# Topological entropy for locally linearly compact vector spaces
and field extensions††thanks: This work was supported by EPSRC Grant N007328/1
Soluble Groups and Cohomology.

Ilaria Castellano
<EMAIL_ADDRESS>
University of Southampton
Building 54, Salisbury Road - 50171BJ Southampton (UK)

###### Abstract

Let $\mathbb{K}$ be a discrete field and $(V,\phi)$ a flow over the category
of locally linearly compact $\mathbb{K}$-spaces. Here we give the formulas to
compute the topological entropy of $(V,\phi)$ subject to the extension or the
restriction of scalars.

##  1 Introduction

Usually, an entropy function over a category $\mathcal{C}$ is an invariant

$\mathrm{h}\colon\mathrm{Flow}(\mathcal{C})\to\mathbb{N}\cup\\{\infty\\},$

of the category $\mathrm{Flow}(\mathcal{C})$ whose objects are the flows111A
flow in $\mathcal{C}$ is a pair $(C,\gamma)$ consisting of an object $C$ of
$\mathcal{C}$ and an endomorphism $\gamma$ of $C$. $(C,\gamma)$ over
$\mathcal{C}$. The first appearance of topological entropy was in 1965 when
Adler, Konheim and McAndrew [1] defined it for continuous self-maps of compact
spaces. Afterwards, Bowen [3] and Hood [6] brought a notion of topological
entropy into the world of uniformly continuous self-maps on uniform spaces
(also known as uniform entropy) opening up the possibility for computing
entropy for any given continuous endomorphism $\phi\colon G\to G$ of a
topological group $G$. E.g., profinite groups and locally compact groups.

Interest in the locally compact case is currently very high. Not only because
locally compact groups are omnipresent in several areas of mathematics but
also because there exists a strict relation between the topological entropy
$h_{top}$ and the scale function defined in [9] whenever one considers
endomorphisms of totally disconnected locally compact groups (see [2, 7] for
details). By analogy with the topological entropy $h_{top}$ for totally
disconnected locally compact groups, the topological entropy
$\mathrm{ent}^{*}$ has been introduced in the category
${}_{\mathbb{K}}\mathrm{LLC}$ of locally linearly compact $\mathbb{K}$-spaces
[5] whenever $\mathbb{K}$ is a discrete field. The main motivation for
studying such an entropy function is to reach a better understanding of
$h_{top}$ by means of the more rigid case of locally linearly compact spaces.

Recall that a topological vector space $V$ over a discrete field $\mathbb{K}$
is said to be locally linearly compact if $V$ admits a local basis at 0
consisting of linearly compact open $\mathbb{K}$-subspaces. In particular, a
topological vector space $E$ is linearly compact when

  1. (LC1)

it is a Hausdorff space in which there is a fundamental system of
neighbourhoods of 0 consisting of linear subspaces of $E$;

  2. (LC2)

any filter base on $E$ consisting of closed linear varieties has a non-empty
intersection.

Notice that any locally linearly compact $\mathbb{K}$-space $V$ is clearly a
totally disconnected locally compact abelian group - whenever $\mathbb{K}$ is
a finite field - and

$h_{top}(V,\phi)=\mathrm{ent}^{*}(V,\phi)\cdot\log|\mathbb{K}|,$

for every continuous endomorphism $\phi\colon V\to V$ (see [5, Proposition
3.9]).

In this paper, we see that both the procedures of restriction and extension of
scalars continue to be available in the context of locally linearly compact
vector spaces (see § 3.3). Namely, given a finite field extension
$\mathbb{F}\leq\mathbb{K}$, there exist functors

$\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(-)\colon{}_{\mathbb{K}}\mathrm{LLC}\to{}_{\mathbb{F}}\mathrm{LLC}\quad\text{and}\quad\mathrm{Ind}^{\mathbb{K}}_{\mathbb{F}}(-)\colon{}_{\mathbb{F}}\mathrm{LLC}\to{}_{\mathbb{K}}\mathrm{LLC}$

forming an adjunction by

$\mathrm{CHom}_{\mathbb{K}}(\mathrm{Ind}^{\mathbb{K}}_{\mathbb{F}}(-),-)\cong\mathrm{CHom}_{\mathbb{F}}(-,\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(-)).$

Therefore it is reasonable to ask whether the topological entropy is effected
by the change of scalar fields and the question is addressed by the formulas

$\mathrm{ent}_{\mathbb{F}}^{*}(\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(V,\phi))=[\mathbb{K}:\mathbb{F}]\cdot\mathrm{ent}_{\mathbb{K}}^{*}(V,\phi)\quad\text{and}\quad\mathrm{ent}_{\mathbb{L}}^{*}(\mathrm{Ind}_{\mathbb{K}}^{\mathbb{L}}(V,\phi))=\mathrm{ent}_{\mathbb{K}}^{*}(V,\phi),$

that are proved in Theorem 4.4 for $\mathbb{F}\leq\mathbb{K}\leq\mathbb{L}$
finite field extensions. Moreover, the condition on finite extension is
necessary. There are no surprises in either the statement or the proof of
Theorem 4.4, but its content does not appear anywhere else. Thus this paper is
intended to be just a complement to [4, 5] where $\mathrm{ent}^{*}$ for
locally linearly compact $\mathbb{K}$-spaces has been introduced and studied.

##  2 Preliminaries and basic properties

Throughout, the topology on arbitrary fields will always be the discrete
topology. We summarise here some of the properties of linearly compact vector
spaces and the topological entropy $\mathrm{ent}^{*}$ (see [5] for
references). We will use them in the rest of the paper sometimes with no
previous acknowledgement.

###  2.1 Linearly compact vector spaces

A topological $\mathbb{K}$-space is said to be linearly topologized if it
admits a local basis at 0 of $\mathbb{K}$-linear subspaces. For linearly
topologized $\mathbb{K}$-spaces $V,V^{\prime}$ and $W$ such that $W\leq V$,
the following hold:

  1. (P1)

If $W$ is linearly compact, then $W$ is closed in $V$.

  2. (P2)

If $V$ is linearly compact and $W$ is closed, then $W$ is linearly compact.

  3. (P3)

Linear compactness is preserved by continuous homomorphisms.

  4. (P4)

If $V$ is discrete, then $V$ is linearly compact if and only if $V$ has finite
dimension.

  5. (P5)

If $W$ is closed, then $V$ is linearly compact if and only if $W$ and $V/W$
are linearly compact.

  6. (P6)

The direct product of linearly compact $\mathbb{K}$-spaces is linearly
compact.

  7. (P7)

An inverse limit of linearly compact $\mathbb{K}$-spaces is linearly compact.

  8. (P8)

If $V$ is linearly compact, then $V$ is complete.

  9. (P9)

Let $V$ be linearly compact. Every continuous $\mathbb{K}$-linear $f\colon
V\to V^{\prime}$ is open onto its image.

  10. (P10)

Linearly compact $\mathbb{K}$-spaces satisfy Lefschetz duality.

  11. (P11)

Every linearly compact $\mathbb{K}$-space is topologically a direct product of
copies of $\mathbb{K}$.

###  2.2 Complete tensor product and profinite plagiarism

Given linearly compact $\mathbb{K}$-spaces $V$ and $W$, a linearly compact
$\mathbb{K}$-space $T$ (together with a bilinear map $b\colon V\times W\to T$)
is a complete tensor product of $V$ and $W$ over $\mathbb{K}$ if it satisfies
the following universal property: for every linearly compact
$\mathbb{K}$-space $Z$ and every continuous bilinear map $\beta\colon V\times
X\to Z$ there exists a continuous bilinear map $\hat{\beta}\colon T\to Z$ such
that

---
$\textstyle{V\times
X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b}$$\scriptstyle{\beta}$$\textstyle{T\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\beta}}$$\textstyle{Z}$
(2.1)

commutes. It is easy to see that if the complete tensor product exists, it is
unique up to isomorphism. We denote it by $V\widehat{\otimes}_{\mathbb{K}}W$.

######  Proposition 2.1.

With the above notation, the complete tensor product
$V\widehat{\otimes}_{\mathbb{K}}W$ exists. In fact, if

$V=\varprojlim_{i\in I}V_{i}\quad\text{and}\quad W=\varprojlim_{j\in J}W_{j},$

where each $V_{i}$ (respectively, $W_{j}$) is a finite-dimensional
$\mathbb{K}$-space endowed with the discrete topology, then

$V\widehat{\otimes}_{\mathbb{K}}W=\varprojlim_{i\in I,j\in
J}(V_{i}\otimes_{\mathbb{K}}W_{j}),$

where $V_{i}\otimes_{\mathbb{K}}W_{j}$ is the usual tensor product as abstract
$\mathbb{K}$-spaces. In particular, $V\widehat{\otimes}_{\mathbb{K}}W$ is the
(linearly compact) completion of $V\otimes_{\mathbb{K}}W$, where
$V\otimes_{\mathbb{K}}W$ has the topology for which a fundamental system of
neighbourhoods of 0 are the kernels of the natural maps

$V\otimes_{\mathbb{K}}W\to V_{i}\otimes_{\mathbb{K}}W_{j}\quad(i\in I,j\in
J).$

###### Proof.

Basically, we use the strategy developed in the proof of [8, Lemma 5.5.1]. For

$V\times W\cong_{top}\varprojlim_{i\in I,j\in J}(V_{i}\times W_{j}),$ (2.2)

there exists a canonical continuous $\mathbb{K}$-bilinear map

$\iota\colon V\times W\to\varprojlim_{i\in I,j\in
J}(V_{i}\otimes_{\mathbb{K}}W_{j}).$ (2.3)

Set $v\hat{\otimes}w:=\iota(v,w)$ for any $(v,w)\in V\times W$. Since every
linearly compact $\mathbb{K}$-space is the inverse limit of its finite-
dimensional $\mathbb{K}$-quotients, it suffices to check the universal
property only for an arbitrary $\mathbb{K}$-space, say $F$, of finite
dimension. Suppose $\beta\colon V\times W\to F$ to be continuous and
$\mathbb{K}$-bilinear. For $\dim_{\mathbb{K}}(F)<\infty$, there is a pair of
indices $(i,j)\in I\times J$ and a continuous $\mathbb{K}$-bilinear map
$\beta_{ij}\colon V_{i}\times Wj\to F$ such that the diagram

$\textstyle{V\times
W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{F}$$\textstyle{V_{i}\times
W_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{ij}}$
(2.4)

commutes222One can check that the dual factorisation does always exist in the
dual category ${}_{\mathbb{K}}\mathrm{Vect}$. Now one uses the universal
property of $V_{i}\otimes_{\mathbb{K}}W_{j}$ to produce a continuous
$\mathbb{K}$-bilinear map $\hat{\beta}_{ij}\colon
V_{i}\otimes_{\mathbb{K}}W_{j}\to F$ such that $\hat{\beta}_{ij}(v_{i}\otimes
w_{j})=\beta(v_{i},w_{j})$. Finally, define $\beta\colon
V\widehat{\otimes}_{\mathbb{K}}W\to F$ to be

$\textstyle{V\widehat{\otimes}_{\mathbb{K}}W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{V_{i}\otimes_{\mathbb{K}}W_{j}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\qquad\hat{\beta}_{ij}}$$\textstyle{F.}$
(2.5)

∎

Moreover, the right-exact covariant functor
$V\widehat{\otimes}_{\mathbb{K}}-\colon{}_{\mathbb{K}}\mathrm{LC}\to{}_{\mathbb{K}}\mathrm{LC}$
is additive and satisfies

$V\widehat{\otimes}_{\mathbb{K}}\mathbb{K}\cong_{top}V,$ (2.6)

where the isomorphism is natural in $V$. Clearly, similar properties hold for
$-\widehat{\otimes}_{\mathbb{K}}W$.

######  Remark 2.2.

Since the complete tensor product is unique up to isomorphism (when it does
exist), the construction above does not depend on the (inverse
limit)-representation of $V$ and $W$.

######  Remark 2.3.

The tensor products here are merely vector spaces, but complete tensor
products can be defined for more general object, e.g., linearly compact
modules over linearly compact commutative rings.

###  2.3 Topological entropy on ${}_{\mathbb{K}}\mathrm{LLC}$

Let $(V,\phi)$ be a flow over ${}_{\mathbb{K}}\mathrm{LLC}$ and denote by
$\mathcal{B}(V)$ the collection of all linearly compact open
$\mathbb{K}$-subspaces of $V$. Then the topological entropy of $(V,\phi)$ is
defined to be

$\mathrm{ent}_{\mathbb{K}}^{*}(V,\phi)=\sup_{U\in\mathcal{B}(V)}H_{\mathbb{K}}^{*}(\phi,U),$
(2.7)

where
$H_{\mathbb{K}}^{*}(\phi,U)=\lim_{n\to\infty}\frac{1}{n}\dim_{\mathbb{K}}(U/(U\cap\phi^{-1}U\cap\ldots\cap\phi^{-n+1}U))$.
Moreover, the linearly compact open $\mathbb{K}$-space
$U\cap\phi^{-1}U\cap\ldots\cap\phi^{-n+1}U$ is called $n$-cotrajectory of
$\phi$ in $U$ and it is denoted by $C^{\mathbb{K}}_{n}(\phi,U)$.

  1. eP1

(Invariance under conjugation) For every topological isomorphism $\alpha\colon
V\to W$ of locally linearly compact $\mathbb{K}$-spaces,
$\mathrm{ent}_{\mathbb{K}}^{*}(W,\alpha\phi\alpha^{-1})=\mathrm{ent}_{\mathbb{K}}^{*}(V,\phi).$

  2. eP2

(Monotonicity) For a closed $\mathbb{K}$-subspace $W$ of $V$ such that
$\phi(W)\leq W$,

$\mathrm{ent}_{\mathbb{K}}^{*}(V,\phi)\geq\max\\{\mathrm{ent}_{\mathbb{K}}^{*}(W,\phi\restriction_{W}),\mathrm{ent}_{\mathbb{K}}^{*}(V/W,\overline{\phi})\\}$

where $\overline{\phi}\colon V/W\to V/W$ is induced by $\phi$,

  3. eP3

(Logarithmic law) If $k\in\mathbb{N}$, then
$\mathrm{ent}_{\mathbb{K}}^{*}(V,\phi^{k})=k\cdot\mathrm{ent}_{\mathbb{K}}^{*}(V,\phi)$

  4. eP4

(Continuity on inverse limits) Let $\\{W_{i}\mid i\in I\\}$ be a directed
system (for inverse inclusion) of closed $\mathbb{K}$-subspaces of $V$ such
that $\phi(W_{i})\leq W_{i}$. If $V=\varprojlim V/W_{i}$, then
$\mathrm{ent}_{\mathbb{K}}^{*}(\phi)=\sup_{i\in
I}\mathrm{ent}_{\mathbb{K}}^{*}(\overline{\phi}_{W_{i}})$, where any
$\overline{\phi}_{W_{i}}\colon V/W_{i}\to V/W_{i}$ is the continuous
endomorphism induced by $\phi$.

  5. eP5

(Change of basis) For every local basis $\mathcal{B}\subseteq\mathcal{B}(V)$
at 0, one has
$\mathrm{ent}_{\mathbb{K}}^{*}(V,\phi)=\sup_{U\in\mathcal{B}}H_{\mathbb{K}}^{*}(\phi,U)$.

Notice that the topological entropy over ${}_{\mathbb{K}}\mathrm{LLC}$ was
originally denoted in [5] simply by $\mathrm{ent}^{*}$. Here we need to keep
in mind the field we are working on.

##  3 Change of fields

Firstly, we face the case of linearly compact vector spaces and then we deduce
from it the general case concerning locally linearly compact spaces.

###  3.1 Restriction of scalars over linearly compact vector spaces

Let $\mathbb{F}\leq\mathbb{K}$ be a field extension of finite degree
$[\mathbb{K}:\mathbb{F}]\in\mathbb{N}$. Since any $\mathbb{K}$-space can be
regarded as $\mathbb{F}$-space via the inclusion
$\mathbb{F}\hookrightarrow\mathbb{K}$, one has a functor of abstract vector
spaces

$\mathrm{res}^{\mathbb{K}}_{\mathbb{F}}(-)\colon{}_{\mathbb{K}}\mathrm{Vect}\to{}_{\mathbb{F}}\mathrm{Vect},$
(3.1)

which is usually called restriction of scalars. Now let $V$ be a linearly
compact $\mathbb{K}$-space, i.e., $V$ is topologically a direct product of
(discrete) 1-dimensional $\mathbb{K}$-spaces (endowed with the Tychonoff
topology). Since restriction commutes with products, the underlying abstract
$\mathbb{F}$-space is

$\mathrm{res}^{\mathbb{K}}_{\mathbb{F}}(V)=\prod\mathrm{res}^{\mathbb{K}}_{\mathbb{F}}(\mathbb{K})\cong\prod\mathbb{F}^{[\mathbb{K}:\mathbb{F}]}.$
(3.2)

From the topological point of view, the abstract $\mathbb{F}$-space
$\mathrm{res}^{\mathbb{K}}_{\mathbb{F}}(V)$ comes equipped with the product
topology of discrete finite-dimensional $\mathbb{F}$-spaces. Therefore $V$ is
a linearly compact $\mathbb{F}$-space. Thus one also obtains a restriction
functor

$\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(-)\colon{}_{\mathbb{K}}\mathrm{LC}\to{}_{\mathbb{F}}\mathrm{LC}.$
(3.3)

for linearly compact vector spaces and field extensions.

Notice that $[\mathbb{K}:\mathbb{F}]<\infty$ is a necessary condition, since a
discrete linearly compact vector space has to be finite-dimensional.

###  3.2 Extension of scalars over linearly compact vector spaces

Let $\mathbb{K}\leq\mathbb{L}$ be a finite field extension, i.e., $\mathbb{L}$
is a linearly compact $\mathbb{K}$-space via multiplication. Firstly, notice
that $V\otimes_{\mathbb{K}}W$ is dense in $V\widehat{\otimes}_{\mathbb{K}}W$
since the complete tensor product is a topological completion of the abstract
tensor product. Indeed $V\widehat{\otimes}_{\mathbb{K}}W$ is topologically
spanned by the set of elements of the form $v\hat{\otimes}w$. Therefore one
can define a structure of topological $\mathbb{L}$-space on the linearly
compact $\mathbb{K}$-space $\mathbb{L}\widehat{\otimes}_{\mathbb{K}}V,$ by
extending the natural action

$l\cdot(l^{\prime}\hat{\otimes}v):=(ll^{\prime})\hat{\otimes}v,\quad\text{for
all}\ l,l^{\prime}\in\mathbb{L},v\in V.$ (3.4)

We denote this topological $\mathbb{L}$-space by
$\mathrm{Ind}^{\mathbb{L}}_{\mathbb{K}}(V)$.

######  Proposition 3.1.

For every linearly compact $\mathbb{K}$-space $V$ and a finite extension
$\mathbb{K}\leq\mathbb{L}$, the induced $\mathbb{L}$-space
$\mathrm{Ind}^{\mathbb{L}}_{\mathbb{K}}(V)$ is topologically isomorphic to

  1. 1.

$\mathbb{L}\otimes_{\mathbb{K}}V$ with the topology for which a local basis at
0 are the kernels of the natural $\mathbb{L}$-maps

$\mathbb{L}\otimes_{\mathbb{K}}V\to\mathbb{L}\otimes_{\mathbb{K}}V_{i},\quad
i\in I,$

where $V=\varprojlim_{i\in I}V_{i}$ for some family $\\{V_{i}\\}$ of finite-
dimensional $\mathbb{K}$-spaces.

  2. 2.

$\prod_{J}\mathbb{L}$ with Tychonoff topology, where $V$ has been regarded as
$\prod_{J}\mathbb{K}$.

In particular, $\mathrm{Ind}^{\mathbb{L}}_{\mathbb{K}}(V)$ is a linearly
compact $\mathbb{L}$-space.

###### Proof.

Since $\mathbb{L}\cong\mathbb{K}^{n}$ for some $n$, (1) follows by (2.6) and
Proposition 2.1.

Finally, (2) follows by (1) and the fact that

$V\cong_{top}\varprojlim_{F\in\mathcal{F}(J)}\prod_{j\in F}\mathbb{K},$

where $\mathcal{F}(J)$ is the directed set of all finite subsets of $J$. ∎

Therefore it is well-defined the induction functor

$\mathrm{Ind}_{\mathbb{K}}^{\mathbb{L}}(-)\colon{}_{\mathbb{K}}\mathrm{LC}\to{}_{\mathbb{L}}\mathrm{LC},$
(3.5)

whenever $\mathbb{L}$ is an extension of $\mathbb{K}$ of finite degree.
Moreover, the induction functor $\mathrm{Ind}^{\mathbb{L}}_{\mathbb{K}}(-)$
turns out to be left-adjoint to the restriction functor
$\mathrm{res}^{\mathbb{L}}_{\mathbb{K}}(-)$ as usual, i.e.,
$\mathbb{L}\widehat{\otimes}_{\mathbb{K}}V$ satisfies the following universal
property: for every linearly compact $\mathbb{L}$-space $W$ and every
continuous $\mathbb{K}$-linear map $f\colon V\to W$ there exists a unique
continuous $\mathbb{L}$-linear map
$\hat{f}\colon\mathbb{L}\widehat{\otimes}_{\mathbb{K}}V\to W$ such that

---
$\textstyle{V\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{\mathbb{L}\widehat{\otimes}_{\mathbb{K}}V\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{f}}$$\textstyle{W}$

commutes.

###  3.3 Restriction and Induction over ${}_{\mathbb{K}}\mathrm{LLC}$

Let $\mathbb{F}\leq\mathbb{K}\leq\mathbb{L}$ be finite field extensions. Here
we define the restriction functor

$\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(-)\colon{}_{\mathbb{K}}\mathrm{LLC}\to{}_{\mathbb{F}}\mathrm{LLC}.$
(3.6)

and the induction functor

$\mathrm{Ind}_{\mathbb{K}}^{\mathbb{L}}(-)\colon{}_{\mathbb{K}}\mathrm{LLC}\to{}_{\mathbb{L}}\mathrm{LLC},$
(3.7)

over the category of locally linearly compact vector spaces relying on the
construction given above for linearly compact vector spaces. With abuse of
notation, those functors shall still be denoted by
$\mathrm{Res}^{*}_{\bullet}(-)$ and $\mathrm{Ind}^{*}_{\bullet}(-)$. It
unlikely causes confusion since the functors defined here coincide with (3.3)
and (3.5) over the corresponding subcategory of linearly compact vector
spaces.

Let $V$ be an object in ${}_{\mathbb{K}}\mathrm{LLC}$. The locally linearly
compact $\mathbb{F}$-space $\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(V)$ is
defined to be the abstract $\mathbb{F}$-space $V$ endowed with the topology
that is locally generated at 0 by

$\\{\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(U)\mid U\ \text{linearly compact
open in}\ V\\},$ (3.8)

(compare with (3.3)). Analogously, $\mathrm{Ind}^{\mathbb{L}}_{\mathbb{K}}(V)$
shall be the abstract $\mathbb{L}$-space $\mathbb{L}\otimes_{\mathbb{F}}V$
together with

$\\{\mathrm{Ind}^{\mathbb{L}}_{\mathbb{K}}(U)\mid U\ \text{linearly compact
open in}\ V\\},$ (3.9)

where each $\mathrm{Ind}^{\mathbb{L}}_{\mathbb{K}}(U)$ (compare with (3.5))
can be identified with a $\mathbb{K}$-subspace of
$\mathbb{L}\otimes_{\mathbb{F}}V$ since $\mathbb{L}$ is free over
$\mathbb{K}$. One can easily check that the functors so defined are additive
and form the adjunction required, i.e.,

$\textup{CHom}_{\mathbb{L}}(\mathrm{Ind}^{\mathbb{L}}_{\mathbb{K}}(V),W)\cong\textup{CHom}_{\mathbb{K}}(V,\mathrm{Res}^{\mathbb{L}}_{\mathbb{K}}(W)).$
(3.10)

Since a functor
$F\colon{}_{\mathbb{K}}\mathrm{LLC}\to{}_{\bullet}\mathrm{LLC}$ induces a
functor of flows, we shall simplify the notation as following:

  1. 1.

$C^{\bullet}_{n}(F(\phi,U)):=C^{\bullet}_{n}(F(\phi),F(U))$;

  2. 2.

$H^{*}_{\bullet}(F(\phi,U)):=H^{*}_{\bullet}(F(\phi),F(U))$;

  3. 3.

$\mathrm{ent}^{*}_{\bullet}(F(V,\phi)):=\mathrm{ent}^{*}_{\bullet}(F(V),F(\phi))$;

where $\bullet\in\\{\mathbb{F},\mathbb{K},\mathbb{L}\\}$.

##  4 Topological entropy after induction and restriction

###  4.1 Linearly compact case

Let $\mathbb{F}\leq\mathbb{K}\leq\mathbb{L}$ be field extensions such that
$[\mathbb{L}:\mathbb{F}]<\infty$. Let $V$ be a linearly compact vector space
and let

${}_{V}\beta\colon\prod_{n=0}^{\infty}V\to\prod_{n=0}^{\infty}V,\quad(v_{n})_{n\in\mathbb{N}}\mapsto(v_{n+1})_{n\in\mathbb{N}},$

denote the left Bernoulli shift of $V$. When $V=\mathbb{K}$ then
${}_{\mathbb{K}}\beta$ is the one-dimensional left Bernoulli shift that has
topological entropy $=1$. Clearly, its image under restriction is nothing but
the ($[\mathbb{K}:\mathbb{F}]$-dimensional) left Bernoulli shift of
$\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(V)=\prod_{n=0}^{\infty}\mathbb{F}^{[\mathbb{K}:\mathbb{F}]}$.
Therefore,

$\mathrm{ent}_{\mathbb{F}}^{*}(\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(V,{}_{\mathbb{K}}\beta))=\mathrm{ent}^{*}_{\mathbb{F}}(\prod_{n=0}^{\infty}\mathbb{F}^{[\mathbb{K}:\mathbb{F}]},{}_{\mathbb{F}^{[\mathbb{K}:\mathbb{F}]}}\beta)=[\mathbb{K}:\mathbb{F}]=[\mathbb{K}:\mathbb{F}]\cdot\mathrm{ent}_{\mathbb{K}}^{*}(V,{}_{\mathbb{K}}\beta),$

by [5, Example 3.13(b)]. On the other hand, the induction gives rise to the
(1-dimensional) left Bernoulli shift on
$\mathrm{Ind}_{\mathbb{K}}^{\mathbb{L}}(V)=\prod_{n=0}^{\infty}\mathbb{L}$,
i.e.,

$\mathrm{ent}_{\mathbb{L}}^{*}(\mathrm{Ind}_{\mathbb{K}}^{\mathbb{L}}(V,{}_{\mathbb{K}}\beta))=\mathrm{ent}^{*}_{\mathbb{L}}(\prod_{n=0}^{\infty}\mathbb{L},{}_{\mathbb{L}}\beta)=1=\mathrm{ent}_{\mathbb{K}}^{*}(V,{}_{\mathbb{K}}\beta).$

The following result shows that the formulas above hold for every flow
$(V,\phi)$ over ${}_{\mathbb{K}}\mathrm{LC}$.

######  Proposition 4.1.

Let $\mathbb{F}\leq\mathbb{K}\leq\mathbb{L}$ such that
$[\mathbb{L}:\mathbb{F}]<\infty$. For every flow $(V,\phi)$ over
${}_{\mathbb{K}}\mathrm{LC}$ one has

$\mathrm{ent}_{\mathbb{F}}^{*}(\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(V,\phi))=[\mathbb{K}:\mathbb{F}]\cdot\mathrm{ent}_{\mathbb{K}}^{*}(V,\phi)\quad\text{and}\quad\mathrm{ent}_{\mathbb{L}}^{*}(\mathrm{ind}_{\mathbb{K}}^{\mathbb{L}}(V,\phi))=\mathrm{ent}_{\mathbb{K}}^{*}(V,\phi).$

For a field $\mathbb{E}$, the collection of all open $\mathbb{E}$-subspaces
$\prod A_{i}$ of $\prod\mathbb{E}$ such that

  1. (G1)

$A_{i}=\mathbb{E}$ for all but finitely many indices,

  2. (G2)

$A_{i}=0$ otherwise;

form a neighbourhood basis at 0 for the Tychonoff topology on
$\prod\mathbb{E}$. We will refer to such a basis as good basis of
$\prod\mathbb{E}$ and its elements will be said to be good.

###### Proof of Proposition 4.1.

For an arbitrary flow $(V,\phi)$ over ${}_{\mathbb{K}}\mathrm{LC}$, let
$U=\prod A_{i}$ be a good $\mathbb{K}$-subspace of $V$. Thus

$\displaystyle
H_{\mathbb{F}}^{*}(\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(\phi,U))$
$\displaystyle=\lim_{n\to\infty}\frac{1}{n}\cdot\mathrm{codim}_{\mathbb{F}}\,C^{\mathbb{F}}_{n}(\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(\phi,U))=$
(4.1)
$\displaystyle=\lim_{n\to\infty}\frac{[\mathbb{K}:\mathbb{F}]}{n}\cdot\mathrm{codim}_{\mathbb{K}}\,C^{\mathbb{K}}_{n}(\phi,U)=[\mathbb{K}:\mathbb{F}]\cdot
H_{\mathbb{K}}^{*}(\phi,U).$ (4.2)

and notice that $\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(U)$ is a good
$\mathbb{F}$-subspace of $V$. Since good subspaces form a neighbourhood basis
at 0, one has that

$[\mathbb{K}:\mathbb{F}]\cdot\mathrm{ent}_{\mathbb{K}}^{*}(V,\phi)\leq\mathrm{ent}_{\mathbb{F}}^{*}(\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(V,\phi)).$
(4.3)

Finally, the equality holds since every good $\mathbb{F}$-subspace of
$\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(V)$ can be realised as the restriction
of a good $\mathbb{K}$-subspace of $V$. In order to prove the second part of
the statement, notice that

$\dim_{\mathbb{K}}(-)=\dim_{\mathbb{L}}(\mathrm{Ind}_{\mathbb{K}}^{\mathbb{L}}(-))\quad\text{and}\quad
C^{\mathbb{L}}_{n}(\mathrm{Ind}^{\mathbb{L}}_{\mathbb{K}}(\phi,U))=\mathrm{Ind}_{\mathbb{K}}^{\mathbb{L}}(C^{\mathbb{K}}_{n}(\phi,U)),\
\text{($U\in\mathcal{B}(V),n\in\mathbb{N}$)}.$

Therefore,

$\displaystyle H^{*}(\mathrm{Ind}^{\mathbb{L}}_{\mathbb{K}}(\phi,U))$
$\displaystyle=\lim_{n\to\infty}\frac{1}{n}\cdot\mathrm{codim}_{\mathbb{L}}\,C^{\mathbb{L}}_{n}(\mathrm{Ind}^{\mathbb{L}}_{\mathbb{K}}(\phi,U))=\lim_{n\to\infty}\frac{1}{n}\cdot\mathrm{codim}_{\mathbb{L}}\,\mathrm{Ind}_{\mathbb{K}}^{\mathbb{L}}(C_{n}(\phi,U))=$
(4.4)
$\displaystyle=\lim_{n\to\infty}\frac{1}{n}\cdot\mathrm{codim}_{\mathbb{K}}\,C_{n}(\phi,U)=H^{*}(\phi,U).$
(4.5)

Same reasoning as above yields
$\mathrm{ent}^{*}(\mathrm{ind}^{\mathbb{L}}_{\mathbb{K}}(V,\phi))=\mathrm{ent}^{*}(V,\phi).$
∎

######  Remark 4.2.

By latter result, it follows that

$\mathrm{ent}_{\mathbb{K}}^{*}(\mathrm{Ind}_{\mathbb{F}}^{\mathbb{K}}(\mathrm{Res}_{\mathbb{F}}^{\mathbb{K}}(V,\phi)))=[\mathbb{K}:\mathbb{F}]\cdot\mathrm{ent}_{\mathbb{K}}^{*}(V,\phi)=\mathrm{ent}_{\mathbb{K}}^{*}(\mathrm{Res}_{\mathbb{K}}^{\mathbb{L}}(\mathrm{Ind}_{\mathbb{K}}^{\mathbb{L}}(V,\phi))).$

This gives back an intuitive method to generate examples of topological flows
with some finite topological entropy $n\in\mathbb{N}$. E.g., let
$(V,{}_{\mathbb{K}}\beta)$ be the 1-dimensional left Bernoulli shift defined
above. One easily generates the $n$-dimensional left Bernoulli shift over
$\mathbb{K}$ by combining induction and restriction with respect to a field
extension of $\mathbb{K}$ of degree $n$.

###  4.2 General case

Recall that every locally linearly compact $\mathbb{K}$-space $V$ can be split
into a topological direct sum of a linearly compact open $\mathbb{K}$-space
$V_{c}$ and a discrete $\mathbb{K}$-space $V_{d}$; namely,
$V\cong_{top}V_{c}\oplus V_{d}$. Such a decomposition is available whenever we
consider a linearly compact open subspace $U$ of $V$. Indeed,
$V\cong_{top}U\oplus V/U$.

Now let $(V,\phi)$ be a flow over ${}_{\mathbb{K}}\mathrm{LLC}$ . Any
decomposition $V\cong_{top}V_{c}\oplus V_{d}$ induces a decomposition

$\phi=\begin{pmatrix}\phi_{cc}&\phi_{dc}\\\ \phi_{cd}&\phi_{dd}\end{pmatrix},$
(4.6)

where $\phi_{\bullet*}:V_{\bullet}\to V_{*}$ is the composition
$\phi_{\bullet*}=p_{*}\circ\phi\circ\iota_{\bullet}$ for
$\bullet,*\in\\{c,d\\}$. Therefore, $\phi_{\bullet*}$ is continuous being
composition of continuous homomorphisms.

######  Lemma 4.3.

Let $\mathbb{F}\leq\mathbb{K}\leq\mathbb{L}$ be finite field extensions and
$V\cong_{top}V_{c}\oplus V_{d}$ a locally linearly compact $\mathbb{K}$-space
together with a continuous endomorphism $\phi\colon V\to V$. Thus

  1. 1.

$\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(V)$ admits a decomposition such that

$\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(V)_{c}=\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(V_{c})\quad\text{and}\quad\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(\phi)_{cc}=\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(\phi_{cc}).$

  2. 2.

$\mathrm{Ind}^{\mathbb{L}}_{\mathbb{K}}(V)$ admits a decomposition such that

$\mathrm{Ind}_{\mathbb{K}}^{\mathbb{L}}(V)_{c}=\mathrm{Ind}_{\mathbb{K}}^{\mathbb{L}}(V_{c})\quad\text{and}\quad\mathrm{Ind}^{\mathbb{L}}_{\mathbb{K}}(\phi)_{cc}=\mathrm{Ind}^{\mathbb{L}}_{\mathbb{F}}(\phi_{cc}).$

###### Proof.

Notice that $\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(V)$ (or
$\mathrm{Ind}^{\mathbb{L}}_{\mathbb{K}}(V)$) can be obtained by restricting
(or extending) $V_{c}$ and $V_{d}$ first in the corresponding categories and
then summing them up in ${}_{\mathbb{K}}\mathrm{LLC}$. Indeed both restriction
functor and induction functor are additive. ∎

######  Theorem 4.4.

Let $\mathbb{F}\leq\mathbb{K}\leq\mathbb{L}$ such that
$[\mathbb{L}:\mathbb{F}]<\infty$. For every flow $(V,\phi)$ over
${}_{\mathbb{K}}\mathrm{LLC}$ one has

$\mathrm{ent}_{\mathbb{F}}^{*}(\mathrm{Res}^{\mathbb{K}}_{\mathbb{F}}(V,\phi))=[\mathbb{K}:\mathbb{F}]\cdot\mathrm{ent}_{\mathbb{K}}^{*}(V,\phi)\quad\text{and}\quad\mathrm{ent}_{\mathbb{L}}^{*}(\mathrm{Ind}_{\mathbb{K}}^{\mathbb{L}}(V,\phi))=\mathrm{ent}_{\mathbb{K}}^{*}(V,\phi).$

###### Proof.

By the previous lemma, it follows directly from Proposition 4.1 and the
reduction to linearly compact vector spaces devolopped in [5]. ∎

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