypersurface of . Let be defined by

It is a holomorphic map on the regular set of . Now let be the projection of on . By Remmert’s mapping theorem, is an analytic subset of .

Since we assumed that does not have persistent parabolic cycles, cannot be all of , so it is a proper analytic subset of . Now define

and

where denotes the set of periodic points of exact period for .

Note that is an open and dense subset of , since it is the complement of a closed proper analytic subset of . By the implicit function theorem and the definition of , the periodic points move holomorphically in , i.e., for any and any , there is a neighbourhood of in and a holomorphic map . This implies that is holomorphic on , as it is clearly holomorphic with respect to . Since it is locally bounded, Riemann’s extension theorem implies that it is in fact holomorphic on all of .

Now notice that for all , divides the multiplicity of every root of the polynomial . Indeed, if is such that , then it is also the case for the other points of the cycle, namely the , . So, for every , there is a unique monic polynomial map such that . Since is globally holomorphic, so is , and for all , is a monic polynomial of degree .

Let us now analyse the zero set of : it is the same as the zero set of . It is clear from the definitions that if then , and that conversely, if and , then . Let us now assume that and that . Since is dense in , there exists a sequence with . Since we have , there exists a sequence of points in such that converges to zero. This means that is in the closure of , thus proving the second item.

Let us now prove the last claim: we assume that . Let and be as above. By compactness of , up to extracting a subsequence we may assume that converges to some , which must satisfy and . Since , the exact period of is some integer dividing . By the implicit function theorem, if 1 were not an eigenvalue of , then for parameters close enough to there would be only one periodic point of period dividing near . But that would contradict the fact that each are distinct points of period for that are all converging to as . ∎

For , we denote by the set defined by

The set is an analytic hypersurface of .

Define now

By the Lelong-Poincaré equation, we have that , where is the (normalized) current of integration on . Likewise, we have

Let also be the sum of the Lyapunov exponents of with respect to its equilibrium measure . We recall here a useful result by Berteloot-Dupont-Molino (see also [berteloot2017distortion]).

Lemma 3.13 ([berteloot2008normalization], Lemma 4.5).

Let be an endomorphism of of algebraic degree . Let and let be the set of repelling periodic points of exact period for , such that . Then for large enough, .

Note that in particular, this implies that , where is the set of periodic points of exact period .

Theorem 3.14 ([berteloot2008normalization]).

Let be an endomorphism of of algebraic degree and let . Then

Actually, the statement appearing in [berteloot2008normalization] involves an average on the set of all repelling cycles of period instead of , but using Lemma 3.13, is is not difficult to see that the two statements are equivalent.

Recall the definition of an Axiom A endomorphism . Let denote the non-wandering set, i.e.

We say that is Axiom A if periodic points are dense in , and is hyperbolic.

We now state the main convergence result of this section.

Theorem 3.15.

Assume that there is at least one parameter such that is Axiom A and is not dense in . We have , the convergence taking place in .

By taking on both sides, we obtain the following equidistribution result as a corollary.

Corollary 3.16.

Under the same assumptions as in Theorem 3.15, for any outside of a polar set we have that

the convergence taking place in the sense of currents.

In order to prove the convergence in Theorem 3.15, in the spirit of [bassanelli2009lyapunov] we first study the convergence of suitable modifications of the potentials .

Definition 3.17.

We define the functions and as follows.

Lemma 3.18.

For any , the sequence of maps converges pointwise and to on .

Proof.

Fix , and let . We have:

which is locally bounded from above. Moreover:

In the last two equalities, we used Theorem 3.14 and Lemma 3.13. Therefore the sequence of maps converges pointwise to on , and since the ’s are plurisubharmonic functions that are locally uniformly bounded from above, by Hartogs lemma, the convergence also happens in . ∎

Lemma 3.19.

For any , the sequence of maps converges pointwise and to on .

Proof.

First notice that, for every , we have

Therefore,

which gives the pointwise convergence. Moreover, is uniformly locally bounded from above, according to Lemma 3.18 and since . Therefore, by Lebesgue’s dominated convergence theorem it converges to . ∎

Proof of Theorem 3.15.

First, note that the sequence does not converge to . Indeed, by assumption there is and such that no cycle of has a Jacobian in . Moreover, since is Axiom A, its cycles move holomorphically for near , which implies that . Therefore the sequence does not converge to .

Let be a psh function such that a subsequence converges to . Let . We have to prove that .

First, let us prove that . Let and be the ball of radius centered at in . Using the submean inequality and the convergence of , we have

Then letting , we have that , which gives the desired inequality.

Now let us prove the opposite inequality. Here we assume additionally that . Let , and let us first notice that

(3)

Indeed, for any we have

and by Fatou’s lemma and the pointwise convergence of we get

which proves (3).

Suppose now to obtain a contradiction that . Since is continuous and is upper semi-continuous, there is and a neighborhood of such that for all ,

We may assume without loss of generality that