# Entropy of holomorphic and rational maps: a survey

###### Abstract

We give a brief survey on the entropy of holomorphic self maps
of compact Kähler manifolds, and rational dominating self maps
of smooth projective varieties. We emphasize the connection
between the entropy and the spectral radii of the induced action of
on the homology of the compact manifold. The main
conjecture for the rational maps states that modulo birational
isomorphism all various notions of
entropy and the spectral radii are equal.

2000 Mathematics Subject
Classification: 28D20, 30D05, 37F, 54H20.

Keywords and phrases: Holomorphic self maps, rational dominating self maps, dynamic spectral radius, entropy.

## 1 Introduction

The subject of the dynamics of a map has been
studied by hundreds, or perhaps thousands, of mathematicians, physicists and
other scientists in the last 150 years. One way to classify the *complexity*
of the map is to assign to it a number ,
which called the *entropy* of . The entropy of should
be an invariant with respect to certain *automorphisms* of
. The complexity of the dynamics of should be reflected by ,
i.e. the larger the more complex is its dynamics.

The subject of this short survey paper is mostly concerned with the entropy of a holomorphic , where is a compact Kähler manifold, and the entropy of a rational map of , where is a smooth projective variety. In the holomorphic case the author [12, 13, 14] showed that entropy of is equal to the logarithm of the spectral radius of the finite dimensional on the total homology group over .

Most of the paper is devoted to the rational map which can be assumed dominating. In this case we have some
partial results and inequalities. We recall three
possible definition of the entropy which are
related as follows: . The analog of the
*dynamical* homological spectral radius are given by
, and , where the three quantities
can be viewed as the volume growth.
It is known that .
is a birational invariant. I.e. let be a smooth
projective variety such that there exists
which is a *birational* map. Then can be lifted
to dominating , and .
However does not have to be equal to .
The main conjecture of this paper are the equalities

(1.1) |

for some birationally equivalent to . For polynomial automorphisms of , which are birational maps of , the results of the papers [16, 36, 8] prove the above conjecture for . Some other examples where this conjecture holds are given in [21, 22].

The pioneering inequality of Gromov [20] uses
basic results in entropy theory, Riemannian geometry and complex manifolds.
Author’s results are using basic results in entropy theory,
algebraic geometry and the results of Gromov, Yomdin [38] and
Newhouse [31]. From the beginning of 90’s the notion of
*currents* were introduced in the study of the dynamics of
holomorphic and rational maps in several complex variables.
See the survey paper [35]. In fact the inequality
proved in [8, 9, 10] and [23],
as well as most of the
results in are derived [21, 22], are
using the theory of currents.

The author believes that in dealing with the notion of the entropy solely, one can cleverly substitute the theory of currents with the right notions of algebraic geometry. All the section of this paper except the last one are not using currents. It seems to the author that to prove the conjecture (1.1) one needs to prove a correct analog of Yomdin’s inequality [38].

We now survey briefly the contents of this paper. §2 deals with the entropy of , where first is a compact metric space and is continuous, and second is compact Kähler and is holomorphic. §3 is devoted to the study of three definitions of entropy of a continuous map , where is an arbitrary subset of a compact metric space . In §4 we discuss rational dominating maps , where is a smooth projective variety. §5 discusses various notions and results on the entropy of rational dominating maps. In §6 we discuss briefly the recent results, in particular the inequality which uses currents.

It is impossible to mention all the relevant existing literature, and I apologize to the authors whose papers were not mentioned. It is my pleasure to thank S. Cantat, V. Guedj, J. Propp, N. Sibony and C.-M. Viallet for pointing out related papers.

## 2 Entropy of continuous and holomorphic maps

The first rigorous definition of the entropy was introduced
by Kolmogorov [27]. It assumes that is a probability
space , where preserves the probability measure
. It is denoted by , and is usually referred under
the following names: *metric* entropy,
*Kolmogorov-Sinai* entropy, or *measure entropy*.
is an invariant under measure preserving invertible
automorphism , i.e. .

Assume that is a compact metric space and a continuous map.
Then Adler, Konheim and McAndrew defined the *topological* entropy
[1]. has a *maximal characterization* in terms of
measure entropies . Let be the Borel sigma algebra
generated by open set in . Denote by the compact
space of probability measures on . Let be the compact set of all -invariant probability measures.
(Krylov-Bogolyubov theorem implies that .)
Then the *variational principle* due to
Goodwyn, Dinaburg and Goodman [18, 7, 17] states
. Hence depends
only to the topology induced by the metric on .
In particular, is invariant under any homeomorphism .

The next step is to consider the case where is a compact smooth
manifold and is a differentiable map, i.e. , where is usually at least . The most remarkable
subclasses of are strongly hyperbolic maps, and in particular
axiom A diffeomorphisms [34]. The dynamics of an Axiom A diffeomorphism
on the nonwandering set
can be coded as a *subshift of a finite type* (SOFT), hence
its entropy is given by the exponential growth of the periodic
points of , i.e. ,
where
the number of periodic points of of period .

It is well known in topology that
can be estimated below by
the Lefschetz number of . Let denote
the total homology group of over , i.e.
,
the direct sum of
the homology groups of of all dimensions with coefficients in .
Then induces the linear operator , where . The *Lefschetz number* of is defined as . Intuitively,
is the algebraic sum of -periodic points of , counted with their multiplicities.

Denote by and the spectral radius of and respectively. Recall that and . Hence

The arguments in [34] yield that for any in the subset of an Axiom A diffeomorphism, ( is defined in [34]), one has the inequality for each . ( is dense in [34, Thm 3.1].) Hence for any one has the inequality [34, Prop 3.3]

(2.1) |

It was conjectured in [34] that the above inequality holds for any differentiable .

Let be the *topological degree* of .
Then .
Hence . It was shown by Misiurewicz and Przytycki [30] that if
then . However this
inequality may fail if .
The entropy conjecture (2.1) for a smooth , i.e. ,
was proved by Yomdin [38]. Conversely, Newhouse [31] showed that
for the *volume growth* of smooth submanifolds of
is an upper bound for . See also a related upper bound in
[32].

This paper is devoted to study the entropy of where is a complex Kähler manifold and is either holomorphic map, or is a projective variety and is a rational map dominating map. We first discuss the case where is holomorphic.

Let be the complex projective space. Then is holomorphic if and only if is a rational map. Hence is the cardinality of the set for all but a finite number of . So in this case. Lyubich [28] showed that . Gromov in preprint dated 1977, which appeared as [20], showed that if is holomorphic then . It is well known in this case .

In [12] the author showed that if is a complex projective variety and is holomorphic, then . Note that one can view a linear operator on , i.e. the total homology group with integer coefficients. Hence can be represented by matrix with integer coefficients. In particular, is an algebraic integer, i.e. the entropy is the logarithm of an algebraic integer. (This fact was observed in [3] for certain rational maps.) In [14] the author extended this result to a compact Kähler manifold.

Examples of the dynamics of biholomorphic maps , where is a compact surface which is Kähler but not necessary a projective variety, are given in [6, 29]. See also [9] for higher dimensional examples. In summary, the entropy of a holomorphic self map of a compact Kähler manifold is determined by the spectral radius of the induced action of on the total homology of .

## 3 Definitions of entropy

In this paper will be always a compact matrix space with the metric . Let be a nonempty set, and assume that is a continuous map with respect the topology induced by the metric on . For and let

So and the sequence is nondecreasing. Hence for each is a distance on . Furthermore, each metric induces the same topology as the metric . For a set is called separated if for any . For any set denote by the maximal cardinality of separated set . Clearly, if , if , and if .

We now discuss a few possible definitions of the entropy of . Let . Then

(3.1) |

We call the *topological entropy* of .
(Note that if for some
and .) Equivalently, can be viewed
as the *exponential growth* of the maximal number of
separated sets (in ).

Clearly if .
Then is the *topological entropy* of .

Bowen’s definition of the entropy of , denoted here as , is given as follows [37, §7.2]. Let be a compact set. Then is a compact set with respect to . Hence . Then is the supremum of for all compact subsets of . I.e.

When no ambiguity arises we let . Clearly, if is compact then . (It is known that for a compact , i.e. [37].)

Since for any it follows that . The following example, pointed out to me by Jim Propp, shows that it is possible that . Let be the closed and the open unit disk respectively in the complex plane. Let and assume that . It is well known that . It is straightforward to show that . Let be a compact set. Let be the closed disk or radius , centered at , such that . Since it follows that .

Our last definition of the entropy of , denoted by , or simply is based on the notion of the orbit space. Let be the space of the sequences , where each . We introduce a metric on :

Then is a compact metric space, whose diameter is twice the diameter of .
The *shift* transformation is given by
.
Then
, i.e.
is a Lipschitz map.
Given then the -*orbit* of , or simply the orbit of ,
is the point
. Denote by ,
the * orbit space*, the
set of all -orbits. Note that .
Hence . , the
restriction of to the orbit space, is “equivalent” to the
map . I.e. let be given by
. Clearly is a homeomorphism.
Then the following diagram is commutative:

Let be the closure of with respect to the metric defined above. Since is compact, is compact. Clearly . Following [12, §4] we define to be equal to the topological entropy of :

When no ambiguity arises we let . Since the closure of is , it is not difficult to show that .

Observe first that if is a compact subset of then is the topological entropy of . Indeed, since is continuous and is compact . Since is a homeomorphism, the variational principle implies that .

We observe next that . Let

Then . Hence . Hence . The arguments of the proof [22, Lemma 1.1] show that . (In [22] is our , and is the topological entropy with respect to the metric . Since and induce the Tychonoff topology on it follows that .)

Our discussion of various topological entropies for is very close to the discussion in [25]. The notion of the entropy can be naturally extended to the definition of the entropy of a semigroup acting on [15]. See [5] for other definition of the entropy of a free semigroup and [11] for an analog of Misiurewicz-Przytycki theorem [30].

## 4 Rational maps

In this section we use notions and results from algebraic geometry
most of which can be found in [19].
Let , sometimes denotes as ,
be the homogeneous coordinates the -dimensional complex projective space .
Recall that a map is called a rational map if
there exists nonzero coprime homogeneous polynomials
of degree such that .
Equivalently lifts to a homogeneous map
.
The set of singular points of
, denoted by ,
sometimes called the *indeterminacy locus* of
, is given by the system .
is closed subvariety of of codimension
at least. is holomorphic if and only if ,
i.e. the above system
of polynomial equations has only the solution .

Let be an irreducible algebraic variety.
It is well known that can be embedded as
an irreducible subvariety of . For simplicity of notation
we will assume that is an irreducible variety of .
So can be viewed as a homogeneous irreducible variety , given as the zero set of homogeneous polynomials
. is called *smooth* if
is a complex compact manifold in the neighborhood of .
A nonsmooth is called a *singular* point.
The set of singular points of , denoted by ,
is a strict subvariety of . is called *smooth* if
. Otherwise is called *singular*.

Let be a
rational map.
Then one can extend
to such that is a strict subvariety of and
. is not unique, but the
can be viewed as . is the
set of the points where is not holomorphic. is
strict projective variety of , (), and each irreducible component of is at least of codimension
. The assumption means that for each
. It is known that , the closure
of , is a homogeneous irreducible subvariety of .
Furthermore either , in this case is called a
*dominating* map, or . In the second case
the dynamics of is reduced to the dynamics of the rational map
. Continuing in the same manner we
deduce that there exists a finite number of strictly descending
irreducible subvarieties
such that is a rational dominating
map. (Note that may be a singular variety.)
Thus one needs only to study the dynamics of a rational
dominating map , where may be a singular variety.

The next notion is the resolution of singularities of and .
An irreducible projective variety birationally equivalent to
if the exists a birational map .
is called a *blow up* of if there exists a birational
map such is holomorphic. is called a blow
down of . Hironaka’s result claims that any irreducible
singular variety has a smooth blow up . Let
be a rational dominating map. Let be a
birationally equivalent to . Then lifts to a rationally
dominating map . Hence to study the dynamics
of one can assume that is rational
dominating map and is smooth.
Hironaka’s theorem implies that there exists a smooth blow up of
such that lifts to a holomorphic map .
Then one has the induced dual linear maps on the homologies and the
cohomologies of and :

We will view the homologies as homologies with coefficients in , and hence the cohomologies , which are dual to , as de Rham cohomologies of differential forms. (It is possible to consider these homologies and cohomologies with coefficients in [12].) Recall that the Poincaré duality isomorphism , which maps a -cycle to closed form. (.) Then one defines and its dual as

It can be shown that do not depend on the resolution of
, i.e. on .
Let be the spectral radii
of respectively. (As noted above can be
represented by matrix with integer entries. Hence is
an algebraic integer.)
Then the *dynamical* spectral radius of is defined as

(4.1) |

(Note that is a limit of algebraic integers, so it may not be an algebraic integer.)

Assume that
is holomorphic. Then are the standard
linear maps on homology and cohomology of . So
and .
It was shown by the author that [12].
This equality followed from the observation that
is the *volume growth* induced by .
View as a submanifold of , is endowed the induced
Fubini-Study Riemannian metric and
with the induced Kähler closed form .
Let be any irreducible variety of complex dimension . Then the volume of
is given by the Wirtinger formula

(4.2) |

(See [12, (2)] and [13, (2.8)].) From the well known equality , for any norm on , it follows that . Newhouse’s result [31] claims that . Combining this inequality with Yomdin’s inequality [38] we deduced in [12]:

(4.3) |

which is a logarithm of an algebraic integer.

Let be the cone generated by the homology classes corresponding to all irreducible projective varieties . Note that . Let be the subspace generated by the homology classes of projective varieties in . Then and denote . Using the theory of nonnegative operators on finite dimensional cones , e.g. [4], it follows that .

Assume again that is rational dominant. Then so and denote . Hence we can define , the volume growth induced by , as in (4.2) [12, 13]. Similar quantities were considered in [33, 3]. It is plausible to assume that and we conjecture a more general set of equalities in the next section.

It was shown in [12] that the results on of Friedland-Milnor [16] imply the inequalities

(4.4) |

for certain polynomial biholomorphisms of , (which are birational maps of .)

It was claimed in [12, pp. 367] that if (4.4) holds then the sequence , converges. (This is probably wrong. One can show that under the assumption (4.4) for all rational dominant maps one has for any .) It was also claimed in [12, Lemma 3] that (4.4) holds in general. Unfortunately this result is false, and a counterexample is given in [23, Remark 1.4]. Note that if holomorphic then equality in (4.4) holds. Hence all the results of [12] hold for holomorphic maps.

## 5 Entropy of rational maps

Let be a rational dominating map. (We will assume that is not holomorphic unless stated otherwise.) In order to define the entropy of we need to find the largest subset such that . Let is the collection of all such that for . Then is open and is a strict subvariety of . Clearly for . Then is set. Let the the closed -Kähler form on . Then is a canonical volume form on . Hence , i.e. has the full volume.

Since is a compact Riemannian manifold, is a compact metric space. Thus we can define the three entropies in §3. So

Assume that is a polynomial dominating map. Then lifts to a rational dominating map , which may be holomorphic. Hence . Assume that is a proper map of . Recall that one point compactification of , denoted by , is homeomorphic to the sphere . Then lifts to a continuous map . Thus we can define the entropy . It is not hard to show that .

Let be the orbit space of , and let be its closure. is closely related to the graph construction discussed in [20, 12, 13, 14, 15] as well as in other papers. Denote by the closure of the set in . Then is an irreducible variety of dimension in . Note that the projection of on the first or second factor of in is . Without a loss of generality we may assume that is smooth.

Otherwise let be a blow up of such that lifts to a holomorphic map . Let . Then is smooth variety of dimension . Note that given by is a blow up of . Lift to . Then is a blow up , hence is smooth.

Let be a closed irreducible smooth variety of dimension such that the projection of on the first or second component is . Define

Note that is an irreducible variety of dimension in for . Note that is a invariant compact subset of , i.e. . Let . , viewed as a submanifold of , is endowed the induced Fubini-Study Riemannian metric and with the Kähler form . Then has the corresponding induced product Riemannian metric, and is Kähler, with the form . Let be volume of the variety . Then the volume growth of is given by

(5.1) |

The fundamental inequality due to Gromov [20]

(5.2) |

Since the paper of Gromov was not available to the general public until the appearance of [20], the author reproduced Gromov’s proof of (5.2) in [13, 14]. Using the above inequality Gromov showed that for any holomorphic .

Let be rational dominating. Then . Hence

(5.3) |

If is smooth then Gromov’s inequality yields that

(5.4) |

###### Conjecture 5.1

Let be a smooth projective variety and be a rational dominating map. Then there exists a smooth projective variety and a birational map , such that the lifting satisfies (1.1).

We now review briefly certain notions, results and conjectures in [14, S3]. Let be as above, and denote by the projection of on the -th component of in for . Since and , then is finite and consists of exactly distinct points for a generic for . One can define a linear map given by . (This is an analogous definition of , where is dominating.) One can show that . Let .

is called a *proper* if each is
finite to one.
Assume that is proper.
Then

(5.5) |

It is conjectured that for a proper

(5.6) |

Note that if is dominating and holomorphic then is proper, and the above conjecture holds.

We close this section with observations and remarks which are not in [14]. Assume that be a rational dominating and smooth. Then is a blow up of , and can be identified with . It is straightforward to show that .

It seems to the author that the arguments given in [14, Proof Thm 3.5] imply that (5.5) holds for any smooth variety of dimension such that . Suppose that this result is true. Let be rational and dominating. Assume that is smooth. Then (5.5) would imply that . Applying the same inequality to and combining it with (5.4) one would able to deduce:

(5.7) |

## 6 Currents

Many recent advances in complex dynamics in several complex variables were
achieved using the notion of a *current*. See for example the
survey article [35]. Recall that on an -dimensional
manifold a current of degree is a linear functional
on all smooth -differential forms with a compact
support, where is a nonnegative integer.

Let be a meromorphic dominating self map of
a compact Kähler manifold of complex dimension , with the Kähler
form .
Let be a pullback of . Then
is a current on .
Define the *p-dynamic degree* of by

It is shown in [8] that the dynamical degrees are invariant with respect to a bimeromorphic map , where is a compact Kähler manifold. (See also [23] for the case where are projective varieties.) Moreover

(6.1) |

Assume that is a projective variety. It can be shown that the dynamic degree is equal to for , which are defined in (4.2), where . Hence

(6.2) |

where is defined in (4.2).
Thus can be viewed as the *algebraic entropy*
of [3]. [24, Lemma 4.3] computes
for a large class of automorphisms of , and see also [10, 23].
Combine (5.4) with (6.1) and (6.2) to
deduce the inequality , which was conjectured in
[13, Conjecture 2.9].

Consider the following example [22, Example 1.4]. Since is proper we have . Clearly is the domain of attraction of the fixed point . Hence . Lift