# Countable Markov shifts with exponential mixing

Mike Todd  Mathematical Institute
University of St Andrews
North Haugh
St Andrews
KY16 9SS
Scotland<EMAIL_ADDRESS>https://mtoddm.github.io and  Boyuan Zhao
Mathematical Institute
University of St Andrews
North Haugh
St Andrews
KY16 9SS
Scotland<EMAIL_ADDRESS>
(Date: 28th August 2024)

###### Abstract.

We give a set of equivalent conditions for a potential on a Countable Markov
Shift to have strong positive recurrence, which is also equivalent to having
exponential decay of correlations. A key ingredient of our proofs is
quantifying how the shift behaves at its boundary.

###### Key words and phrases:

Countable Markov shifts, Exponential mixing

###### 2020 Mathematics Subject Classification:

37A25, 37D05, 37B10, 37D25

##  1\. Introduction

A standard approach in ergodic theory and dynamical systems is to show that a
dynamical system $f:X\to X$ with invariant probability measure $\mu$,
asymptotically behaves like an independent, identically distributed process:
how quickly this happens can be understood as the rate of decay of
correlations, or rate of mixing, for $(X,f,\mu)$. Generally, uniformly
expanding systems with some compactness properties can be shown to have
exponential rate decay of correlations. Recent research has focussed on
estimating the rate of decay of correlations for systems with non-uniformly
expansion and/or lack of compactness.

Here we will consider dynamical systems $f:X\to X$ with measure
$\mu=\mu_{\phi}$, which is an equilibrium state for some potential
$\phi:X\to\overline{\mathbb{R}}$ . Note that the first task in this setting is
to show that such $\mu_{\phi}$ exists. In the case of interval maps $f:I\to I$
with finitely many branches, on each of which $f$ is smooth, we might consider
$\phi=-\log|Df|$, so an equilibrium state $\mu_{\phi}$ is an absolutely
continuous (with respect to Lebesgue) invariant measure. For example, if $f$
is unimodal and satisfies the Collet-Eckmann condition (exponential growth of
derivative along the critical orbit), then $\mu_{\phi}$ exists and the system
has exponential decay of correlations: indeed in [NS] these two conditions
were shown to be equivalent, alongside a number of other conditions, one of
which was that
$\chi_{per}:=\inf\left\\{\log|Df^{n}(x)|/n:f^{n}(x)=x,n\in\mathbb{N}\right\\}$
has $\chi_{per}>0$. See [R] for related results in the multimodal case.

In the same dynamical setting, but in the case of Hölder potentials $\phi$,
various conditions have been applied to ensure the existence of an equilibrium
states, which also usually imply that the system also has exponential decay of
correlations. A classic such example is the condition that
$\sup\phi-\inf\phi<h_{top}$, where $h_{top}$ denotes topological entropy,
which was used for interval maps in [HK, BT1] for example (in the former a
specification property was assumed, the latter dealt with multimodal maps of
the interval). A more general condition, which was used to show uniqueness of
equilibrium states in [DKU], is $\sup\phi<P(\phi)$ where $P(\phi)$ is the
pressure of $\phi$. A weaker condition of _hyperbolicity_ , that there is some
$n\in\mathbb{N}$ such that $\sup_{x\in X}S_{n}\phi(x)/n<P(\phi)$, was used in
[IR] in the case of complex rational maps, and in [LR1] in the interval map
case, to show existence and uniqueness of equilibrium states. Then it was
shown in [LR2] that for many multimodal maps (satisfying a growth condition on
critical orbits) _all_ Hölder potentials are hyperbolic. Related conditions
for interval maps were given in [LSV, Def 3.4], under which the authors called
the potential ‘contracting’: added to a covering condition this implies
exponential decay of correlations.

In the case of symbolic dynamical systems, nonuniform hyperbolicity can be a
consequence of an infinite alphabet: here we will discuss Countable Markov
Shifts. In this setting, though we have a Markov structure which makes many
issues straightforward, these structures are flexible, so that for example a
very strong condition like $\sup\phi<P(\phi)$ is not sufficient to guarantee
existence of equilibrium states (see Section 5.3). There are however a number
of equivalent conditions guaranteeing exponential decay of correlations given
in [CS], in particular Strong Positive Recurrence (SPR) and the Spectral Gap
Property (SGP): these can be shown to imply exponential decay of correlations.
In this paper we combine the types of ideas used in that paper with those in
[NS] to show various equivalent conditions for (SPR), in particular that
$\chi_{per}(\phi):=\sup\left\\{\frac{1}{n}S_{n}\phi(x):\sigma^{n}(x)=x,n\in\mathbb{N}\right\\}$
has $\chi_{per}(\phi)<P(\phi)$. The main idea is to use entropy at infinity,
see [ITV] (see also an equivalent notion in [B]) to control returns to compact
parts of our system. This idea can also be seen in [DT, Lemma 2.17, Theorem
7.14], where the entropy at infinity for the symbolic version of the finitely
branched interval maps is zero.

We also give other conditions equivalent to the one mentioned above, one being
‘contraction at infinity’: this rules out standard examples in the renewal
shift, for example, where (SPR) fails due to very weak contraction in the
boundary of the system. We also give examples to show that that our conditions
are sharp and moreover, other conditions related to those in [NS] are not
equivalent to ours, see the discussion in Section 2.4 and the examples in
Section 5.

The paper is organised as follows. In Section 2 we give our main definitions
and results, Theorems 2.6 and 2.7. In Section 3 we prove Theorem 2.6 and in
Section 4 we prove Theorem 2.7. Finally in Section 5 we give examples to show
applications and sharpness of our results.

Acknowledgements. MT was partially supported by the FCT (Fundação para a
Ciência e a Tecnologia) project 2022.07167.PTDC. BZ acknowledges the support
of a Chinese Scholarship Council grant.

##  2\. Definitions and main theorems

In this section we give the general setting for our work and then state the
main results.

Notation. If $A_{n},B_{n}\in(0,\infty)$, we write $A_{n}\asymp B_{n}$ if there
is some $C>0$ such that $B_{n}/C\leq A_{n}\leq CB_{n}$. For a set $E$ we write
$\\#E$ to denote its cardinality.

###  2.1. Countable Markov Shifts, basic setup

Here we outline the Countable Markov Shift setting, for which we follow the
works of Sarig, eg [S1]. Let $\mathcal{A}$ be a countable alphabet and $M$ an
$\mathcal{A}\times\mathcal{A}$ transition matrix with entries 0 and 1. Our
shift space is given by

$\Sigma:=\left\\{(x_{0},x_{1},\dots)\in\mathcal{A}^{\mathbb{N}_{0}}:M_{x_{i},x_{i+1}}=1\text{
for }i\geq 0\right\\}.$

Our dynamics is the usual shift
$\sigma(x_{0},x_{1},\ldots)=(x_{1},x_{2},\ldots)$.

Let

$\Sigma_{k}:=\left\\{w=w_{0},\ldots,w_{k-1}:M_{w_{i},w_{i+1}}=1\text{ for
}0\leq i<k-1\right\\}$

be the set of all _admissible words_ with length $k$. Let
$\Sigma^{*}:=\cup_{k\geq 0}\Sigma_{k}$ (this means that we include the empty
set here). Given $w=w_{0},\ldots,w_{k-1}$ let $[w]_{i}=w_{i}$ for $0\leq i\leq
k-1$. If our shift allows it, then the Markov property allows us to
concatenate words $v\in\Sigma_{k},w\in\Sigma_{k^{\prime}}$ to obtain
$vw\in\Sigma_{k+k^{\prime}}$: infinite concatenations correspond to points in
$\Sigma$.

We use $|\cdot|$ to denote the length of an admissible word: so if
$w\in\Sigma_{k}$ then $|w|=k$. Given $x_{0},\dots,x_{k-1}\in\Sigma_{k}$, the
corresponding $k$-_cylinder_ is

$[x_{0},\dots,x_{k-1}]:=\left\\{y\in\Sigma:y_{0}=x_{0},\ldots,y_{k-1}=x_{k-1}\right\\}.$

The set of all such $k$-cylinders is denoted by $\mathcal{C}_{k}$. We also
define the symbolic metric $d(\cdot,\cdot)$ by

$d(x,y)=2^{-t(x,y)},\hskip 8.53581ptt(x,y):=\min\\{n\geq 0:x_{n}\neq
y_{n}\\},$

so $k$-cylinders have diameter $2^{-(k-1)}$.

The system is said to be _topologically transitive_ if for each
$a,b\in\mathcal{A}$, there exists $N_{ab}$ such that
$[a]\cap\sigma^{-N_{ab}}[b]\neq\emptyset;$ the system is _topologically
mixing_ if, moreover, $[a]\cap\sigma^{-n}[b]\neq\emptyset$ for any $n\geq
N_{ab}$.

If a point $x$ is periodic with period $k$, it can be denoted by
$x=(\overline{x_{0},\dots,x_{k-1}}).$

Define the _$k$ -th variation_ of a potential
$\phi:\Sigma\rightarrow\mathbb{R}$ by

$var_{k}(\phi):=\sup\left\\{|\phi(x)-\phi(y)|:x_{0}=y_{0},\dots,x_{k-1}=y_{k-1}\right\\}.$

Then $\phi$ is said to have _summable variations_ if the _distortion constant_
$B_{\phi}:=\sum_{k\geq 2}var_{k}(\phi)$ is finite, and _weakly Hölder
continuous_ if there exists $\theta\in(0,1)$ and $C>0$ such that
$var_{k}(\phi)\leq C\theta^{k}$ for all $k\geq 2$.

###  2.2. Pressure, recurrence and contraction

For each $a\in\mathcal{A}$, we let
$\varphi_{a}(x):=\mathbbm{1}_{[a]}\inf\\{k\geq 1:x_{k}=a\\}$ be the first
return time to $[a]$ and define

$Z_{n}(\phi,a):=\sum_{\sigma^{n}x=x}e^{S_{n}\phi(x)}\mathbbm{1}_{[a]}(x),\hskip
8.53581ptZ_{n}^{*}(\phi,a):=\sum_{\sigma^{n}x=x}e^{S_{n}\phi(x)}\mathbbm{1}_{[\varphi_{a}=n]}(x).$

The _Gurevich pressure_ of $\phi$ is then given by

$P_{G}(\phi):=\limsup_{n\rightarrow\infty}\frac{1}{n}\log{Z_{n}(\phi,a)}$

which is easily seen to be independent of the state $a$, and if the system is
topological mixing, the $\limsup$ can be replaced by $\lim$. The _topological
entropy_ $h_{top}$ of $(\Sigma,\phi)$ is the quantity $P(0)$. By variational
principle, see [IJT, Theorem 2.10] for a version in the generality considered
here,

$P_{G}(\phi)=P(\phi):=\sup\left\\{h_{\nu}(\sigma)+\int\phi\,d\nu:\nu\in\mathcal{M}(\Sigma),\int\phi\,d\nu>-\infty\right\\},$

where $\mathcal{M}(\Sigma)$ is the space of all $\sigma$-invariant Borel
probability measures on $\Sigma$. A probability measure is said to be an
_equilibrium state_ for $\phi$ if it realises the supremum above. If $m$ is a
measure such that whenever $A\subset\Sigma$ is measurable and
$\sigma:A\to\sigma(A)$ is bijective,
$m(\sigma(A))=\int_{A}e^{-\phi}\leavevmode\nobreak\ dm$, then $m$ is _$\phi$
-conformal_.

The potential $\phi$ is said to be _recurrent_ if $\sum_{n\geq
1}e^{-nP(\phi)}Z_{n}(\phi,a)$ diverges and _transient_ otherwise. In the
recurrent case, $\phi$ is further said to be _positive recurrent_ if
$\sum_{n\geq 1}ne^{-nP(\phi)}Z_{n}^{*}(\phi,a)<\infty$, and _null recurrent_
otherwise. Note that there are $(\phi-P(\phi))$-conformal measures when $\phi$
is recurrent, as in [S2, Theorem 1].

We finish this subsection by highlighting the key definitions of this paper.
The following notion uses inducing: given $a\in\mathcal{A}$, consider
$\overline{\sigma}_{a}(x)=\sigma^{\varphi_{a}(x)}(x)$ for $x\in[a]$. Then the
induced version of $\phi$ is
$\overline{\phi}=\overline{\phi}_{a}:=\sum_{i=0}^{\varphi_{a}-1}\phi\circ\sigma^{i}$.
The pressure of $\overline{\phi}$ for the induced system
$([a],\overline{\sigma}_{a})$ is given as above.

######  Definition 2.1 (Strong Positive Recurrence).

Following the conventions and notation in [S2], for a given state
$a\in\mathcal{A}$ define the quantity

$p_{a}^{*}[\phi]:=\sup\left\\{p\in\mathbb{R}:P(\overline{\phi+p})<\infty\right\\},$
$\Delta_{a}[\phi]:=P\left(\overline{\phi+p_{a}^{*}[\phi]}\right).$

The shift system $(\Sigma,\sigma,\phi)$ is said to be _strongly positive
recurrent (SPR)_ if $\Delta_{a}[\phi]>0$ for some $a\in\mathcal{A}$.

Equivalently, for some $a\in\mathcal{A}$,

$\limsup_{n\rightarrow\infty}\frac{1}{n}\log Z_{n}^{*}(\phi,a)<P(\phi),$ (2.1)

see for example [RS, Lemma 2.1] (note that SPR potentials are also positive
recurrent).

Next, we assume our system satisfies a condition that regulates the excursions
to infinity. We first require some more notation. Here we rewrite
$\mathcal{A}=\mathbb{N}$ (while the following notions do rely on the precise
identification of $\mathcal{A}$ with $\mathbb{N}$ here, but the results we get
from them do not). Then for each $q\in\mathbb{N}$, define

$[\leq q]:=\bigcup_{a=1}^{q}[a],\text{ and }[>q]:=\bigcup_{a>q}[a].$

######  Definition 2.2 (Entropy and contraction at infinity).

For each $n$, $M$ and $q$, define the set of $n+1$-cylinders

$B(n,M,q):=\left\\{[x_{0},\dots,x_{n}]\in\mathcal{C}_{n+1}:x_{0},x_{n}\leq
q,\,\\#\\{k\leq n:x_{k}\leq q\\}\leq\frac{n+1}{M}\right\\}$

and write $z_{n}(M,q):=\\#B(n,M,q)$. The _entropy at infinity_ $h_{\infty}$ as
in [ITV, Def 1.2] is defined via

$h_{\infty}(M,q):=\limsup_{n\rightarrow\infty}\frac{1}{n}\log
z_{n}(M,q),\hskip
8.53581pth_{\infty}(q):=\liminf_{M\rightarrow\infty}h_{\infty}(M,q),$
$h_{\infty}:=\liminf_{q\rightarrow\infty}h_{\infty}(q).$

Also, define the following quantities:

$z_{\phi,n}(M,q):=\sup\left\\{\frac{1}{n}S_{n}\phi(x):x\in B(n,M,q)\right\\},$
$\delta_{\phi,\infty}(M,q):=\limsup_{n\to\infty}z_{\phi,n}(M,q),\
\delta_{\phi,\infty}(q):=\liminf_{M\to\infty}\delta_{\phi,\infty}(M),$
$\delta_{\phi,\infty}:=\liminf_{q\to\infty}\delta_{\phi,\infty}(q).$

Then the system is said to have _contraction at infinity (CI)_ , if
$\delta_{\phi,\infty}<P(\phi)$.

Finally we give a topological property of our shift, an idea first mentioned
in [IV, Definition 4.9].

######  Definition 2.3 (The $\mathcal{F}$ property).

Our system has the _$\mathcal{F}$ -property_ if for any state
$a\in\mathcal{A}$ and every $n\in\mathbb{N}$, the number of periodic points in
$[a]$ with period $n$ is finite.

The $\mathcal{F}$-property holds when $(\Sigma,\sigma)$:

  * •

is locally compact (i.e. for every $i\in\mathcal{A}$,
$\sum_{j\in\mathcal{A}}M_{i,j}<\infty$);

  * •

has $h_{top}<\infty$.

From the examples given in Section 5, we can see that a system may satisfy the
$\mathcal{F}$-property whilst failing all the above conditions. Note that If
$\phi$ is uniformly bounded from below and $P(\phi)<\infty$ then the
$\mathcal{F}$-property must also hold.

######  Remark 2.4.

The main fact we will use about the $\mathcal{F}$-property in this paper is
that, by topological transitivity, given $q\in\mathbb{N}$ and
$N\in\mathbb{N}$, there are finitely many paths starting and ending in $[\leq
q]$ of length $N$.

######  Remark 2.5.

One consequence of the $\mathcal{F}$-property is that for any $q,M$, if $n$ is
large enough then $B(n,M,q)\neq\emptyset$. If it fails, then for example there
will be some $q\in\mathbb{N}$ with infinitely many loops of some length $N$
based at $q$: if $M>N$, then $B(n,M,q)$ does not see any of these, in spite of
these loops contributing, in some sense, to the boundary. Note that this issue
would also appear in the definition of $h_{\infty}$ in [B].

###  2.3. Main Theorems

######  Theorem 2.6.

Let $(\Sigma,\phi)$ be a topologically transitive CMS with the
$\mathcal{F}$-property, $\phi$ a potential with summable variations and the
pressure $P(\phi)<\infty$. Then the following are equivalent.

  1. (UCS)

Uniformly contracting structure:

$\chi_{per}(\phi):=\sup\left\\{\frac{1}{n}S_{n}\phi(x):\sigma^{n}(x)=x\right\\}<P(\phi);$

  2. (CRC)

Compact returns contract: Given $q\in\mathbb{N}$, there exist
$C_{q}\in\mathbbm{R}$ and $\lambda_{q}>0$ such that if $x\in\Sigma$ has
$x_{0},x_{n}\leq q$ then

$S_{n}\phi(y)\leq C_{q}-n\lambda_{q};$

  3. (CI)

Contraction at infinity: $\delta_{\phi,\infty}<0$.

All of the conditions here are new in this context, as far as we are aware. As
mentioned in the introduction, (UCS) can be compared to $\chi_{per}>0$ in [NS]
since in that case $\phi=-\log|Df|$, so $\chi_{per}=-\chi_{per}(\phi)$, and
moreover $P(\phi)=0$. We also note that it is easy to show that
$\chi_{per}(\phi)=\sup\\{\int\phi\leavevmode\nobreak\
d\mu:\mu\in\mathcal{M}(\Sigma)\\}$. We use the word ‘contraction’ for our
potential in line with [LSV]. If the assumptions of Theorem 2.6 are slightly
strengthened, then the following holds.

######  Theorem 2.7.

Let $(\Sigma,\sigma,\phi)$ be a topologically transitive CMS, $\phi$ a
potential with summable variations, and entropy at infinity $h_{\infty}=0$.
Then (UCS) holds if and only if SPR holds.

Note that we are also able to prove related results when $h_{\infty}>0$, such
as Theorem 4.1

This theorem provides a new characterisation of SPR which, in the locally
Hölder potential case, is known to be equivalent to various other conditions,
for example the spectral gap property, which gives exponential decay of
correlations/mixing (see [CS]). We also note that in [CS, Theorem 2.2’] SPR
was shown to be open and dense in appropriate topologies. So in the setting of
the above theorem (UCS) is also open and dense. We observe that openness of
such a condition is elementary, but denseness requires more work.

###  2.4. Other conditions on contraction

As in the introduction, inspired by, for example [NS], one might propose
further conditions and hope that they are equivalent to those in our main
theorems. Here we suppose $P(\phi)=0$.

The following conditions involve hitting compact parts

  * (A)

There exist $q\in\mathbb{N}$, $C\in\mathbbm{R}$, $\varepsilon>0$ such that if
$x\in\Sigma$ has $x_{n}\leq q$ then $S_{n}\phi(x)\leq C-n\varepsilon$.

  * (B)

There exist $q\in\mathbb{N}$, $C\in\mathbbm{R}$, $\varepsilon>0$ such that if
$x\in\Sigma$ has $x_{0}\leq q$, then $S_{n}\phi(x)\leq C-n\varepsilon$.

Here (A) can be interpreted as ‘orbits which land in a compact part experience
contraction’ and (B) can be interpreted as ‘orbits which start in a compact
part experience contraction’.

The following condition involves avoiding compact parts

  * (C)

There exist $q\in\mathbb{N}$, $C\in\mathbbm{R}$, $\varepsilon>0$ such that if
$x\in\Sigma$ has $x_{k}>q$ for $k=0,\ldots,n-1$, then $S_{n}\phi(x)\leq
C-n\varepsilon$.

Supposing there is a $\phi$-conformal measure $m_{\phi}$, we might also hope
to show that measures of cylinders decay exponentially: below we give a weak
condition of this type, which by conformality of $m_{\phi}$, a measure we
assume to exist here, is essentially equivalent to (A) above.

  * (D)

There exist $q\in\mathbb{N}$, $\lambda>1$ and $K>0$ such that for all
$q^{\prime}\leq q$, if $C_{n}\in\mathcal{P}_{n}$ has
$\sigma^{n}(C_{n})=[q^{\prime}]$, then $m_{\phi}(C_{n})\leq K\lambda^{-n}$;

Finally we might try to get contraction like in (CI), but not assuming that
the orbits have to both start and end in a compact part $[\leq q]$: for
example only end in $[\leq q]$ which is related to condition (A) above, or
only start in $[\leq q]$, which is related to condition (B) above.

The above conditions can be shown to imply (UCS), but we show in Section 5.2
that they are strictly stronger, i.e., we have examples which satisfy (UCS),
but fail all of these conditions. We also note that the conditions of [LSV]
may fail, but (UCS) hold.

##  3\. Proof of Theorem 2.6

Throughout this paper, since we are always have $P(\phi)<\infty$, by
subtracting a constant from $\phi$ we can assume without loss of generality
that $P(\phi)=0$. Moreover, from here on, unless stated otherwise, we assume
our alphabet $\mathcal{A}$ is $\mathbb{N}$.

Given $a,b\in\mathbb{N}$, define

$\ell(a,b):=\min\left\\{k:\exists w\in\Sigma_{k},[w]_{0}=a\text{ and
}wb\in\Sigma_{k+1}\right\\}.$

This is finite by topological transitivity. Given $q\in\mathbb{N}$, let

$\ell(q):=\sup_{a,b\leq q}\ell(a,b),$

and note that this is also finite.

By the Markov property, for any $x\in\Sigma$ and $n\in\mathbb{N}$, for
$\ell=\ell(x_{n},x_{0})\geq 0$ and a word $w(x_{n},x_{0})\in[x_{n}]$ of length
$\ell$ such that

${w}(x_{n},x_{0})x_{0},\dots,x_{n-1}\in\Sigma_{n+\ell},$ (3.1)

i.e., a periodic point (there may be more than one of these, but here we just
fix one arbitrarily).

######  Lemma 3.1.

Supposing that $\phi$ has summable variations. Defining

$\underline{C}_{q}(\phi):=\min_{a,b\leq
q}\inf_{y\in[w(a,b)]}\left\\{S_{\ell(a,b)}\phi(y)\right\\},$

we have $\underline{C}_{q}(\phi)>-\infty$.

###### Proof.

This follows since there are finitely many word pairs $a,b\leq q$ to consider
and $S_{\ell(a,b)}\phi$ is finite on each $[{w}(a,b)]$. ∎

######  Lemma 3.2.

For a topologically transitive CMS, $\phi$ of summable variations, (UCS)
implies (CRC).

###### Proof.

Let $q\in\mathbb{N}$ be given and pick $\lambda_{q}>0$ small enough such that
$\chi_{per}(\phi)<-2\lambda_{q}<0$. Suppose $x\in\Sigma$ has $x_{0},x_{n}\leq
q$. Then for $z=\left(\overline{{w}(x_{n},x_{0})x_{0},\ldots,x_{n-1}}\right)$
defined as above,

$\displaystyle\underline{C}(q)+S_{n}\phi(x)$ $\displaystyle\leq
S_{n+\ell}\phi(z)+2B_{\phi}<2B_{\phi}-\left(n+\min_{a,b\leq
q}\ell(a,b)\right)\lambda_{q},$

where $\underline{C}(q)>-\infty$ as in Lemma 3.1. Hence
$S_{n}\phi(x)<C_{q}-n\lambda_{q}$ where
$C_{q}=\max\left\\{0,2B_{\phi}-\underline{C}_{q}(\phi)\right\\}$. ∎

It is immediate, given Remark 2.5, that (CRC) implies (CI), so it only remains
to show that (CI) implies (UCS) to complete the proof of Theorem 2.6.

The following lemma gives us a way of going from proofs of the results in this
paper for non-positive potentials to general potentials $\phi$.

######  Lemma 3.3.

There exists $\rho:\Sigma\to\mathbb{R}$ bounded on partition sets such that
for $\phi^{\prime}:=\phi+\log\rho-\log\rho\circ\sigma$, we have
$\phi^{\prime}\leq 0$. Moreover, $\phi^{\prime}$ has summable variations (or
is locally Hölder) if $\phi$ has summable variations (or is locally Hölder).

This is essentially [S2, Lemma 1], but we sketch parts of the proof here for
completeness.

###### Proof.

If $\phi$ is recurrent, then $\rho$ is the eigenfunction of the transfer
operator $\mathcal{L}_{\phi}$ associated to $\phi$. If $\phi$ is transient,
take $\rho=\sum_{n\geq 1}\mathcal{L}_{\phi}^{n}\mathbbm{1}_{[a]}$. The
regularity follows as in [S2, Lemma 1], though there the shift is assumed
topologically mixing: $\rho$ remains finite and non-positive in both cases
under topological transitivity. ∎

The previous lemma provides a convenient upper bound on the potential for our
proofs. We next show that $\phi^{\prime}$ inherits (CI) from $\phi$.

######  Lemma 3.4.

If $\delta_{\phi,\infty}<0$, then $\delta_{\phi^{\prime},\infty}<0$.

###### Proof.

By definition, there exist $\varepsilon>0$ and
$N_{\varepsilon},M_{\varepsilon},q_{\varepsilon}$ such that

$z_{\phi,n}(M,q_{\varepsilon})<-2\varepsilon$ (3.2)

for all $n>N_{\varepsilon}$ and $M>M_{\varepsilon}$. Then for every
$n>N_{\varepsilon}$ large enough that
$\frac{n+\ell(q_{\varepsilon})}{2M}<\frac{n}{M}$, for every $x\in
B(n,2M,q_{\varepsilon})$, as in (3.1), there exists an admissible word
${w}={w}(x_{n-1},x_{0})$, and a periodic point $y$ of period
$n^{\prime}=n+|{w}|$, such that $y\in B(n^{\prime},M,q_{\varepsilon})$,
$y=\left(\overline{x_{0},\dots,x_{n-1}{w}(x_{n-1},x_{0})}\right)$ and by
summable variations,

$S_{n}\phi^{\prime}(x)\leq
S_{n^{\prime}}\phi^{\prime}(y)-\underline{C}_{q}(\phi^{\prime})+B_{\phi^{\prime}}=S_{n^{\prime}}\phi(y)-\underline{C}_{q}(\phi^{\prime})+B_{\phi^{\prime}},$

where $\underline{C}_{q}(\phi^{\prime})$ is defined as in Lemma 3.1. Then
(3.2) implies

$S_{n}\phi^{\prime}(x)<-2n^{\prime}\varepsilon-
C_{q}(\phi^{\prime})+B_{\phi^{\prime}}$

and by choosing $n$ large, this implies that for all $M>M_{\varepsilon}$,
$z_{\phi,n}(2M,q_{\varepsilon})<-\varepsilon$, and consequently
$\delta_{\phi^{\prime},\infty}(q_{\varepsilon})<-\varepsilon$. Since this
inequality holds for all $q>q_{\varepsilon}$, we conclude that
$\delta_{\phi^{\prime},\infty}<0$. ∎

Note that this lemma also holds for any $\psi=\phi+\xi-\xi\circ\sigma$,
provided $\xi$ has summable variations. We will later use the following lemma
to show that we cannot have a sequence of periodic measures whose integrals of
$\phi$ converge to zero which moreover converge to a probability measure.

######  Lemma 3.5.

There are is no $q\in\mathbb{N}$ and sequence $(x^{k})_{k}$ of periodic points
of period $p_{k}$ such that $\frac{1}{p_{k}}S_{p_{k}}\phi(x^{k})\to 0$ and
$\nu_{k}([\leq q])\to 1$ as $k\to\infty$, where
$\nu_{k}=\frac{1}{p_{k}}\sum_{i=0}^{p_{k}-1}\delta_{\sigma^{i}x^{k}}$.

In the proof we make use of the notion of a sequence of measures
$(\mu_{n})_{n}$ _converging on cylinders_ to a measure $\mu$: this means that
for any $C\in\mathcal{C}_{k}$, $\mu_{n}(C)\to\mu(C)$ as $n\to\infty$, see [IV]
for more details.

###### Proof.

We will show that if we assume that the lemma is false then there is an
equilibrium state $\nu$ having zero measure-theoretic entropy, which is a
contradiction.

Let $\phi^{\prime}$ be as in Lemma 3.4, and it is easy to see
$S_{n}\phi(x)=S_{n}\phi^{\prime}(x)$ for all $n\in\mathbb{N}$ and all
$\sigma^{n}x=x$. Suppose there is such a $q$ and sequence of periodic points
as in the lemma. Note that $\int\phi^{\prime}\leavevmode\nobreak\ d\nu_{k}\to
0$. By [IV, Theorem 1.2], $\mathcal{M}_{\leq 1}(\Sigma)$ the space of shift-
invariant sub-probability measures on $\Sigma$ is compact with respect to the
convergence on cylinders topology, i.e., there is $\nu\in\mathcal{M}_{\leq
1}(\Sigma)$ such that $\nu_{k}\to\nu$ (up to subsequences) on cylinders. Our
assumption implies that $\nu$ is a probability measure. Hence, as
$(\nu_{k})_{k}$, $\nu$ are probability measures, [IV, Lemma 3.17] implies that
the convergence also holds in the weak-* topology. In particular if
$\phi^{\prime}_{L}(x)=\max\\{\phi^{\prime}(x),-L\\}$ then $\phi^{\prime}_{L}$
is continuous and bounded and $\int\phi_{L}^{\prime}\leavevmode\nobreak\
d\nu_{n_{k}}\to\int\phi_{L}^{\prime}\leavevmode\nobreak\ d\nu$.

######  Claim.

Given $L>0$, for any $\varepsilon>0$ there exists $K_{\varepsilon}^{\prime}$
such that $k\geq K_{\varepsilon}^{\prime}$ implies

$\left|\int\phi_{L}^{\prime}\leavevmode\nobreak\
d\nu_{n_{k}}-\int\phi^{\prime}\leavevmode\nobreak\
d\nu_{n_{k}}\right|<\varepsilon/4.$

###### Proof of Claim..

Since

$\int\phi^{\prime}\leavevmode\nobreak\
d\nu_{n_{k}}=\int\phi_{L}^{\prime}\leavevmode\nobreak\
d\nu_{n_{k}}+\int_{\\{\phi^{\prime}<-L\\}}\phi^{\prime}\leavevmode\nobreak\
d\nu_{n_{k}},$

if the claim is false then there is $\varepsilon>0$ such that for any
$N\in\mathbb{N}$ we can find $k\geq N$ such that
$\left|\int_{\\{\phi^{\prime}<-L\\}}\phi^{\prime}\leavevmode\nobreak\
d\nu_{n_{k}}\right|\geq\varepsilon/4$, i.e.,
$\int_{\\{\phi^{\prime}<-L\\}}\phi^{\prime}\leavevmode\nobreak\
d\nu_{n_{k}}\leq-\varepsilon/4$. But since $\phi^{\prime}\leq 0$, this means
$\int\phi^{\prime}\leavevmode\nobreak\ d\nu_{n_{k}}\leq-\varepsilon/4$,
contradicting the fact that $\int\phi^{\prime}\leavevmode\nobreak\
d\nu_{n_{k}}\to 0$. ∎

Now given $L>0$, take $K_{\varepsilon}\geq K_{\varepsilon}^{\prime}$, for
$K_{\varepsilon}^{\prime}$ as in the claim and such that
$\left|\int\phi^{\prime}\leavevmode\nobreak\
d\nu_{n_{k}}\right|<\varepsilon/4$ and
$\left|\int\phi_{L}^{\prime}\leavevmode\nobreak\
d\nu-\int\phi_{L}^{\prime}\leavevmode\nobreak\
d\nu_{n_{k}}\right|<\varepsilon/2$ whenever $k\geq K_{\varepsilon}$. Then

$\displaystyle\begin{split}\left|\int\phi_{L}^{\prime}\leavevmode\nobreak\
d\nu\right|&\leq\left|\int\phi_{L}^{\prime}\leavevmode\nobreak\
d\nu-\int\phi_{L}^{\prime}\leavevmode\nobreak\
d\nu_{n_{k}}\right|+\left|\int\phi_{L}^{\prime}\leavevmode\nobreak\
d\nu_{n_{k}}\right|\\\
&<\frac{\varepsilon}{2}+\left|\int\phi^{\prime}\leavevmode\nobreak\
d\nu_{n_{k}}-\phi^{\prime}_{L}\leavevmode\nobreak\
d\nu_{n_{k}}\right|+\left|\int\phi^{\prime}\leavevmode\nobreak\
d\nu_{n_{k}}\right|\\\
&<\frac{\varepsilon}{2}+\frac{\varepsilon}{4}+\frac{\varepsilon}{4}=\varepsilon.\end{split}$
(3.3)

Now the Monotone Convergence Theorem implies
$-\int\phi_{L}^{\prime}\leavevmode\nobreak\
d\nu\nearrow-\int\phi^{\prime}\leavevmode\nobreak\
d\nu=-\int\phi\leavevmode\nobreak\ d\nu$ as $L\to\infty$. Moreover (3.3) and
weak* convergence of $\nu_{n_{k}}$ to $\nu$ imply
$\left|\int\phi\leavevmode\nobreak\ d\nu\right|<\varepsilon$ for all
$\varepsilon$, i.e., $\int\phi\leavevmode\nobreak\ d\nu=0$, a contradiction. ∎

######  Proposition 3.6.

Under the assumptions of Theorem 2.6, (CI) implies (UCS).

###### Proof.

Suppose (UCS) fails. Then there exists a sequence of periodic points
$x^{1},x^{2},\dots$ with periods $p_{1},p_{2},\dots$ and with Birkhoff
averages $s_{n}=\frac{1}{p_{k}}S_{p_{k}}\phi(x^{n})\leq 0$ for all $n$ and
$\lim_{n\to\infty}s_{n}=0$. By the definition of (CI) and Lemma 3.4, for all
$\varepsilon>0$ such that $\delta_{\phi^{\prime},\infty}<-\varepsilon<0$,
there exist $N_{\varepsilon}$, $M_{\varepsilon},q_{\varepsilon}$ such that for
all $n>N_{\varepsilon}$, $M>M_{\varepsilon}$, $q>q_{\varepsilon}$ and all
$x\in B(n,M,q)$, $S_{n}\phi^{\prime}(x)<-n\varepsilon$.

Given $q,N\in\mathbb{N}$, let $\mathcal{A}_{[\leq q],N}$ be the set of words
$w\in\Sigma_{N}$ such that $[w]_{0}\leq q$ and $[w]_{i}>q$ for $i=1,\ldots,N$
and $wq^{\prime}\in\Sigma_{N+1}$ for some $q^{\prime}\leq q$, i.e., the _first
return words to $[\leq q]$ of length $N$_. Let $\mathcal{A}_{[\leq
q]}:=\cup_{N\geq 1}\mathcal{A}_{[\leq q],N}$.

Given $x\in\Sigma$ such that $x_{0},x_{n}\leq q$ we can decompose
$x_{0},\ldots,x_{n-1}=w_{1}v_{1}w_{2}v_{2}\ldots w_{k}v_{k}$, where
$w_{i}\in\mathcal{A}_{[\leq q]}$ and
${v}_{i}\in\\{\emptyset\\}\cup\cup_{N}\\{1,\ldots,q\\}^{N}$ for all $1\leq
i\leq k$. Let $x(q)$ denote the set of words $w_{i}$ in this decomposition,
counting with multiplicity.

Given $q>q_{\varepsilon}$, define the proportion function $\zeta(\cdot)$

$\zeta(x^{n}):=\frac{1}{p_{n}}\sideset{}{{}^{\prime}}{\sum}_{\left\\{{w}\in
x^{n}(q)\right\\}}|{w}|.$ (3.4)

Here $\sum^{\prime}$ means that we count with multiplicity, i.e., if $w$
appears $k$ times in the decomposition of $x^{n}$ we sum $|w|$ $k$ times.

Notice that since and symbol $a\leq q$ appears just once in each
${w}\in\mathcal{A}_{[\leq q],N}$, so long as $(N+1)M_{\varepsilon}\geq 1$,
$S_{N}\phi^{\prime}(x)<-N\varepsilon$ for $x\in[{w}]$. Hence, since we can
assume that $(N_{\varepsilon}+1)M_{\varepsilon}\geq 1$, we can show that
$\lim_{n\to\infty}\zeta(x^{n})=0$: if there exists $\eta>0$ such that
$\lim_{n\to\infty}\zeta(x^{n})\geq\eta$, by the non-positivity of
$\phi^{\prime}$,

$\displaystyle\begin{split}\lim_{n\rightarrow\infty}s_{n}&\leq\lim_{n\rightarrow\infty}\frac{1}{p_{n}}\sideset{}{{}^{\prime}}{\sum}_{\left\\{{w}\in
x^{n}(q)\right\\}}\sup_{x\in[{w}]}S_{|{w}|}\phi^{\prime}(x)\\\
&\leq\lim_{n\rightarrow\infty}\frac{1}{p_{n}}\sideset{}{{}^{\prime}}{\sum}_{\left\\{{w}\in\mathcal{A}_{[\leq
q],N},N>N_{\varepsilon}\right\\}}\sup_{x\in[{w}]}S_{|{w}|}\phi^{\prime}(x)\\\
&\leq\lim_{n\rightarrow\infty}-\eta\varepsilon<0,\end{split}$ (3.5)

contradicting our assumption that $\lim_{n\to\infty}s_{n}=0$.

Next since by the $\mathcal{F}$-property and topological transitivity,
$\\#\\{\cup_{N\leq N_{\varepsilon}}\mathcal{A}_{[\leq q],N}\\}<\infty$ and the
following quantity is also finite:

$q^{\prime}(q,N):=\min\left\\{q^{\prime}\in\mathbb{N}:\text{if
}{w}\in\cup_{N\leq N_{\varepsilon}}\mathcal{A}_{[\leq q],N}\text{ then
}w\in[\leq q^{\prime}]^{|w|}\right\\}.$ (3.6)

Therefore, since $\lim_{n\to\infty}\zeta(x^{n})=0$,

$\lim_{n\to\infty}\nu_{n}\left([\leq q^{\prime}]\right)=1\text{ where
}\nu_{k}=\frac{1}{p_{k}}\sum_{i=j}^{p_{k}-1}\delta_{\sigma^{i}x^{k}},$

which this contradicts Lemma 3.5, hence (CI) implies (UCS). ∎

##  4\. Proof of Theorem 2.7

Assuming now (UCS) holds, then (CI) follows from Theorem 2.6 and in the next
theorem we then show the system is (SPR) under slightly more general
assumptions than those in Theorem 2.7.

######  Theorem 4.1.

Let $(\Sigma,\phi)$ be a topologically transitive CMS with the
$\mathcal{F}$-property, $\phi$ a potential of summable variations satisfying
(CI) and assume that $\delta_{\phi,\infty}+h_{\infty}<0$. Then $\phi$ is
(SPR).

We provide an example in Section 5.3 with $\delta_{\phi,\infty}+h_{\infty}=0$
but the systems fails to be SPR, so the condition
$\delta_{\phi,\infty}+h_{\infty}<0$ above is sharp.

######  Lemma 4.2.

For all $\varepsilon>0$, there exists $q$ and $K_{q}\geq 0$ and such that if
$x_{0},x_{n}\leq q$ and $x_{k}>q$ whenever $1\leq k\leq n-1$, then

$S_{n}\phi(x)<K_{q}+n\left(\delta_{\phi,\infty}+\varepsilon\right).$

###### Proof.

Let $\varepsilon>0$ be given. By definition there exists $q$ such that
$\delta_{\phi,\infty}(q,M)<\delta_{\phi,\infty}+\frac{\varepsilon}{2}$ for all
$M$ large. Then there exists $N_{\varepsilon}$ such that for all
$n>N_{\varepsilon}$ if $x\in\Sigma$ is such that $x_{0},x_{n}\leq q$, but
$x_{k}>q$ for $i=1,\dots,n-1$, then

$\frac{1}{n}S_{n}\phi(x)<\delta_{\phi,\infty}(q,M)+\frac{\varepsilon}{2}<\delta_{\phi,\infty}+\varepsilon.$

Since the $\mathcal{F}$-property implies that for each $n$ the number of words
of length $n$ which start and end at $[\leq q]$ are finite, also using
summable variations,

$K_{q}:=\max\left\\{\max_{n\leq
N_{\varepsilon}}\sup\left\\{S_{n}\phi(x):x_{0},x_{n}\leq
q\right\\},0\right\\}$

is finite and satisfies the lemma. ∎

Given $q\in\mathbb{N}$ as in the lemma, let $Y=[\leq q]$ and define
$\tau_{Y}:Y\to\mathbb{N}$ by $\tau_{Y}(x):=\inf\\{n\geq 1:\sigma^{n}(x)\in
Y\\}$. Then let $F:Y\to Y$ be the first return map $F=\sigma^{\tau_{Y}}$. Let
$m_{Y}$ be the conditional conformal measure on $Y$. The set of $k$-cylinders
is denoted by $\mathcal{C}_{k}^{F}$.

Define the corresponding induced potential
$\hat{\phi}_{Y}=\sum_{i=0}^{\tau_{Y}-1}\phi\circ\sigma^{i}$ and note that this
has summable variations (in fact $var_{1}\hat{\phi}_{Y}<1$), so
$B_{\hat{\phi}}<\infty$. By for example [S2, Theorem 2], $P(\hat{\phi})\leq
0$, so setting $\overline{\phi}=\hat{\phi}-P(\hat{\phi})$ we have a normalised
potential and there is an $\overline{\phi}$-conformal measure $m_{Y}$ and an
equivalent invariant measure $\mu_{Y}$, see [S3, Theorem 1]; also
$B_{\overline{\phi}}=B_{\hat{\phi}}<\infty$. Note that if $\phi$ is recurrent
then $\hat{\phi}=\overline{\phi}$. We also define
$S^{F}_{n}\overline{\phi}_{Y}=\sum_{i=0}^{n-1}\overline{\phi}_{Y}\circ F^{i}.$

######  Lemma 4.3.

There is $C_{1}>0$ and $\varepsilon>0$ such that if $Z\in\mathcal{C}_{p}^{F}$
and

$\sum_{i=0}^{p-1}\tau_{Y}(F^{i}(Z))=n$

then

$m_{Y}(Z)\leq
C_{1}^{p}\exp{\left(n(\delta_{\phi,\infty}+\varepsilon)\right)}.$

###### Proof.

Writing $\tau_{Y}(F^{i}(Z))=\tau_{i}$, $Z$ is an $(n+1)$-cylinder with respect
to the $\sigma$ of the form

$[z_{0},\dots,z_{\tau_{1}-1},z_{\tau_{1}},\dots,z_{\tau_{p}-1},z_{\tau_{p}}]$

where $z_{\tau_{i}}\leq q$ for $i=1,\dots,p$. By conformality and Lemma 4.2,

$\displaystyle m_{Y}\left([z_{0}]\right)$
$\displaystyle=\int_{[z_{0},\dots,z_{\tau_{p}-1}]}e^{-S_{n}\phi(x)+pP(\hat{\phi})}\,dm_{Y}$
$\displaystyle\geq m_{Y}([z_{0},\dots,z_{\tau_{p}-1}])e^{-\sup
S_{n}\phi(x)+pP(\hat{\phi})}.$

Hence

$m_{Y}([z_{0},\dots,z_{\tau_{p}-1}])\leq\exp\left(p(K_{q}-P(\hat{\phi}))+n(\delta_{\phi,\infty}+\varepsilon)\right)m_{Y}([z_{0}]),$

so setting $C_{1}=e^{K_{q}-P(\hat{\phi})}$, we are finished. ∎

###### Proof of Theorem 4.1.

The proof is similar to that of [DT, Theorem 7.14]. As we will see, by (2.1)
it suffices to show the inducing scheme on some 1-cylinder $[a]$ has an
exponential tail.

Pick $\varepsilon>0$ such that

$\delta_{\phi,\infty}+h_{\infty}<-4\varepsilon,$

choose $q$ satisfying Lemma 4.2 and such that for all large $M$,

$\delta_{\phi,\infty}+h_{\infty}(M,q)<-3\varepsilon.$ (4.1)

So for $(Y,F)$ as above, which must also satisfy Lemma 4.3, by topological
transitivity there exists $N$ such that for all $Z\in\mathcal{C}_{1}^{F}$,

$Y\subset\bigcup_{j=1}^{N}F^{j}(Z).$ (4.2)

Pick some 1-cylinder $Y_{0}=[a]$ with $m_{Y}(Y_{0})>0$ and let $m_{Y_{0}}$ be
the conditional conformal measure here.

######  Claim 1.

There is some uniform constant $\beta>0$ such that for
$Z_{n}\in\mathcal{C}_{n}^{F}$,

$\frac{m_{Y}\left(x\in Z_{n}:F^{j}(x)\notin
Y_{0},j=n,\ldots,n+N-1\right)}{m_{Y}(Z_{n})}<e^{-\beta}.$

###### Proof of 1..

By (4.2), for each $[b]\subset Y$ there is some cylinder $A\subset[b]$ and
$0\leq k\leq N-1$ such that $F^{k}(A)=[a]$. Write $k=k(A)$ here. Denote the
collection of such cylinders by $\mathcal{B}$. In particular there is some
$A\in\mathcal{B}$ such that $A\subset F^{n}(Z_{n})$. Letting
$A^{\prime}=F^{-n}A\cap Z_{n}$, and it suffices to find a lower bound for
$\frac{m_{Y}(A^{\prime})}{m_{Y}(Z_{n})}$, independent of
$Z_{n}\in\mathcal{C}_{n}^{F}$ and $A\in\mathcal{B}$.

Then $\min_{A\in\mathcal{B}}\inf_{x\in A}S_{k(A)}^{F}\,\overline{\phi}_{Y}(x)$
is bounded from below by summable variations (similarly to Lemma 3.1).

By conformality, for any $C\subseteq Y$, if $F^{m}:C\to F^{m}C$ is injective,
$m_{Y}(F^{m}C)=\int_{C}\exp(-S_{m}^{F}\,\overline{\phi}_{Y})dm_{Y}$, hence

$\displaystyle\frac{m_{Y}(A^{\prime})}{m_{Y}(Z_{n})}$
$\displaystyle\geq\frac{m_{Y}(A)}{m_{Y}(F^{n}Z_{n})}\exp{\left(-\sup_{Z_{n}}S_{n}^{F}\,\overline{\phi}_{Y}+\inf_{A^{\prime}}S_{n}^{F}\,\overline{\phi}_{Y}\right)}$
$\displaystyle\geq\frac{m_{Y}([a])}{m_{Y}(Y)}e^{-B(\overline{\phi}_{Y})}e^{\inf_{A}S^{F}_{k}\,\overline{\phi}_{Y}}>0,$

uniformly, as required. ∎

######  Claim 2.

For each $k\geq 1$,

$\frac{m_{Y}\left(x\in Y_{0}:F^{j}(x)\notin
Y_{0},j=1,\ldots,nN\right)}{m_{Y}(Y_{0})}<e^{-k\beta}.$

###### Proof of 2..

This claim is proved by induction. As $Y_{0}=[a]$ can be written as a union of
$1$-cylinders with respect to $F$, 1 and the fact that for all positive
numbers $a,b,c,d$,
$\frac{a+c}{b+d}\leq\max\left\\{\frac{a}{b},\frac{c}{d}\right\\}$, together
implies

$\frac{m_{Y}\left(x\in Y_{0}:F^{j}(x)\notin
Y_{0},j=1,\ldots,N\right)}{m_{Y}(Y_{0})}<e^{-\beta}.$

Assume inductively that for each $i\geq 1$,

$\frac{m_{Y}\left(x\in Y_{0}:F^{j}(x)\notin
Y_{0},j=1,\ldots,iN\right)}{m_{Y}(Y_{0})}<e^{-i\beta}.$

Defining the set

$\mathcal{Z}_{i}:=\left\\{Z\in\mathcal{C}^{F}_{iN+1}:Z\subset
Y_{0},\,F^{j}(Z)\notin Y_{0},\,j=1,\ldots,iN\right\\},$

by 1 and the inequality above:

$\displaystyle\frac{m_{Y}\left(x\in Y_{0}:F^{j}(x)\notin
Y_{0},j=1,\ldots,(i+1)N\right)}{m_{Y}(Y_{0})}$
$\displaystyle=\frac{1}{m_{Y}(Y_{0})}\sum_{Z\in\mathcal{Z}_{i}}m_{Y}(Z)\frac{m_{Y}\left(x\in
Z:F^{j}(x)\notin Y_{0},j=iN+1,\ldots,(i+1)N\right)}{m_{Y}(Z_{i(N+1)})}$
$\displaystyle<\frac{1}{m_{Y}(Y_{0})}\sum_{Z\in\mathcal{Z}_{i}}m_{Y}(Z)e^{-\beta}\leq
e^{-\beta}\frac{m_{Y}\left(x\in Y_{0}:F^{j}(x)\notin
Y_{0},\,j=1,\dots,iN\right)}{m_{Y}(Y_{0})}$
$\displaystyle<e^{-(i+1)\beta}.\qed$

Letting $T=\gamma n$ for some $\gamma\in(0,1)$ to be determined later, we can
split the set $\left\\{x\in Y_{0}:\tau_{a}(x)\right\\}$ depending on whether
$x$ visits $Y$ more or less than $T$ times in its first $n$ symbols, which can
be written

$\displaystyle m_{Y_{0}}\left(\\{\tau_{a}=n\\}\right)$
$\displaystyle\leq\,m_{Y_{0}}\left(\left\\{\tau_{a}(x)=n,\sum_{j=0}^{T}\tau_{Y}(F^{j}(x))>n\right\\}\right)$
$\displaystyle\quad+m_{Y_{0}}\left(\left\\{\tau_{a}(x)=n,\sum_{j=0}^{T}\tau_{Y}(F^{j}(x))\leq
n\right\\}\right)$ $\displaystyle=:I+II.$

By 2,

$\displaystyle I\leq\sum_{p=T}^{n}\exp{\left(-\frac{p}{N}\beta\right)}\leq
C_{2}\exp{\left(-\frac{T}{N}\beta\right)},$

for some $C_{2}\in\mathbb{R}$. The number of $n$-cylinders with respect to
$\sigma$ which spend a proportion $\gamma\leq 1/M$ of their $\sigma$-iterates
up to $n$ in $Y$ is no more than $\\#B(n,M,q)$. Moreover, for all large $n$,
$\\#B(n,M,q)\leq C_{3}e^{n(h_{\infty}(M,q)+\varepsilon)}$ for some $C_{3}>0$,
so combining this with Lemma 4.3 and (4.1) we get

$\displaystyle II$ $\displaystyle\leq
C_{1}^{T}\exp\left(n(\delta_{\phi,\infty}+\varepsilon)\right)\\#B(n,M,q)$
$\displaystyle\leq
C_{1}^{T}C_{3}\exp{\left(n(\delta_{\phi,\infty}+h_{\infty}(M,q)+2\varepsilon)\right)}\leq
C_{1}^{T}C_{3}\exp{(-n\varepsilon)}.$

Then choosing
$\gamma=\min\left\\{\frac{1}{M},\frac{\varepsilon}{2\log{C_{1}}}\right\\}$,
both $I$ and $II$ are exponentially small so that

$\limsup_{n\to\infty}\frac{1}{n}\log{m_{Y_{0}}(\\{\tau_{a}=n\\})}<0.$

As $m_{Y_{0}}$ is conformal,

$m_{Y_{0}}(\\{\tau_{a}=n\\})\asymp\sum_{\sigma^{n}x=x,\varphi_{a}(x)=n}e^{S_{n}\phi(x)-i(x)P(\hat{\phi})}\geq
Z_{n}^{*}(\phi,a),$

where $i(x)$ corresponds to the number of hits to $Y$ before $Y_{0}$. Hence
(2.1) holds and the system is strongly positive recurrent. ∎

Theorem 4.1 means that (UCS) implies (SPR). For the other direction of Theorem
2.7, it suffices to prove the statement under topological mixing since we can
use _spectral decomposition_ , a tool to reduce arguments on topologically
transitive to topologically mixing. Briefly speaking, if $(\Sigma,\sigma)$ is
a topologically transitive CMS with period $p$, the alphabet is divided into
$p$ equivalence classes $\\{\mathcal{A}_{1}\dots,\mathcal{A}_{p-1}\\}$ and
$\Sigma=\biguplus_{i=0}^{p-1}\Sigma_{i}$,
$\Sigma_{i}=\left\\{x\in\Sigma:x_{0}\in\mathcal{A}_{i}\right\\}.$ Then
$(\Sigma_{i},\phi_{p},\sigma^{p})$ is conjugate to a topological mixing CMS
and most statements (especially those in this paper) proved for
$(\Sigma_{i},\phi_{p},\sigma^{p})$ remain valid for the original CMS. For more
detailed discussion, see for example [RS, §2.2,§6].

######  Proposition 4.4.

Under the assumptions of Theorem 2.7, (SPR) implies (UCS).

###### Proof.

By (SPR) there is $a\in\mathcal{A}$ such that $\Delta_{a}[\phi]>0$. First by
[S2, Lemma 3], $P(\phi)=0$ implies the induced pressure on $[a]$,
$P(\overline{\phi})$, is zero, and (SPR) implies that there exists
$\varepsilon_{a}>0$ such that

$P(\overline{\phi+2\varepsilon_{a}})<\infty.$ (4.3)

Moreover, as in (2.1), there exists $N_{a}\in\mathbb{N}$ such that for all
$n>N_{a}$, all $x$ such that $\varphi_{a}(x)=n$,

$\frac{1}{n}S_{n}\phi(x)<-\varepsilon_{a}.$ (4.4)

Suppose by contradiction that $\chi_{per}(\phi)=0$; take $\phi^{\prime}$ as in
Lemma 3.4 and $\chi_{per}(\phi^{\prime})=\chi_{per}(\phi)$. Then there exists
a sequence of periodic points $x^{1},x^{2},\dots,$ with period
$p_{1},p_{2},\dots$ and Birkhoff averages

$s_{n}=\frac{1}{p_{n}}S_{p_{n}}\phi(x^{n})=\frac{1}{p_{n}}S_{p_{n}}\phi^{\prime}(x^{n})>-\varepsilon_{a}\text{
and }\lim_{n\to\infty}s_{n}=0.$ (4.5)

Case 1. Suppose there exists $x\in\left\\{x^{1},\dots\right\\}$ such that
$\forall k\geq 0$, $x_{k}\neq a$. Then as in (3.1), by topological
transitivity, there are words ${v}$, ${w}$ of length $\ell_{1}=\ell(a,x_{0})$
and $\ell_{2}=\ell(x_{n-1},a)$ respectively such that $[{v}]_{0}=a,$
$vx_{0}\in\Sigma_{|v|+1}$, $x_{n-1}w\in\Sigma_{|w|+1}$,
${w}a\in\Sigma_{|w|+1}$, hence

${v}x_{0},\dots,x_{n-1}{w}\in\Sigma_{\ell_{1}+n+\ell_{2}}.$

Moreover, for each $k\in\mathbb{N}$ and $n_{k}=kn+\ell_{1}+\ell_{2}$ there is
a periodic point $y(k)\in[a]$ with $\varphi_{a}(y(k))=n_{k}$ of the form:

$y(k)=\left(\overline{{v}\left(x_{0},\dots,x_{n-1}\right)^{k}{w}}\right)$

where $\left(x_{0},\dots,x_{n-1}\right)^{k}$ means the string is repeated $k$
times. By summable variations, there exists a constant $C>0$ such that for all
$k$,

$Z_{n_{k}}^{*}(\phi,a)\geq\exp{\left(S_{n_{k}}\phi(y(k))\right)}\geq\exp{\left(C-kn\varepsilon_{a}\right)}.$

Then as in [S2, (5)], $\left|P(\overline{\phi+p})-\log\sum_{k\geq
1}e^{kp}Z_{k}^{*}(\phi,a)\right|\leq B_{\phi}$, therefore,

$\infty=\log{\sum_{k=1}^{\infty}e^{n_{k}\varepsilon_{a}+C}e^{S_{n_{k}}\phi(y(k))}}\leq
C+\log\sum_{n=1}^{\infty}e^{n\varepsilon_{a}}Z^{*}_{n}(\phi,a)\leq
P(\overline{\phi+\varepsilon_{a}})+B_{\phi},$

which is a contradiction to (4.3) since $B_{\phi}<\infty$.
Case 2. Now suppose all $x\in\\{x^{1},\dots\\}$ contain state $a$. Without
loss of generality one can suppose $x^{i}_{0}=a$ for all $i$ by periodicity.
Let $\mathcal{A}_{a}\subset\Sigma^{*}$ be words $w$ where $[w]_{i}=a$ if and
only if $i=0$, and moreover ${w}a\in\Sigma_{|w|+1}$, the set of _first return
words_ to $a$.

For all $n,$

$x^{n}=\left(\overline{{w}_{0}\dots w_{k_{n}-1}}\right)\text{ for some
$k_{n}\geq 1$, }{w}_{i}\in\mathcal{A}_{a}\text{ and
}\sum_{j=0}^{k_{n}-1}|{w}_{j}|=p_{n};$

That is, each $x^{n}$ can be decomposed into several first return words.

As $S_{m}\phi(x)=S_{m}\phi^{\prime}(x)$ for any periodic point with period
$m$, (4.4) implies that for all first return word $\alpha$ with length longer
than $N_{a}$,

$\sup_{x\in[{\alpha}]}S_{m}\phi^{\prime}(x)<-m\varepsilon_{a}.$

Letting $\mathcal{A}_{a,>k}:=\\{w\in\mathcal{A}_{a}:|w|>k\\}$, re-define the
proportion function similarly to (3.4),

$\zeta:\left\\{x^{1},\dots\right\\}\rightarrow[0,1],$
$\zeta(x^{n})=\frac{1}{p_{n}}\sideset{}{{}^{\prime}}{\sum}_{\left\\{{w}\in
x^{n}(a)\cap\mathcal{A}_{a,>N_{a}}\right\\}}|{w}|,$

where $\Sigma^{\prime}$ again means that we count with multiplicity. Then
repeating (3.5) with $r=\varepsilon_{a}$ this definition ensures
$\lim_{n\to\infty}\zeta(x^{n})=0$ since otherwise we contradict the property
(4.5) of our periodic points. By the $\mathcal{F}$-property, we can define the
function $q_{a}$ by

$q_{a}(N):=\min\left\\{q\in\mathbb{N}:\text{if }{w}\in\mathcal{A}_{a}\text{
and }|{w}|\leq N\text{ then }[w]_{i}\leq q\text{ for
}i=0,\ldots,|w|-1\right\\}.$

The sequence of probability measures

$\nu_{n}=\frac{1}{p_{k}}\sum_{j=0}^{p_{n}-1}\delta_{\sigma^{j}x^{n}}$

satisfies $\lim_{n}\nu_{n}([\leq\\!q_{a}(N_{a})])=1.$ But since
$\lim_{k\rightarrow\infty}s_{n_{k}}=0$, we have a contradiction to Lemma 3.5,
hence such sequence of periodic points does not exist. ∎

This concludes the proof of Theorem 2.7.

##  5\. Examples

The conditions for our main results are weak, so there are many CMS satisfying
these, but in this section we focus on a set of examples which simultaneously
represent a broad class of CMS for which our theory applies as well as
exhibiting edge cases which demonstrate the sharpness of our results.

Our examples take the form of ‘bouquet’ Markov graphs, see [Ru1, Ru2]. We note
that we could have used other shift models, eg $S$-gap shifts, to demonstrate
the sharpness of our results, but the bouquet setup covers these as well as
all other topologically transitive CMS, as we note at the end of Section 5.5.

Some of the examples which inspired this work come from codings for dynamical
systems, particularly in the case of interval maps $f:[0,1]\to[0,1]$: here a
subset $Y\subset[0,1]$ is taken and some return time
$\tau:Y\to\mathbb{N}\cup\\{\infty\\}$ is chosen so that $F=f^{\tau}$, an
_inducing scheme_ defines a Markov map on $Y$, i.e., there is a partition
$\\{Y_{i}\\}_{i}$ such that $\tau|_{Y_{i}}$ is some constant $\tau_{i}$ and
$F(Y_{i})$ is a collection of elements of this partition. We can associate
bouquet Markov graph to this, described in more detail below, which then
defines a shift, and we can take the potential $\phi:I\to[-\infty,\infty]$ and
lift this to the symbolic model. For example, such a coding can be done for
general multimodal maps of the interval, as shown in, for example, [BT2,
Theorem 3], or more classical and specific inducing schemes like those given
in [BLS] (which include Collet-Eckmann maps). Note that our framework here
does not give new results in the specific setups in the references mentioned.

###  5.1. Bouquet setup

Following [Ru1, Ru2], let $a:\mathbb{N}\to\mathbb{N}_{0}$ with $a(1)=1$. We
define our set of vertices as

$V:=\\{r\\}\cup\bigcup_{n=1}^{\infty}\left\\{v_{k}^{n,i}:1\leq i\leq a(n),\
1\leq k\leq n-1\right\\},$

where all vertices with distinct labels above are distinct vertices. We call
$r$ the _root_. For notational convenience write $v_{0}^{n,i}=v_{n}^{n,i}=r$.
Then the only allowed transitions in our Markov graph are $v_{k}^{n,i}\mapsto
v_{k+1}^{n,i}$ for $0\leq k\leq n-1$. This defines a _bouquet_ of loops: with
$a(n)$ disjoint _simple_ loops (from $r$ back to $r$) of length $n$. The
resulting shift space which we refer to as a _bouquet shift_ is
$\Sigma=\Sigma_{V}$: it has $a(n)$ periodic cycles of prime period $n$.

Below we will make various choices of $(a(n))_{n}$ and potentials
$\phi:\Sigma_{V}\to\mathbb{R}$. Our analysis will be via first returns to
$[r]$. Note that [Ru1, Ru2] were concerned with measures of maximal entropy
(in which case we set $\phi\equiv-h_{top}(\sigma)$), rather than the more
general setting of equilibrium states that we are interested in here.

For various calculations in this section we must replace $V$ with
$\mathbb{N}$. This can be done by listing the elements of $V$ in the natural
order, starting with $r$. Before moving to more specific examples we prove a
useful lemma, which applies to all bouquet shifts.

######  Lemma 5.1.

$h_{\infty}=\limsup_{n\to\infty}\frac{1}{n}\log a(n)$.

###### Proof.

Suppose that $\limsup_{n\to\infty}\frac{1}{n}\log a(n)=\log\lambda$, which we
may assume is finite, as otherwise the conclusion is immediate. Then for
$\varepsilon>0$ there is $C>0$ such that for an infinite sequence of
$n\in\mathbb{N}$,

$\frac{1}{C}\lambda^{n(1-\varepsilon)}\leq a(n)\leq
C\lambda^{n(1+\varepsilon)},$

and indeed the upper bound holds for all $n\in\mathbb{N}$.

We first show that $h_{\infty}\geq\limsup_{n\to\infty}\frac{1}{n}\log a(n)$.
Notice that for $M,q\in\mathbb{N}$, if $n$ is large enough so that $(n+1)M\geq
1$ and so that any of the simple loops of length $n$ only intersect $[\leq q]$
at the root $1$ (for example if $n>n_{q}$ where
$q\leq\sum_{k=1}^{n_{q}}a(k)$), we have $z_{n}(M,q)\geq
a(n)\geq\frac{1}{C}\lambda^{n(1-\varepsilon)}$ for an infinite sequence of
such $n$, hence the required lower bound holds.

For the upper bound, let

$\mathcal{P}(q):=\left\\{u\in\Sigma^{*}:[u]_{i}\leq q\text{ for
}i=0,\ldots|u|-1\text{ and }u1\in\Sigma_{|u|+1}\right\\},$
$\mathcal{G}(q):=\left\\{v\in\Sigma^{*}:[v]_{0},[v]_{|v|-1}=1\right\\}.$
$\mathcal{S}(q):=\left\\{w\in\Sigma^{*}:[w]_{i}\leq q\text{ for
}i=0,\ldots|w|-1\text{ and }1w\in\Sigma_{|w|+1}\right\\}$

be the set of $q$-prefixes, $q$-good words, and $q$-suffixes respectively.
Then any cylinder $[x_{0},\ldots,x_{n}]\in B(n,M,q)$ can be decomposed so that
$x_{0},\ldots,x_{n}=uvw$ where
$u\in\mathcal{P}(q),v\in\mathcal{G}(q),w\in\mathcal{S}(q)$.

Then

$\displaystyle z_{n}(M,q)$
$\displaystyle\leq\\#\left\\{uvw\in\Sigma_{n+1}:u\in\mathcal{P},v\in\mathcal{G},w\in\mathcal{S}\right.$
$\displaystyle\hskip 99.58464pt\left.\text{ and }\\#\\{0\leq
i\leq|v|-1:[v]_{i}=1\\}\leq\frac{|v|}{M}\right\\}$

which can be bounded above by
$\\#\mathcal{P}(q)\\#\mathcal{S}(q)\tilde{z}_{n}(M,q)$ where

$\displaystyle\tilde{z}_{n}(M,q)$ $\displaystyle\leq\sum_{k\leq
n/M}\sum_{i_{1}+\cdots+i_{k}=n}a(i_{1})\cdots a(i_{k})$ $\displaystyle\leq
C\sum_{k\leq n/M}\sum_{i_{1}+\cdots+i_{k}=n}\lambda^{n(1+\varepsilon)}\leq
C2^{\frac{n}{M}}\lambda^{n(1+\varepsilon)}=C\left(2^{\frac{1}{M}}\lambda^{1+\varepsilon}\right)^{n},$

so taking appropriate limits, and noting that
$\\#\mathcal{P}(q),\\#\mathcal{S}(q)<\infty$, we obtain
$h_{\infty}=\log\lambda$. ∎

###  5.2. (UCS) is a weak condition

Here we will use a simple set of examples to compare (UCS) with other
conditions of this type.

Set $a(n)=1$ for all $n$, $\phi|_{[rv_{1}^{n,1}]}=-n\log 2$ and $\phi=0$
otherwise. For the first return map to $[r]$ the induced potential
$\overline{\phi}:[r]\to\mathbb{R}$, takes the value $-n\log 2$ on the vertex
corresponding to the loop of length $n$. Then

$P(\overline{\phi})=\log\left(\sum_{n\geq 1}\frac{1}{2^{n}}\right)=0.$

Since, moreover, $\sum_{n\geq 1}\frac{n}{2^{n}}<\infty$, $\phi$ is positive
recurrent and has $P(\phi)=0$.

The system $(\Sigma_{V},\sigma,\phi)$ clearly satisfies (UCS) since for the
periodic point $x_{n}$ of prime period $n$, $\frac{1}{n}S_{n}\phi(x_{n})=-\log
2$. On the other hand, the hyperbolicity condition as in [IR, LR1] fails since
for any $n$, there is a point $y_{n}\in[v_{1}^{n,1}v_{2}^{n,1}\cdots
v_{n-1}^{n,1}]$ such that $S_{n}\phi(y_{n})=0$. Moreover, (A), (C) and (D)
also fail for the same reasons, as well as a condition like (CI) where orbits
are not assumed to start in a compact part. Finally, regarding the conditions
of [LSV], this would require $\sum_{C\in\mathcal{C}_{1}}\sup_{x\in
C}e^{\phi(x)}<\infty$ as well as a condition like hyperbolicity to hold, both
of which fail here.

We can modify the example to $\phi|_{[v_{n-1}^{n,1}r]}=-n\log 2$ and 0
otherwise, to obtain the same induced system as above, but here we see that we
can fail (B), as well as condition like (CI) where orbits are not assumed to
end in a compact part, whilst satisfying (UCS). If we wished to fail all of
these conditions apart from (UCS), for a loop of length $n$ we could put the
weight halfway along the loop.

######  Remark 5.2.

If we wanted $\phi$ to be uniformly bounded, then the above examples can be
smoothed out, eg putting weight $-2\log 2$ on $n/2$ of the vertices in the
loop of length $n$ (suitably adjusting for when $n$ is odd).

###  5.3. Example the showing the sharpness of Theorem 4.1

Here we give a class of examples where (UCS) holds, but
$\delta_{\phi,\infty}+h_{\infty}=0$ and (SPR) fails, so that the condition
$\delta_{\phi,\infty}+h_{\infty}<0$ in Theorem 4.1 is necessary.

Let $a(n)=2^{n}$ and $C,\beta>0$ to be chosen later. Now define
$\phi|_{[rr]}=\log{C}$, $\phi|_{[rv_{1}^{n,i}]}=\log{C}-n\log 2-\beta\log n$
and $\phi=0$ otherwise (as in Remark 5.2 we could also spread this potential
out if desired).

First observe that $Z_{n}(\phi,[r])\geq C2^{n}2^{-n}n^{-\beta}$, so
$P(\phi)\geq 0$.

Taking the first return map to $[r]$ the induced potential $\overline{\phi}$
corresponding to loops of length $n$ takes the value $\log C-n\log 2-\beta\log
n$. Then

$P(\overline{\phi})=\log\left(C\sum_{n}a(n)e^{-n\log 2-\beta\log
n}\right)=\log\left(C\zeta(\beta)\right)$

where $\zeta$ denote the Riemann zeta function. We use the ideas of Hofbauer
and Keller presented in [IT, Section 4.1], generalised to this setting (see
also the ideas of [Ru1, Table 1]).

  * (a)

If $\beta>1$ and we choose $C=1/\zeta(\beta)$ then the pressure of the induced
system is zero, $\phi$ is recurrent and $P(\phi)=0$.

  * (b)

If $\beta>1$ and $C>1/\zeta(\beta)$, or $\beta\in(0,1]$, then the pressure of
the induced system is positive and this is not interesting for our purposes
(note this would imply $P(\phi)>0$).

  * (c)

If $\beta>1$ and $C<1/\zeta(\beta)$ then $\phi$ is transient and $P(\phi)=0$.

We will now assume that we are in case (a).

Since

$C\sum_{n}na(n)e^{-n\log 2-\beta\log n}=C\sum_{n}n^{1-\beta},$

the system is positive recurrent, and we have an equilibrium state
$\mu_{\phi}$ here if $\beta>2$ (if $\beta\in(1,2]$ then $\phi$ is null
recurrent), and a conformal measure $m_{\phi}$. Since $h_{top}(\sigma)$ must
solve

$1=\sum_{n}a(n)e^{-nh_{top}(\sigma)}=\sum_{n}2^{n}e^{-nh_{top}(\sigma)},$

we see that $h_{top}(\sigma)=\log 4$.

The fact that $h_{\infty}=\log 2$ follows from Lemma 5.1. We next show that
$\delta_{\phi,\infty}=-\log 2$. That $z_{n,\phi}(M,q)\geq-\log
2+\frac{1}{n}(\log C+\beta\log n)$ for $n+1>M$ and $n>n_{q}$ is immediate from
the definition, so $z_{n,\phi}(M,q)\geq-\log 2$. For the upper bound, the
proof is similar to, though simpler than, that of Lemma 5.1: if we consider
$v\in\mathcal{G}(q)$ as defined there, then for $x\in[v]$,
$\frac{1}{|v|}S_{|v|}\phi(x)\leq-\log 2$ and since the finite behaviour
contributed by any prefixes and suffixes disappears in the limit,
$\delta_{\phi,\infty}=-\log 2$ and so $\delta_{\phi,\infty}+h_{\infty}=0$.

We see here that $Z_{n}^{*}(\phi,r)=C/n^{\beta}$ so (SPR) fails. Hence Theorem
4.1 is sharp in the sense that we can satisfy (UCS), but if
$\delta_{\phi,\infty}+h_{\infty}<0$ does not hold then we can fail (SPR). Note
also that if $\phi$ was null recurrent or, as in case (c) above, transient, we
would also fail these conditions in a more dramatic way.

###  5.4. Infinite entropy case

If $h_{top}(\sigma)=\infty$ then $h_{\infty}=\infty$, so
$h_{\infty}+\delta_{\phi,\infty}<0$ doesn’t make sense, and anything can
happen. For example, take $a(n)=2^{2^{n}}$ and set $\phi|_{[rr]}=\log C$,
$\phi|_{[rv_{1}^{n,i}]}=\log C-2^{n}\log 2-\psi(n)$ and $\phi=0$ otherwise,
where $\limsup_{n}\frac{1}{n}\log|\psi(n)|<\infty$ and $C$ is chosen so that
$P(\overline{\phi})=0$. Then we can easily ensure (UCS) by making $\psi$ not
large for small $n$, but we can also pick $\psi(n)$ so that the system
satisfies SPR, or choose it so that it doesn’t.

###  5.5. Relation of bouquets to inducing schemes and general shifts

At the beginning of this section we described interval maps $(I,f)$ with an
inducing scheme $(Y=\cup_{i}Y_{i},F=f^{\tau})$. If we have $F(Y_{j})=Y$ for
all $i$, which is the case for the examples mentioned above, then we identify
$Y$ with $r$, let $a(n)$ be the number of $j$ with $\tau_{j}=n$, and associate
a loop $r\mapsto v_{1}^{n,i_{j}}\mapsto v_{2}^{n,i_{j}}\mapsto\cdots\mapsto
v_{n-1}^{n,i_{j}}\mapsto r$ with each such $j$.

We can project a sequence $(x_{0},x_{1},\ldots)\in\Sigma$ to $x\in I$ by a
projection $\pi$ as follows. Suppose that $x\in Y$ has
$F^{\ell}(x)\in[rv_{1}^{n_{\ell},i_{\ell}}]$ for all $\ell\geq 0$ some
$n_{\ell},i_{\ell}$. Then there will be a corresponding sequence
$(x_{0},x_{1},\ldots)\in\Sigma$ given by
$(r,v_{1}^{n_{0},i_{0}},v_{2}^{n_{0},i_{0}},\ldots,v_{n_{0}-1}^{n_{0},i_{0}},r,v_{1}^{n_{1},i_{1}},\ldots)$.
So let $\pi(x_{0},x_{1},\ldots)=x$ here. If $x_{0}=v_{k}^{n,i_{j}}$ for $k>1$
then consider $y\in Y$ the projection of the sequence
$(r,v_{1}^{n,i_{j}},\ldots,v_{k-1}^{n,i_{j}},x_{0},x_{1},\ldots)$ and let
$\pi(x_{0},x_{1},\ldots)=f^{k}(y)$.

If $\phi:I\to[-\infty,\infty]$ is a potential, then this lifts to a potential
on the bouquet shift $\phi\circ\pi$. The regularity of the lifted potential
depends on the regularity of the original one and the choice of inducing
scheme. For some specific cases of multimodal maps where $\phi=-\log|Df|$ and
there is an inducing scheme so that $\phi$ lifts to a potential of summable
variation, see for example [BLS, Proposition 4.1] which considers multimodal
maps with different rates of growth of derivative along critical orbits. In
this case Collet-Eckmann maps yield symbolic models satisfying (UCS) along
with our other equivalent properties, while non-Collet-Eckmann maps fail all
of these.

We can extend a version of the coding used above for any topologically
transitive CMS $(\Sigma,\sigma)$: we can pick a 1-cylinder and take first
returns to it and then use the induced system to recode the system via a
bouquet with the root being the 1-cylinder selected. Hence the bouquet setup
captures the behaviour of any topologically transitive CMS.

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# Spin-dependent multiple reentrant localization in an antiferromagnetic helix
with transverse electric field: Hopping dimerization-free scenario

Sudin Ganguly<EMAIL_ADDRESS>Department of Physics, School of Applied
Sciences, University of Science and Technology Meghalaya, Ri-Bhoi-793101,
India  Kallol Mondal<EMAIL_ADDRESS>School of Physical
Sciences, National Institute of Science Education and Research Bhubaneswar,
Jatni, Odisha-752050, India  Homi Bhabha National Institute, Training School
Complex, Anushaktinagar, Mumbai-400094, India  Santanu K. Maiti
<EMAIL_ADDRESS>Physics and Applied Mathematics Unit, Indian
Statistical Institute, 203 Barrackpore Trunk Road, Kolkata-700108, India

###### Abstract

Reentrant localization (RL), a recently prominent phenomenon, traditionally
links to the interplay of staggered correlated disorder and hopping
dimerization, as indicated by prior research. Contrary to this paradigm, our
present study demonstrates that hopping dimerization is not a pivotal factor
in realizing RL. Considering a helical magnetic system with antiferromagnetic
ordering, we uncover spin-dependent RL at multiple energy regions, in the
absence of hopping dimerization. This phenomenon persists even in the
thermodynamic limit. The correlated disorder in the form of Aubry-André-Harper
model is introduced by applying a transverse electric field to the helical
system, circumventing the use of traditional substitutional disorder.
Described within a tight-binding framework, present work provides a novel
outlook on RL, highlighting the crucial role of electric field,
antiferromagnetic ordering, and the helicity of the geometry.

###### pacs:

72.80.Vp, 72.25.-b, 73.23.Ad,

Introduction: The metal-to-insulator transition is an inevitable consequence
of disorder. Despite this, the single-particle wave function displays
significant distinctions in behavior when exposed to uncorrelated (random)
disorder compared to correlated (quasiperiodic) one. In the presence of random
disorder, Anderson localization al1 ; al2  takes place. The scaling theory g4
of Anderson localization predicts that single-particle states in one- and two-
dimensional systems will exhibit exponential spatial localization, even when
subjected to extremely weak disorder. Consequently, this leads to the absence
of a single-particle mobility edge mott  (SPME). However, an energy-dependent
mobility edge can exist in three-dimensional systems. In the context fo
Anderson localization, extensive investigations have been conducted on
numerous fascinating systems across various branches of physics al2 ; al3 ;
al4 ; al5 ; al6 ; al7 ; al8 .

In contrast to the uncorrelated disorder, correlated disorder provides the
advantage of a sharply-defined critical point for the extended-localized phase
transition aah2 ; fls1 ; fls2 , as well as exhibiting fractal eigenmodes fb1 ;
fb2  and critical behavior fls1 ; crtb  in low-dimensional cases. Among the
variety of quasiperiodic models aah2 ; fls1 ; fls2 ; fb1 ; fb2 ; crtb ; cd1 ;
cd2 ; cd3 ; cd4 ; cd5 , the Aubry-André-Harper (AAH) model aah1 ; aah2  stands
out as the most widely recognized and versatile example. The AAH model, akin
to Anderson localization, lacks SPME due to the presence of a distinctly
defined critical point characterizing the extended-localized phase transition
aah2 ; fls1 ; fls2 . Nonetheless, researchers have explored various
generalizations of the standard AAH model to overcome this limitation. These
extensions encompass diverse features, such as exponential short-range hopping
gaah1 , flatband networks gaah2 , higher dimensions gaah3 , power-law hopping
gaah4 , flux-dependent hopping gaah5 , and nonequilibrium generalized AAH
models gaah6 , among others. Furthermore, the feasibility of AAH systems has
been demonstrated through experimental realizations utilizing cold atoms and
optical waveguides exaah1 ; exaah2 . These experimental implementations
provide crucial platforms for studying the behavior of AAH models in
controlled settings, offering valuable opportunities to explore and validate
theoretical predictions in the realm of quantum simulation and condensed
matter physics.

Conventionally, it is known that once a state is localized, it continues to
remain localized even when the disorder strength is increased. However, recent
findings have overturned the long-standing belief. Recently, researchers
predicted band-selective localization/delocalization transitions in 1D tight-
binding chain reentrant9  and a spin chain with antiferromagnetic nearest-
neighbor (NN) coupling reentrant10  in the presence of an interpolating AAH-
Fibonacci on-site potential modulation. Experimental validation was performed
using cavity-polariton devices for the 1D tight-binding chain reentrant9 .
Analogous theoretical outcomes emerged in a 1D NN chain, attributed to the
interplay between dimerized hopping and staggered on-site AAH disorder
reentrant1 . This reappearance of delocalized states from localized ones, as
the staggered AAH disorder strength increases, was termed as reentrant
localization (RL). The incorporation of the dimerized 1D chain and staggered
on-site AAH disorder have resulted in several studies aiming to achieve RL
reentrant2 ; reentrant3 ; reentrant4 ; reentrant5 ; reentrant6 ; reentrant7 ;
reentrant8 . Drawing upon the insights provided in Refs. reentrant1 ;
reentrant2 ; reentrant3 ; reentrant4 ; reentrant5 ; reentrant6 ; reentrant7 ;
reentrant8 , the present study poses a fundamental inquiry: Does RL arise
exclusively from the interplay between hopping dimerization and staggered on-
site AAH disorder?

In the antiferromagnetic helix (AFH), we assume that the neighboring magnetic
moments are aligned along $\pm z$ directions, and since in our framework we
consider $\sigma_{z}$ as diagonal, the full TB Hamiltonian of the AFH can be
decoupled as a sum of up and down spin sub-Hamiltonians helec6 . Due to the
anti-parallel configuration of successive magnetic moments, the staggered
condition in site energies is easily achieved. Moreover, in the presence of
the helicity of the geometry, site energies are correlated in the well-known
AAH form upon the application of a transverse electric field helec1 ; helec2 ;
helec3 ; helec4 ; helec5 ; helec6 ; helec7 . So, the helicity, AFM ordering
and electric filed make the system a unique one. In such a system, probably
for the first time we explore spin-dependent RL phenomena at multiple
energies, completely in the absence of hopping dimerization. The localization
behavior is examined by inspecting various aspects, including the single-
particle energy spectrum, inverse participation ratio (IPR), participation
ratio (PR), and other relevant measures.

The key findings of the present work are: (i) spin-dependent RL occurs in the
presence of electric field in AFH, where hopping dimerization is not a key
factor, (ii) total of three RLs are observed, and (iii) The three instances of
RL are robust as they persist in the thermodynamic limit.

System and theoretical framework: The schematic diagram of the proposed setup
is illustrated in Fig. 1. It depicts a right-handed AFH comprising $N$
magnetic sites, where the magnetic moments of successive sites are aligned in
opposite directions $(\pm z)$. An external electric field of magnitude $E_{g}$
is applied perpendicular to the helix axis.

Figure 1: (Color online). Schematic diagram of an antiferromagnetic right-
handed helix. Red balls denote the magnetic sites, where each ball represents
a magnetic moment with an arrow indicating its direction. $E_{g}$ is the
electric field, applied perpendicular to the helix axis. $l_{1}$, $l_{2}$, and
$l_{3}$ are the first, second, and third neighbor distances, respectively.
$\Delta z$ is the stacking distance, the distance along $z$-axis between two
neighboring sites. $\phi=n\Delta\phi$, where $\Delta\phi$ is the twisting
angle between the neighboring sites and $n$ is the site index.

In the AFH, each magnetic site $n$ is characterized by a net spin
$\langle\mathbf{S}_{n}\rangle$. The orientation of these spin vectors is
described by the polar angle $\theta_{n}$ and the azimuthal angle
$\varphi_{n}$. When an incoming electron interacts with these local magnetic
moments, the standard spin-moment exchange interaction $J$ is considered. To
incorporate this interaction, a spin-dependent scattering (SDS) parameter is
introduced at each site $n$ as
$\contour[1]{black}{$\mathpzc{h}$}_{n}=J\langle\mathbf{S}_{n}\rangle$ sds .
The magnitude of SDS parameter is denoted as
$|\contour[1]{black}{$\mathpzc{h}$}|$, assumed to be isotropic, i.e.,
$\contour[1]{black}{$\mathpzc{h}$}_{n}=\mathpzc{h}~{}~{}\forall~{}n$. It is
noteworthy that, for the current investigation, interactions between
neighboring magnetic moments are neglected and left for future study.

The tight-binding Hamiltonian for the AFH system in the presence of an
electric field is expressed as

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\bm{c}_{n}^{\dagger}\left(\bm{\epsilon}_{n}-\contour[1]{black}{$\mathpzc{h}$}_{n}\cdot\bm{\sigma}\right)\bm{c}_{n}$
(1) $\displaystyle+$
$\displaystyle\sum_{n=1}^{N}\sum_{m=1}^{N-n}\left(\bm{c}_{n}^{\dagger}\bm{t}_{m}\bm{c}_{n+m}+h.c.\right),$

where, $\bm{c}_{n}^{\dagger}$, $\bm{c}_{n}$, $\bm{\epsilon}_{n}$, $\bm{t}_{m}$
are the two-component quantities for the spin-1/2 case and they read as

$\displaystyle\bm{c}_{n}^{\dagger}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}c_{n\uparrow}^{\dagger}&c_{n\downarrow}^{\dagger}\end{pmatrix},\;\,\quad\bm{c}_{n}=\begin{pmatrix}c_{n\uparrow}\\\
c_{n\downarrow}\end{pmatrix},$ $\displaystyle\bm{\epsilon}_{n}$
$\displaystyle=$
$\displaystyle\text{diag}\left(\epsilon_{n},\epsilon_{n}\right),\quad\bm{t}_{m}=\text{diag}\left(t_{m},t_{m}\right).$
(2)

Here ${c}_{n\alpha}^{\dagger}\left(c_{n\alpha}\right)$ are the creation
(annihilation) operator at the $n$th site with spin
$\alpha(=\uparrow,\downarrow)$. $\epsilon_{n}$ is the on-site potential at
site $n$ and $t_{m}$ is the hopping integral between the sites $n$ and $n+m$.

In the presence of an external electric field, perpendicular to the helix
axis, the site energy modifies as helec1

$\epsilon_{n}=eV_{g}\cos{(n\Delta\phi-\beta)},$ (3)

where $e$ is the electronic charge and $V_{g}$ corresponds to the gate voltage
associated to the applied electric field $E_{g}$ with $2V_{g}=2E_{g}R$.
$\beta$ is the angle between the positive $x$-axis and the incident electric
field. Such a modulation of the on-site potential (Eq. 3) can be mapped to the
AAH model aah1 ; aah2  with a suitable choice of $\Delta\phi$ helec3  and thus
a correlated disorder can be introduced in the helix with a disorder strength
$V_{g}$. The inclusion of SDS leads to the effective site energy matrix
$\left(\bm{\epsilon}_{n}-\contour[1]{black}{{$\mathpzc{h}$}}_{n}\cdot\bm{\sigma}\right)$,
where $\bm{\sigma}$ is the usual Pauli matrices.

The second term of Eq. 1 is associated with electron hopping in different
magnetic sites, where $t_{m}$ reads as helec1 ; helec5

$t_{m}=t\,{\rm{e}}^{-\left(l_{m}-l_{1}\right)/l_{c}}.$ (4)

Here $l_{m}$ is the Euclidean distance between sites $n$ and $n+m$, $l_{1}$
and $l_{c}$ are the nearest-neighbor distance and decay constant,
respectively. In terms of radius $R$ (Fig. 1) of the helix, twisting angle
$\Delta\phi$, and stacking distance $\Delta z$, $l_{m}$ takes the form

$l_{m}=\sqrt{\left[2R\sin{\left(\frac{m\Delta\phi}{2}\right)}\right]^{2}+\left[m\Delta
z\right]^{2}}.$ (5)

To analyze the localization behavior, two commonly used measures are the
inverse participation ratio (IPR) and its complementary counterpart, the
normalized participation ratio (NPR). For the $n$th normalized eigenstate,
they are defined as def-ipr1 ; def-ipr2

$\text{IPR}_{n}=\sum_{i}\lvert\psi_{n}^{i}\rvert^{4},\quad\text{NPR}_{n}=\left(N\sum_{i}\lvert\psi_{n}^{i}\rvert^{4}\right)^{-1}.$
(6)

In the case of a highly extended state, the IPR approaches to zero, while NPR
tends to unity. Conversely, for a strongly localized state, the IPR
approximately approaches to unity, and NPR to zero.

The earlier defined IPRn and NPRn can be adjusted to characterize the
parameter space region where localized and delocalized states coexist, by
calculating their averages over a specified subset of states labeled as
$N_{L}$, defined as def-ipr2 , and they are

$\langle\text{IPR}\rangle=\sum_{n}^{N_{L}}\frac{\text{IPR}_{n}}{N_{L}},\hfill\langle\text{NPR}\rangle=\sum_{n}^{N_{L}}\frac{\text{NPR}_{n}}{N_{L}}.$
(7)

When all $N_{L}$ states are localized (delocalized),
$\langle\text{IPR}\rangle$ tends to unity (zero) and
$\langle\text{NPR}\rangle$ tends to zero (unity). In the parameter range where
both $\langle\text{IPR}\rangle$ and $\langle\text{NPR}\rangle$ remain finite,
the Hamiltonian’s spectrum shows an intermediate phase. Here, spatially
extended and localized eigenstates coexist alongside SPME.

Results and Discussion: First, we investigate the spin-resolved IPR
characteristics of individual states (defined in Eq. 6), of the AFH in
presence of a transverse electric field. For the helical system, the modified
on-site energies induced by the electric field can be mapped into a correlated
disordered systemhelec1 ; helec2 ; helec3 ; helec4 ; helec5 ; sudin-prb . This
mapping effectively relates the modified on-site energies to AAH disorder,
with $V_{g}$ playing the role of the AAH disorder strength. We consider the
magnetic moments aligned along the $\pm z$ directions, allowing the
Hamiltonian of the AFH to be decoupled into up and down spin components
$\left(H=H_{\uparrow}+H_{\downarrow}\right)$. In the absence of an electric
field, $H_{\uparrow}$ and $H_{\downarrow}$ exhibit symmetry, resulting in zero
spin-splitting effect. However, the application of an electric field disrupts
this symmetry and a finite mismatch arises between $H_{\uparrow}$ and
$H_{\downarrow}$ along their respective energy channels sudin-prb .

We consider a right-handed helix, characterized by specific structural
parameters that render the system a short-range hopping helix. The chosen
parameters are radius $R=8\,$Å, stacking distance $\Delta z=4.3\,$Å, twisting
angle $\Delta\phi=\pi\left(\sqrt{5}-1\right)/4$, and decay constant
$l_{c}=0.8\,$Å. The selection of $\Delta\phi$ results in on-site energies
resembling an incommensurate potential, akin to the AAH disorder.

The number of sites is taken as $N=1598$ to make the net magnetization zero
($N$ is close to a Fibonacci number 1597). We choose the NNH strength
$t=1\,$eV and $\mathpzc{h}=0.9$. The direction of the electric field is
assumed to be parallel to the positive $x$-axis, that is $\beta=0$. It should
be noted that the parameter $\beta$ does not have an impact on the
localization properties reentrant9 .

It is important to note that as the system size approaches
$N\rightarrow\infty$, the values of IPR or NPR for distinct spin species
converge, leading to analogous results both for the up down-spin channels.
Therefore, here we concentrate exclusively on the up-spin case, without loss
of any generality, and exactly identical scenario will be obtained for the
down spin electrons.

As mentioned earlier, the present system Hamiltonian can be separated into up
and down spin components. In Fig. 2(a), we depict the energy spectra for the
up-spin channels, showcasing the variation with the gate voltage $V_{g}$
(measured in units of Volts). Each energy point on the plot is assigned a
specific color based on its corresponding IPR value, which is calculated
according to Eq. 6. To highlight the localization transition, the colorbar
employs dark gray to represent the lowest 10$\%$ of the maximum IPR values,
emphasizing extended states. The remaining portion of the color spectrum
ranges from white to dark red,

Figure 2: (Color online). Density plot of IPR for the up spin channel. (a) The
energy spectrum vs gate voltage $V_{g}$. (b) The energy index vs gate voltage
$V_{g}$. The color map shows IPR values of different energy eigenstates.
$t=1\,$eV, $\mathpzc{h}=0.9$, $\beta=0$, and $N=1598$. The three instances of
RL are denoted with RL1, RL2, and RL3 in (a). Their corresponding regions are
marked with gree ellipses.

visually representing the increasing degree of localization. Below the
threshold of approximately $V_{g}=0.5$, nearly all states exhibit an extended
nature, as noticed by the dark gray coloration. However, past the point of
$V_{g}\sim 1$, all states undergo a complete localization. The localization
persists until around $V_{g}\sim 1.5$, and subsequently, we identify three
occurrences of RL from the color-coded IPR values. The three instances of RL
are highlighted by green ellipses within the approximate $V_{G}$-window: RL1
from 1.5 to 1.6, RL2 from 1.78 to 1.8, and RL3 from 1.9 to 2. A more
insightful perspective can be gained by examining the individual eigenstates’
IPR, as shown in Fig. 2(b), where the three instances of RLs are more
prominently visible.

To observe the mixed-phase zone, we plot the behavior of
$\langle\text{IPR}\rangle$ and $\langle\text{NPR}\rangle$ as a function of
$V_{g}$ as shown in Fig. 3 for the up spin case, represented by the the red
and green colors, respectively. The averaging for $\langle\text{IPR}\rangle$
and $\langle\text{NPR}\rangle$ is performed over a subset of eigenstates
within the range of $30\%$ to $70\%$ of the total states as depicted in Fig.
2(b). All the system and other parameter values remain unchanged with those in
Fig. 2. In Fig. 3, both $\langle\text{IPR}\rangle$ and
$\langle\text{NPR}\rangle$ become finite for $0.7>V_{g}>1.2$, indicating a
critical region with a coexistence of extended and localized states.

Figure 3: (Color online). $\langle\text{IPR}\rangle$ (red curve) and
$\langle\text{NPR}\rangle$ (green curve) as a function of $V_{g}$ for a subset
of states ranging from 30 to 70$\%$ of the eigenstates of Fig. 2(b). All the
system parameters remain the same as described in Fig. 2. (Inset)
$\langle\text{NPR}\rangle$ versus $V_{g}$ plot for system sizes $N=1598$,
2584, 4182, 6766, 10946, 17712 and $N\rightarrow\infty$ represented by light
to dark green color.

Beyond $V_{g}\sim 1.2$, all states become fully localized. Subsequently, both
$\langle\text{IPR}\rangle$ and $\langle\text{NPR}\rangle$ attain finite values
in the previously mentioned three RL regions, namely, for $1.5>V_{g}>1.6$,
$1.78>V_{g}>1.8$, and $1.9>V_{g}>2$. Therefore, the system hosts as a total of
four SPMEs. To mitigate potential finite-size effects, we examine the behavior
of $\langle\text{NPR}\rangle$ across various system sizes, specifically,
$N=1598$, 2584, 4182, 6766, 10946, and 17712. Using the evaluated data, we
extrapolate $\langle\text{NPR}\rangle$ as $N\rightarrow\infty$. The
corresponding result is presented in the inset of Fig. 3. The same subset of
eigenstates as that of the main plot of Fig. 3 is used for the evaluation of
$\langle\text{NPR}\rangle$. A gradient of green color, transitioning from
light to dark, is employed to represent the behavior of
$\langle\text{NPR}\rangle$ as a function of $V_{g}$ for the system sizes in
ascending order. For $N\rightarrow\infty$, $\langle\text{NPR}\rangle$
converges to zero in all the three reentrant localized regions. This clearly
demonstrates the robustness of the occurrence of the three RLs with respect to
system size and rules out any finite-size effects.

Figure 4: (Color online). Density plot of $\eta$ in the (a) $V_{g}$-$h$, (b)
$V_{g}$-$t$, and (c) $V_{g}$-$l_{c}$ planes. The system size and all the other
parameters remain the same as described in Fig. 2.

Parameter space: With the confidence that the three RLs are not due to any
finite-size effect, we explore the parameter space to identify the conditions
under which these three RLs exist. To do so, we compute $\eta$, defined as eta

$\eta=\text{log}_{10}\left[\langle\text{IPR}\rangle\times\langle\text{NPR}\rangle\right].$
(8)

In the mixed phase zone, both $\langle\text{IPR}\rangle$ and
$\langle\text{NPR}\rangle$ are finite and $\mathcal{O}(1)$, yielding $\eta$
within $-2\leq\eta\leq-1$. Conversely, in localized (extended) regime,
$\langle\text{NPR}\rangle$ $\left(\langle\text{IPR}\rangle\right)$ tends
toward $\sim N^{-1}$, and $\eta<-\text{log}_{10}N$. For instance, at $N\sim
10^{3}$, $\eta<-3$. Hence, the quantity $\eta$ serves as a clear discriminator
between fully extended or localized phase and a mixed phase.

First, we explore the behavior of $\eta$ in the phase space of $V_{g}$ and
$\mathpzc{h}$ for the up spin channel as shown in Fig. 4(a). All other
parameters remain constant as indicated in Fig. 2. In the colorbar of $\eta$,
steel blue color corresponds to the extended or localized phase, while the
brown color represents the mixed phase. In Fig. 4(a), the localized or
extended phases are determined by analyzing the $\langle\text{IPR}\rangle$ and
$\langle\text{NPR}\rangle$ values. All three RLs emerge within the
$\mathpzc{h}$-range of approximately 0.75 to 0.96. The third RL ceases to
exist beyond $\mathpzc{h}\sim 0.96$. Additionally, no instances of RL are
present beyond $\mathpzc{h}\sim 1.2$. This observation strongly suggests that
RL occurrence is possible when $\mathpzc{h}$ is comparable to the numerical
value of $V_{g}$.

Second, we study the $\eta$-behavior in the phase space of $V_{g}$ and $t$ for
the up spin channel as shown in Fig. 4(b). All other parameters kept unchanged
as indicated in Fig. 2. As both $V_{g}$ and $t$ increase from zero to a finite
value, the width of the first critical region expands. The first RL emerges
around $t\sim 0.8$. The second RL becomes noticeable around $t\sim 0.9$, and
the third one around $t\sim 1$. The first RL diminishes beyond $t\sim 1.1$,
the second one around $t\sim 1.15$, and the third one for $t\sim 1.25$. All
three RLs are pronounced within the range of approximately $1<t<1.1$. This
range is comparable to the numerical value of $\mathpzc{h}$, which is fixed at
$\mathpzc{h}=0.9$, consistent with the previous analysis. After $t\sim 1.25$,
all RLs vanish and merge into the first critical region.

So far, all the results were discussed for the scenario of short-range
hopping. To investigate the transition from short-range hopping to long-range
hopping and understand the localization behavior, we examine the
$\eta$-behavior within the $V_{g}$ and $l_{c}$ phase space, as illustrated in
Fig 4(c). The variation of the decay constant spans from 0.1 Å to 8.5 Å,
thereby inducing a shift from a short-range hopping regime to a long-range
hopping scenario. In accordance with Fig. 2, the remaining parameters are
maintained at their specified values without any alterations. For low values
of $l_{c}$ (primarily SRH) within the range of 0.1 to 2 Å, the $V_{g}$-window
associated with the first critical region remains nearly constant and it
expands as $l_{c}$ increases. The three RLs are present right from
$l_{c}=0.1\,$Å. The first RL converges with the first critical region at
approximately $l_{c}\sim 0.75\,$Å, but the second and third RLs persist with
increasing $l_{c}$. The second RL unites with the first critical region while
the third one survives till $l_{c}\sim 4\,$Å. Ultimately, the third RL
disappears at approximately $l_{c}\sim 5\,$Å, leaving only one critical region
beyond that point. Another notable observation is that from the first RL, a
secondary RL emerges within a brief $V_{g}$ window with $l_{c}$ between 3 to 4
Å, and subsequently integrates with the second RL. A similar scenario is
observed for the second RL, wherein another secondary RL originates from
$l_{c}\sim 2\,$Å and then dissipates into the localized region beyond
$l_{c}\sim 3\,$Å.

Colclusion: The present investigation has revealed the occurrence of multiple
spin-dependent reentrant localization in a helical system when subjected to an
electric field with net zero magnetization, characterized by antiferromagnetic
ordering of the moments. Crucially, this accomplishment has been realized
without incorporating any hopping dimerization scenario, a factor upon which
previous studies attributing the occurrence of RL had relied. Our study of
spin-resolved IPR values across various energy states has revealed the
presence of three distinct RLs. The robustness of these three RLs has been
affirmed through a comprehensive analysis of averaged IPR and NPR values in
the limit as the system size approaches infinity ($N\rightarrow\infty$). The
investigation of $\eta$-behavior unravels that RL become apparent when the
electric field strength $V_{g}$ is on a comparable scale to the value of
$\mathpzc{h}$. Additionally, in the $V_{g}$-$l_{c}$ plane, it is notable that
RLs vanish as the system transitions from the short-range hopping to the long-
range hopping case. Furthermore, our investigation has identified two
secondary RLs stemming from the first and second RLs, which however warrants
further in-depth exploration.

Overall, exploring the practical implications of spin-dependent reentrant
localization without hopping dimerization could lead to innovative
applications in quantum technologies. Further, investigating the behavior of
RL in the $V_{g}$-$l_{c}$ plane holds promise for understanding and
controlling localization phenomena in helical systems, as the system undergoes
systematically from a short-range hopping to a long-range hopping regime.

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