# COVID-19 Vaccine Hesitancy and Information Diffusion: An Agent-based
Modeling Approach

Pooria Taghizadeh Naderi  University of Tehran  Ali Asgary  York University
Jude Kong  York University  Jianhong Wu  York University  Fattaneh Taghiyareh
University of Tehran

## Abstract

Despite the unprecedented success in rapid development of several effective
vaccines against the Cov-SARS-2, global vaccination rollout efforts suffer
from vaccine distribution inequality and vaccine acceptance, leading to
insufficient public immunity provided by the vaccine products. While a major
current focus in vaccine acceptance research is how to model and inform
vaccine acceptance based on social-demographic parameters, characteristics of
vaccine acceptance are not well understood and in particular it is not known
whether and how information diffusion influences vaccine acceptance. This
study examines how information diffusion can change vaccine acceptance by
developing a comprehensive computational model with agent-based simulation
technique to overcome the modeling and quantification complexity associated
with socio-demographics, vaccine types, population statistics, and information
diffusion. Our analyses, calibrated by the vaccine acceptance survey data from
the provinces and territories of Canada, provide clear evidence that
propagation of information can greatly influence vaccine acceptance rates. The
results illustrate that spread of negative messages about the COVID-19
vaccines can cause significant vaccine hesitancy that challenges the goal of a
high public immunity provided by the vaccines. Our findings might help solve
the vaccine hesitancy problem by focusing more on individuals’ opinions and
behavior.

##  1 Introduction

Since the beginning of the COVID-19 pandemic, the vaccine has been considered
a key solution to end the crisis. While many vaccine developers have been
working towards developing effective vaccines, vaccine development and
production alone are not enough to provide public immunity. Adequate number of
people must be fully vaccinated to create immunity in the population. Vaccine
hesitancy poses a challenge Malik et al. (2020); Higgins et al. (2021); Loomba
et al. (2021); Bloom et al. (2020); Salali and Uysal (2020) and is
increasingly becoming a vital factor in the vaccination process. There are
intensive discussions about vaccine acceptance among policymakers, academic
researchers, and the general public. Several studies, such as Kwok et al.
(2021); Gagneux-Brunon et al. (2021); DeRoo et al. (2020); Detoc et al.
(2020), have assessed vaccine hesitancy in nations, and it is apparent that
there is a considerable unwillingness in different countries that may
negatively impact the vaccination rate for the Cov-SARS-2. Unlike the previous
pandemics and diseases, information and misinformation about vaccines are
spread rapidly among the public through mass media and social media during the
COVID-19 pandemic. Therefore, understanding the flow of information within
social media platforms can help predicting how a product is accepted or
refused in the society and what elements can promote its acceptance or wider
use Naderi and Taghiyareh (2021).

An individual’s decision to be vaccinated or not is influenced by personal,
social, and political factors, including information received from family
members, friends, peers and users of online social networks. In order to
develop effective vaccination strategies and interventions, it is critical to
learn how social influences impact vaccine acceptance. Several factors such as
vaccine safety, disease severity, and the vaccine price play a key role in an
individual’s vaccination decision. However, few studies have been conducted to
assess how individuals’ beliefs and social ties influence vaccine acceptance.
In particular, individuals’ beliefs may change after exposure to feedback and
comments from those family members, friends, coworkers and other users sharing
the social media platform who have received the vaccine.

The main goal of this study is to examine how different factors, particularly
social influences, impact individuals’ decisions to take a vaccine and vaccine
propagation in a community using agent-based modeling and analysis. We believe
that we have designed an innovative simulation toolkit to study vaccine
acceptance that can be used in different societies. In our platform, we have
incorporated mechanisms to quantify the importance of information diffusion.
Our research and simulation tool could be a useful aid for decision-makers
because they give them a reliable platform to examine and test their
decisions. using the simulation tool, it is possible to measure how promoting
positive messages through social media, advertisements, and national media
campaigns might increase vaccine acceptance in the region or the country.

##  2 Background and literature review

In response to the COVID-19 pandemic, many studies have been conducted to
understand vaccine acceptance, vaccine hesitancy, and vaccine social
influence. Based on our literature review, most of these studies focus on
COVID-19 vaccine acceptance in general (28 articles), followed by much
empirical research on COVID-19 vaccine acceptance in a specific country (16
articles). Very few studies have used modeling and simulation approaches for
examining vaccine acceptance and propagation (3 articles).

###  2.1 Vaccine acceptance in general

Empirical research results show that despite vaccine availability, vaccination
hesitancy has typically been observed uncontrolled and varies widely among
different countries Sallam (2021). UK residents, for instance, displayed the
lowest hesitancy level of 10.1%, while Kuwaitis exhibited a very high
hesitancy level of 76.4% Sallam (2021); Lazarus et al. (2021). These studies
revealed that individuals’ characteristics, including age, gender, income, and
education contribute to vaccine hesitancy Murphy et al. (2021); Daly and
Robinson (2021). Other studies suggested that in the early period of vaccine
availability in the market, its hesitancy was much higher Fadda et al. (2020).
The study Latkin et al. (2021) also showed that a person’s trust in the
COVID-19 vaccine and factors contributing to mistrust may play a role in
vaccine acceptance in different societies. According to Latkin et al. Latkin
et al. (2021), trust is low when people believe or perceive that vaccines were
being distributed too soon without adequate testing, data, or proof of
success. Mistrust is also linked with uncertainty and lack of knowledge about
the long-term side effects, and people who adhere to natural immunity distrust
vaccines as a whole or do not like them. According to Latkin et al. (2021),
there are also concerns about the efficacy of the COVID-19 vaccines. The study
Callaghan et al. (2021) concluded that individuals who do not plan to get
vaccinated against COVID-19 do not believe that the vaccine will be safe or
effective. Other less significant reasons for vaccine refusal include lack of
insurance or financial resources. Several groups believe that they have
already contracted COVID-19 and therefore refuse to take vaccine. This study
Callaghan et al. (2021) also confirmed that women and African Americans have
greater hesitancy regarding COVID-19 vaccination, and they explore the reasons
why women and African Americans are less apt to refuse vaccination. It has
been reported that women did not pursue vaccination as often as men because
they did not consider the vaccine safe or believed that it would not be
effective. Moreover, based upon the statistical analysis of differences among
races, blacks are more likely than whites to decline vaccination for reasons
included in their survey, with statistical signals for safety concerns,
effectiveness, inadequate insurance, and financial resources. Another aspect
of vaccine acceptance that has not been studied in the past is the average
vaccination time. In a survey Muqattash et al. (2020), participants were asked
about vaccination time and its effect on vaccination decisions, and the
results indicated that average vaccination times affect vaccine acceptance
rates. Studies showed that vaccine prices play a significant role in vaccine
hesitancy, and in some countries, free vaccination is the result García and
Cerda (2020); Yin et al. (2021); Muqattash et al. (2020). García and Cerda
(2020) aimed to estimate the individual’s willingness to pay (WTP) in response
to a hypothetical COVID-19 vaccine. This study determined that the vaccine’s
price could affect vaccine acceptance in the society. Their results concur
with prior research, which found that the average family income could affect
vaccine resistance. Many people believe that the government should finance the
vaccine.

###  2.2 Misinformation

Most studies have focused on individual characteristics like gender, race,
education, job, and vaccine attributes like price and expiry dates. However,
another crucial aspect that has received less attention is the misinformation
that participants receive through social media or their family members and
friends. A research by García et al. García and Cerda (2020) involved a
randomized controlled trial in the UK and the USA to determine how online
misinformation on COVID-19 vaccines influences individuals’ intent to protect
themselves and others. They asked participants to fill a survey and answer
questions related to their intention before being exposed to misinformation.
Individuals intent to get vaccinated is influenced by misinformation based on
different socioeconomic factors. For example, the UK results showed that
unemployed persons are more susceptible to misinformation than employed
individuals. Further, whites and lower-income groups in the USA are more
vulnerable to misinformation than other racial and ethnic groups.
Surprisingly, trusting celebrities is associated with greater resilience to
misinformation in the UK than controls. Conversely, trusting family members
and friends in the USA is associated with lower resistance to misinformation.
The studyChou and Budenz (2020) discussed several methods of leveraging
emotion in vaccine communication for the COVID-19 vaccines, both in the short
term to address vaccine hesitancy and in the long term to promote vaccine
confidence. They discovered that misinformation about vaccines could be
eliminated by making society aware of how misinformation campaigns use
negative emotions and beliefs.

The effects of misinformation campaigns on vaccination intentions have been
documented in Loomba et al. (2021). However, they have not seemed proportional
to how information has reached a person in social media platforms as a complex
mix of what media has shown and what others have shared. In the current
environment of online social network structures, which result in homophilic
social interaction, selective exposure and creation of echo chambers or
polarization of opinions can amplify or dampen the spread of misinformation.
Current studies introduce many different aspects of vaccine hesitancy which we
summarize them in Table 1. However, they were limited by the fact that they do
not create a detailed model that would demonstrate similar behaviors to what
would occur in the real world in the case of the COVID-19 pandemic.

Table 1: A snapshot of the main influences on vaccination decision-making among individuals Hesitancy reasons |  Reference
---|---
Too new, worried about unforeseen effects, Vaccine fast-tracking |  Latkin et al. (2021); Paul et al. (2021)
Compromised immune systems, preference for natural immunity |  Latkin et al. (2021); Paul et al. (2021)
Profit distrust, concerns about commercial profiteering |  Latkin et al. (2021); Paul et al. (2021)
Current government distrust |  Latkin et al. (2021)
Vaccine contents, Vaccine skepticism, Vaccine refusal in general, General mistrust of vaccine benefits, General skepticism, Vaccine being unnecessary, trust in research and in vaccines |  Latkin et al. (2021); Ni et al. (2021); Goldman et al. (2020); Paul et al. (2021); Palamenghi et al. (2020)
Virus strain/mutation |  Latkin et al. (2021); Loomba et al. (2021)
Doubt in efficacy, Doubt in efficacy, Vaccine Won’t be Effective |  Latkin et al. (2021); Goldman et al. (2020); Callaghan et al. (2021)
Side effects, Side effects/safety concerns, Vaccine Won’t be Safe, Perceived Contraindication |  Ni et al. (2021); Goldman et al. (2020); Callaghan et al. (2021)
Perceived low risk of disease, Perceived child not at risk to contract COVID19, Already had COVID-19 |  Ni et al. (2021); Goldman et al. (2020); Callaghan et al. (2021)
Difficulty or inconvenience in accessing vaccination services |  Ni et al. (2021)
Vaccinate if more information available |  Goldman et al. (2020)
Lack of Insurance, Lack of Financial Resources |  Callaghan et al. (2021)

##  3 Materials and Method

We develop an agent-based model to describe the vaccine acceptance and the
influence of communication, vaccine characteristics, and socio-demographic
attributes of individuals. We use AnyLogic Software version 8.7.3. We simulate
if and when people accept vaccines and the model assesses the impacts of
different communication channels on people’s vaccine acceptance behavior.
Simulation can run for any period to capture the dynamics of vaccine
acceptance over time and demonstrate the latest information about vaccine and
disease spread. The simulation has been developed to be run for any country
given the initial parameter values is available at country and regional levels
within a country. In this paper we apply the model to Canada and its 13
provinces and territories to demonstrate the simulation results. Our agent
based model consists of four agent types: Person, Country, Vaccine and
Communication.

###  3.1 Person Agent

The person agent is the major agent in this ABM. Figure 1 shows the main
statechart for the Person agent. Once a person agent is created a logistic
regression function runs that indicates the initial vaccine acceptance of the
person based on the agent’s attributes such as age, gender, education, income,
and race (Table 2). The logistic regression (Eq.1) used in this study is based
on the previous studies on vaccine hesitancy in different countries and
regions Loomba et al. (2021); Lazarus et al. (2021); Borriello et al. (2021);
Alqudeimat et al. (2021). For each demographic feature, $OR_{i}$ indicates the
odds ratio, while CO represents the coefficient value that represents the sum
of all $OR_{i}$, and P stands for vaccine acceptance probability.

$CO=\sum\log(OR_{i})\\\ P=\frac{e^{CO}}{{e^{CO}}+1}$ (1)

Result: Next state

$P,N,Opinion=0$

while  _ $state\in\\{Belief,Disbelief\\}$ _ do

m = new message

if  _m > 0_ then

P = P + 1

else

N = N + 1

end if

$Opinion=\frac{P}{P+N}$

if  _$state=Belief$ and $Opinion<\theta$_ then

return Hesitant

else

if  _$state=Disbelief$ and $Opinion>\theta$_ then

return Hesitant

else

Continue

end if

end if

end while

return Belief

Algorithm 1 Information diffusion algorithm for disbelief state

We assume that all persons are in a hesitant state with the vaccine at the
beginning. Their status changes to either Disbelief or Belief states based on
the probability we calculated in Eq.1. A person’s Hesitant, Disbelief and
Belief states are influenced by the information received from different
sources including friends, media and government. In order to measure positive
and negative information diffusion, we used a threshold-based algorithm Singh
et al. (2018), which counts the number of positive and negative information
users may receive in each step. Based on the algorithm 1, we decide to change
the user state from the Belief or the Disbelief to Hesitant at an appropriate
rate. This algorithm requires two input parameters, Teta and InfoEffect. Teta
is used to control when a person changes his/her state, and InfoEffect
measures how quickly an agent moves from the Belief state or the Disbelief
state to the Hesitant state. For example, in the case where Teta = 0.3, you
may see that the person changes his/her state from the Disbelief to the
Hesitant if the number of positive messages exceeds 30 percent of the total
messages. A person that is in the Belief state is going to take the vaccine
(first dose in cases of two doses vaccine) when it becomes available to her or
him. Also, with the addition of the Vaccine Passport (or certificate)
parameter to the Person agent, the vaccine passport can now be considered
mandatory. Regardless of being in Disbelief, in this case, if the vaccine
passport parameter is set to 1, person agents would consider getting the
vaccine. If the person experiences side effects, he/she may send negative
information to others that may influence their vaccine acceptance level. If
the person becomes infected after the first or second dose, he or she may send
negative information to others. On the other hand, if this person becomes
immune, he or she will send a positive message to others. We assume that the
information diffusion between individuals that are connected can change
vaccine acceptance in the society. In the simulation tool, users can set the
parameters according to their study area information and then investigate
different scenarios.

Figure 1: Person agent state chart Table 2: Summary of Agent person parameters Person agent attributes |  Description |  Attribute values/range
---|---|---
Age |  Age of the person |  18-29; 30-39; 40-49;50+
Sex |  Sex of the person |  Male, Female
Education |  Education level of the person |  High school and less; University/college degree
Income |  Income of the person |  Low; Medium; High
Race |  Racial background of the person |  White, black, Asian, other
Contact rate |  Average of daily contacts between the person and others |  1
Vaccine passport |  Whether having vaccine passport is used in the country or not |  Binary, 0, 1
Vacacine passport effect rate |  How much effective is vaccine passport rule |  Vaccination speed in the country /10
DedicatedVaccine Type |  Type of vaccine provided to the person |  E.g. Pfizer, Moderna
Acceptance |  Vaccine acceptance level by the person |  0-1

###  3.2 Province agent

The country/Province agent is linked with the person agent and holds the
demographic, vaccination, COVID-19, and vaccine hesitancy information. Custom
distributions are used for age, education, gender, race, income, and vaccine
distribution. All parameter values are extracted from a database that can be
updated as new information becomes available. The agent province parameters
consist of ten items. The following operators made up Equation 1: Sex OR,
Education OR, Income OR, and Race OR. Furthermore, we created a random network
between agents using Link Per Agent and Population Total to simulate real-life
connections between agents. Additionally, we explored information diffusion
using Teta, the InfoEffect, and Media Effects. A GIS map is used to
demonstrate the vaccine Person agents are living in the province agent and
thus receive relevant attributes from their host province.

###  3.3 Vaccine agent

Vaccine agent represents vaccine and its attributes and are linked with the
Person and Country agents. Vaccine parameters are also stored in the database
and are updated as new information about vaccines become available. Vaccine
parameters include name, manufacturer, efficacy, origin, number of doses
required, development approval, type, and side effects. In each case,
different types of vaccines are distributed among individuals with a different
percentage. In Canada, Pfizer, Moderna, and Oxford/AstraZeneca vaccines are
mostly used. Nevertheless, the data on all vaccines are available in the model
database and LABEL:covidTracker if they need to be considered.

###  3.4 Communication agent

The communication agent is essential in this model. We assume that the Person
agent’s vaccine acceptance is influenced by different communication agents,
including Radio, Newspapers, Television, Social media, Government Agencies,
and Health Professionals. Based on the simulation, a communication agent can
either be active or inactive. Additionally, an agent can send positive or
negative messages, affecting individual users based on the Effect Rate
parameter, and become dormant for some time. Various communication agents have
different effect rates, and we considered this in our Communication agent. For
instance, social media can have a more significant influence than radio.

###  3.5 Experiment setup

As our model can be applied to any region with a population, we selected
Canadian provinces and territories to validate it. We need to provide data
such as vaccination type distributions and population distributions in order
to run the simulation. As a primary distribution for Canada, we used Little.
(2021) and statcan. In order to calculate individuals’ acceptance of the
vaccine, we needed an odd ratio of vaccine acceptance or hesitancy, so we used
Lunsky et al. (2021) as the input for the logistic function. Based on reported
vaccination data in Canada, we calibrated Dose 1 speed, Dose 2 speed, and
Information Effect. Next, in a Monte Carlo simulation, we determined how the
stochastic component of the model can impact the outcome. Based on reported
vaccination data in Canada, we calibrated Dose 1 speed, Dose 2 speed, and
Information Effect. This data consists of 144 rows and Date, Total
Vaccination, and At least One Dose column. Each row stands for one day; total
vaccination presents the cumulative number of fully vaccinated individuals,
and At least One Dose is the sum of the first dose and second dose until that
day. Next, in a Monte Carlo simulation, we determined how the stochastic
component of the model can impact the outcome.

###  3.6 Calibration

As previously mentioned, the model aims to simulate vaccine acceptance, so
that the total vaccination percentage serves as a comparison parameter between
the simulation and the actual data. We compared our total vaccination rate per
day for each province and territory in Canada by calibrating first dose speed,
second dose speed, and InfoEffectRate on VaccineTracker Little. (2021) data.
These features were calibrated using Anylogic, as shown in Table 3. It appears
that the second dose speed is considerably lower than the first, leading to a
relatively insufficient level of vaccination in Canada.

Table 3: Model Calibration for Canada total vaccination Province | Info Effect Rate | First Dose speed | Second Dose Speed
---|---|---|---
British Columbia | 0.2 | 0.033 | 3.08E-4
Alberta | 0.2 | 0.066 | 3.606E-4
Saskatchewan | 0.2 | 0.066 | 3.606E-4
Manitoba | 0.2 | 0.066 | 3.606E-4
Ontario | 0.2 | 0.066 | 3.606E-4
Quebec | 0.4 | 0.014 | 1.776E-4
New Brunswick | 0.2 | 0.066 | 3.606E-4
Nova Scotia | 0.2 | 0.061 | 3.588E-4
Prince Edward Island | 0.2 | 0.072 | 3.66E-4
Newfoundland and Labrador | 0.2 | 0.028 | 3.063E-4
Yukon | 0.2 | 0.018 | 0.004
Northwest Territories | 0.2 | 0.06 | 0.005
Nunavut | 0.2 | 0.046 | 0.004

##  4 Results

This paper examines how to model vaccine acceptance based on various factors,
including socio-demographics, vaccine types, and information diffusion. To
begin the simulation, we determined the vaccine acceptance based on an
individual’s socio$-$demographic characteristics. Next, we looked at how
information propagation and media exposure affect vaccine acceptance.
Therefore, our initialization method must be validated by comparing our
results with similar surveys conducted in Canada. Generally, our preliminary
results indicate that vaccine acceptance is similar to surveys and information
diffusion parameters significantly affect vaccine acceptance. As we enroll all
users with a hesitation state, we can measure hesitancy for each province by
counting agents in the Disbelief state (Figure.1) after 90 days. Since March
1st is the first day in our datasetLittle. (2021) and the surveyTang et al.
(2021) was conducted between January and March 2021, we waited 90 days. We
show that the vaccine hesitancy among agents is similar to the data collected
in Canada, suggesting that the logistic function implemented works as
expected. The simulation was also performed without the information diffusion
effect to illustrate the differences in outcomes. Figure.2 illustrates that
the simulation results are comparable to the surveys with a RMSE of 0.802, and
similar behavior was observed in all provinces and territories, with no sudden
or unexpected changes. It can be observed from Figure.2 that there is a
significant gap in Alberta between survey and simulation because we use 4,472
agents for the Alberta population, but their sample size is 301 for Alberta,
which leads to these differences. Clearly, the information diffusion removal
(gray line) increases the error and strongly confirms the effect of
information diffusion on vaccine acceptance. Our comparison was limited to one
similar research study about the hesitancy/acceptance in Canadian provinces
Tang et al. (2021), therefore, it cannot be directly compared with other
surveys.

Figure 2: A comparison of hesitancy rate between the simulation and ground-
truth data

###  4.1 Vaccine hesitancy and socio$-$demographic

In order to make sure our simulation does not have any specific bias, we
decided to analyze the acceptance of vaccines in each category of provinces
and territories based on the socio-demographic characteristics. We calculated
hesitancy percentages for each group and compared them with data from the
previous studyTang et al. (2021). Figure 3 shows that the model estimation
matches the reported data by Tang et al. (2021), and the RMSE for each
demographic attribute is less than 5.06. In most cases, such as the
University, High school, Female, and Male, simulation revealed a similar
hesitancy percentage. Age categories’ hesitancy percentage is slightly lower
than the value we expected, and there is certainly room for improvement. It
should be noted that we were unable to find any data regarding race and income
in Canada.

Figure 3: A comparison of hesitancy rate between the simulation and ground-
truth data in different categories of socio-demographic variables

(a) $y=x$

(b) $y=3sinx$

(c) $y=5/x$

(d) $y=5/x$

(e) $y=5/x$

(f) $y=5/x$

(g) $y=5/x$

(h) $y=5/x$

(i) $y=5/x$

(j) $y=5/x$

(k) $y=5/x$

(l) $y=5/x$

(m) $y=5/x$

Figure 4: Predicting full vaccination in comparison with actual full
vaccination in Canada’s provinces

###  4.2 Fully Vaccinated prediction (calibration)

It is crucial to compare the speed of vaccination in Canada with our
simulation results because this work aimed to test the effect of information
diffusion on vaccination. A Canada vaccination report dataset Little. (2021)
consists of two columns: Total Vaccination and Fully Vaccinated. All of this
data is manually compiled from credible sources in near real-time as new
information becomes available. Despite 315 days of Canada vaccination data, we
chose to run our model for 144 days since the last 144 days had a non-zero
value for the fully vaccinated population. We used this dataset and the
Anylogic calibration method to tune the Dose1Speed and the Dose2Speed
parameters of our model. Figure 4 indicates a good correlation between our
model and vaccination rates across provinces. It can be observed from Figure 4
that the patterns of simulation and real-world data are essentially the same
in NB, PE, Northwest, and Manitoba. Similar behavior was observed in the
majority of cases, with no sudden changes. However, the simulation
underestimated the number of fully vaccinated individuals in NL and Quebec,
but it follows the dataset pattern. The Quebec result(Figure 4(m)) is slightly
lower than the dataset value, and we think the reason is that the Quebec
dataset is not updated every day.

(a) x=Days , y= Number of injections

(b) x=Days , y= Number of users in thedisbelief state

Figure 5: A robustness test of the simulation with monte-Carlo. (a) The number
of agents in the Immune state on a scale of 1000. (b) The number of Agents in
the Disbelief state on a scale of 1000

(a) x=Days , y=Number of injections, Teta is varied

(b) x=Days , y=Number of users in the disbelief state, Teta is varied

(c) x=Days , y=Number of injections, TInfoEffect is varied

(d) x=Days , y=Number of users in the disbelief state, InfoEffect is varied

Figure 6: Teta (a,b) and InfoEffect (c,d) varied from zero to one to compare
the immunity rate in Canada simulation. (a,c) The number of agents in the
Immune state on a scale of 1000. (b,d) The number of Agents in the Disbelief
state on a scale of 1000.

###  4.3 Monte Carlo - User states

According to the Monte Carlo experiment, the results indicate that uncertainty
did not significantly impact the model’s behavior (see Figure 5.) For each
iteration of the Monte-Carlo simulation, we counted the individuals in each
state. Based on the results of the Monte-Carlo experiment, it is evident that
agents’ behavior is affected by information effects, Dose1Speed, and
Dose2Speed parameters despite the stochastic nature of the simulation. The
results of other simulations that are not part of the Monte-Carlo simulation
are also trustworthy, and the results of the model run multiple times will say
the same.

###  4.4 Information Diffusion Effects

As mentioned early, our primary goal in this paper is to construct a
computational platform and use it to simulate and analyze the effect of
information propagation on vaccine acceptance. Therefore, we compared the
number of people who received at least one dose vaccine according to different
scenarios by varying the Teta and InfoEffect parameters. Generally speaking,
both parameters resulted in a change in vaccination rates. First, we fixed all
parameters except the InfoEffect, and then we increased its value from zero to
one by 0.25 each step, and we repeated the same scenario for the Teta
parameter. As listed in Figure6, this parameter affects the vaccination
process as expected. When the Info Effect is equal to zero, the number of
vaccinated people is entirely different from the real-world data. The
population that has received at least one dose for Canada is nearly 26
million, and our simulation returns the same number when the Info Effect is
not zero. However, when we set the Info Effect equal to zero, we get nearly 15
million. Comparing Figure6(c) and Figure 6(d) shows that the number of agents
in Disbelief states is much higher when we ignore the Info Effect parameter.
This result proves that the information propagation could change people’s
minds and encourage them to get vaccinated. We observe from Figure6(a),6(b)
that the Teta parameter also affects the vaccination rate similar to the Info
Effect parameters. This result confirms previous findings and shows that the
Teta value could considerably change the outcomes. The number of vaccinated
agents increased to nearly 30 million after setting Teta to 0 and relaxing our
model. Conversely, the vaccinated agents are close to 5 million when the Teta
is 1.

##  5 Discussion and Conclusion

As outlined in the introduction, previous research simulated the vaccination
focused on parameters such as socio-demographics or vaccine types, but they
ignored the effect of information diffusion. Through agent-based modeling,
this study examined how various factors, such as social influences, affect
vaccination decisions. We defined Person, Province, Communication and Vaccine
agent to model the vaccination rate. This model can be used for any country to
predict the vaccination rate when enough vaccine doses have been administered.
The simulation can be used to determine which part of the society ignores
vaccines more often than others. This model can also assist decision-makers in
taking into account media advertisements that focus on the positive aspects of
vaccines. The first part of the findings, including vaccine hesitancy and
fully vaccinated prediction, validates the usefulness of the simulation as a
platform to explore the effects of information diffusion, and these values
correlate reasonably well with Tang et al. (2021). Moreover, the Monte Carlo
test examined the model’s robustness, and with a very minor non-alignment, it
was found to be robust. The most remarkable result to emerge from the data is
that information diffusion affects the vaccination rate, and modelers should
consider this behavior in their model or simulation for vaccination. We
believe that this is the first time that information diffusion is considered
an effective factor in a vaccination simulation context. This study has gone
some way towards enhancing our understanding of modeling vaccine acceptance.
The present findings might help to solve the vaccine hesitancy problem in
different countries. Our method can be applied to any country as long as the
parameters are initialized correctly. Our data suggest that favorable
information propagation in social networks could increase vaccine acceptance
and speed up the vaccination rate. As our findings come from odd ratios
reported by Lunsky et al. (2021), these findings should be treated with
caution. We are aware that our research may have two limitations. The first is
our regression model dependent on the reported odds ratios, and the second is
that some important factors such as COVID-19 rate, vaccine effectiveness, and
side effects are not well studied. These limitations highlight the difficulty
of collecting data and the complexity of the model.

## Acknowledgement

The research of AA, JK and JW has been funded by the Canadian Institute of
Health Research (CIHR) 2019 Novel Coronavirus (COVID-19) rapid research
program, and the Natural Science and Engineering Research Council of Canada’s
Emerging Infectious Disease Modelling Initiative (MfPH). JW is a member of the
Ontario COVID-19 Modelling Consensus Table, sponsored by the Ontario Ministry
of Health, Ontario Health, and Public Health Ontario. AA and JW are both
members of the External Modelling Expert Panel of the Public Health Agency of
Canada, and JW is a member of the Canada’s Expert Pande on COVID Submittee on
Modeling of the Chief Science Advisor of Canada.

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# Odometers in non-compact spaces

Godofredo Iommi  Facultad de Matemáticas, Pontificia Universidad Católica de
Chile (UC), Avenida Vicuña Mackenna 4860, Santiago, Chile
<EMAIL_ADDRESS>www.mat.uc.cl/$\sim$giommi and  Mario Ponce
Facultad de Matemáticas, Pontificia Universidad Católica de Chile (UC),
Avenida Vicuña Mackenna 4860, Santiago, Chile<EMAIL_ADDRESS>
###### Abstract.

We define an _odometer_ in the Baire space. That is the non-compact space of
one sided sequences of natural numbers. We go on to prove that it is
topologically conjugated to the dyadic odometer restricted to an appropriate
non-compact subset of the space of one sided sequences on two symbols. We
extend the action of the odometer to finite words. It turns out that,
arranging the finite words in appropriate binary tree, this extension runs
through the tree from left-right and top-down. Several well known binary trees
made out of rational numbers, such as the Kepler tree or the Calkin and Wilf
tree, are recovered in this way. Indeed, it suffices to identify finite words
with rational numbers by means of certain continued fractions. The odometric
action allow us to recover, in a unified way, several counting results for the
rational numbers. Moreover, associated to certain interval maps with countably
many branches we construct their corresponding odometers. Explicit formulas
are provided in the cases of the Gauss map (continued fractions) and the Renyi
map (backward continued fractions).

###### Key words and phrases:

Odometer, Baire space, continued fractions

###### 1991 Mathematics Subject Classification:

37E05, 37B10, 37A35

We would like to thank Aníbal Velozo for interesting discussions on the topic
of this article and Giovanni Panti for informing us of [BI]. GI was partially
supported by Proyecto Fondecyt 1230100. MP was partially supported by Proyecto
Fondecyt 1220032

##  1\. Introduction

Odometers form an important class of dynamical systems. They capture and
exemplify how complicated simple dynamical systems can be. A well known
example is the _dyadic odometer_. In this case, the phase space is that of one
sided sequences of zeroes and ones. The action is given by the addition by
$\underline{1}=(1,0,0,\dots)$ with the _carry over_ rule (see section 2 for
details). For this system, every orbit is dense and it has a unique invariant
measure. Interestingly, every continuous interval map $f:[0,1]\to[0,1]$ of
zero topological entropy with an invariant non-atomic measure is metrically
isomorphic to the dyadic odometer (see, for example, [KH, Theorem 15.4.2]).
The dyadic odometer is related to binary expansions. More general odometers
related to other numeration systems, such as Cantor series, can also be
constructed [GLT]. We refer to the survey [D] by Downarowicz, for a wealth of
properties and examples. In all the above settings, the phase space is
compact. In this note we provide a simple and explicit definition of an
_odometer_ in the Baire space ${\mathbb{N}}^{{\mathbb{N}}}$, see Definition
3.1. This space is endowed with the natural product topology and it is a non
compact space (actually, not even locally compact). Our definition is similar
to the _addition by one with carry over_ but with countably many symbols. The
similarity is not only formal, we actually prove that it is topologically
conjugated to the dyadic odometer restricted an appropriate non-compact subset
of the space of one sided sequences on two symbols (see Theorem 2.4).
Therefore, the map we define is minimal, uniquely ergodic, has zero
topological entropy, pure discrete spectrum and, of course, all other
properties of the dyadic odometer that are preserved under topological
conjugacy. Recall that the Baire space is dynamical model for a wide range of
continued fractions. Therefore, this enable us to define odometers in
numerical systems with countably many symbols. We note that, recently, certain
substitution systems have been defined in countable alphabets [DFMV, F].

Odometers are counting devices. Perhaps, one of the first manifestation of
this is the counting of dyadic rationals. Indeed, it is possible to extend the
action of the dyadic odometer to finite words. The binary expansion of real
numbers allows for the identification of finite words in the alphabet
$\\{0,1\\}$ with rational dyadic numbers. It turns out that the orbit of the
word $(1)$, with respect to the dyadic odometer, provides a bijection between
the sets of dyadic numbers and ${\mathbb{N}}\cup\\{0\\}$. A standard way to
see this is to order the dyadic rationals in a binary tree and to define a
function that runs through the tree from left-right and top-down. This is can
be achieved using the dyadic odometer as explained in section 3.4.1. We extend
the action of the odometer on the Baire space to finite words on a countable
alphabet, see section 3. Several continued fraction systems allow us to
identify finite words on a countable alphabet with rational numbers.
Proceeding as in the case of the dyadic tree, it is possible to (re)obtain
several well known trees of rationals. For example, the Kepler tree, which was
defined by Kepler in 1619, can be obtained in this way identifying finite
words with rational numbers using the continued fraction expansion, see
section 3.4.2. Also, the Calkin and Wilf tree, see section 3.4.3, is obtained
using the same method and identifying finite words with rational numbers using
the backward continued fraction expansion. Several results on counting
rationals that is, providing explicit bijections between
${\mathbb{N}}\cup\\{0\\}$ and rationals in $(0,1)$, can now be recovered just
considering the orbit of the root of the tree with respect to the appropriate
extension of the action of the odometer to the set of finite words.

The Baire space has strong universality properties. For example, Alexandrov
and Urysohn proved that the Baire space is the unique, up to homeomorphism,
nonempty zero-dimensional Polish space, for which all compact subsets have
empty interior. This, and similar, properties allows for the extension of the
definition of _non-compact_ odometer to a wide range of spaces. For example,
it is possible to define odometers in non-locally compact symbolic spaces
different from the Baire space. These spaces occur naturally in the coding of
certain dynamical systems.

More concretely, for a class of interval maps with countably many branches we
define the corresponding odometer. We consider first the case of the Gauss
map, related to the continued fraction expansion. We obtain an explicit
formula for its odometer which is defined in terms of the Fibonacci numbers
(Proposition 5.10). Also, we study the odometer associated to the Renyi map,
which is related to the backward continued fraction expansion. It turns out
that the corresponding odometer is a map that has been used to construct
explicit bijections between the natural and the rational numbers (Proposition
5.14). Bonanno and Isola [BI, Theorem 2.3] proved that this map is
topologically conjugated to the dyadic odometer and we recovered the same
result using backward continued fractions in [IP].

##  2\. Odometer in the Baire space

In this section we define an odometer in a non-compact symbolic space. We
prove that it shares all the dynamically relevant properties of its compact
counterpart. Endow the set $\\{0,1\\}$ with the discrete topology and
$\Sigma_{2}=\\{0,1\\}^{{\mathbb{N}}}$, with the product topology. This is a
compact metrizable space.

######  Definition 2.1.

The dyadic odometer is the map $O:\Sigma_{2}\to\Sigma_{2}$ defined by

$O((\underbrace{1,\dots,1,}_{k\textrm{-times}}0,w_{k+2},w_{k+3},\dots))=(\underbrace{0,\dots,0,}_{k\textrm{-times}}1,w_{k+2},w_{k+3},\dots),$

and $O((1,1,\dots))=(0,0,\dots)$.

The map $O$ is also known as von Neumann-Kakutani odometer, adding machine or
$2-$odometer. The dynamics of this system is well understood, see [Aa,
pp.25-26] and [B, Section 4.5.2]. We enumerate a sample of properties:

######  Proposition 2.2.

The dyadic odometer satisfies:

  1. (a)

For every $n\in{\mathbb{N}}$ and $w\in\Sigma_{2}$ we have,

$\left\\{((O^{k}w)_{1},\dots,(O^{k}w)_{n}):0\leq k\leq
2^{n}-1\right\\}=\\{0,1\\}^{n}.$

  2. (b)

The map $O$ is minimal.

  3. (c)

The map $O$ is uniquely ergodic and its unique invariant measure is the
$(1/2,1/2)-$Bernoulli measure.

  4. (d)

The map $O$ has zero topological entropy.

  5. (e)

The map $O$ has purely discrete spectrum.

  6. (f)

The map $O$ is equicontinuous.

Let ${\mathbb{N}}$ be the set of natural numbers,
${\mathbb{N}}_{0}={\mathbb{N}}\cup\\{0\\}$ and $\Sigma_{\geq
0}={\mathbb{N}}_{0}^{{\mathbb{N}}}$. Endow ${\mathbb{N}}_{0}$ with the
discrete topology and $\Sigma_{\geq 0}$ with the product topology. The space
$\Sigma_{\geq 0}$ is not compact nor locally-compact. As in descriptive set
theory, the space $\Sigma_{\geq 0}$ is called _Baire set_.

######  Definition 2.3.

The $0-$_odometer_ is the map $O_{0}:\Sigma_{\geq 0}\to\Sigma_{\geq 0}$
defined by

$O_{0}((w_{1},w_{2},w_{3},\dots))=(\underbrace{0,\dots,0,}_{w_{1}\textrm{-times}}(w_{2}+1),w_{3},\dots).$

We will prove that the map $O_{0}$ is topologically conjugated to the
restriction of $O$ to an appropriate non-compact subset of $\Sigma_{2}$. Let

$\displaystyle\Sigma_{\bar{1}}=\left\\{(w_{i})_{i}\in\Sigma_{2}:\text{ there
exists }k\in{\mathbb{N}}\text{ s.t. for every }i\geq k,w_{i}=1\right\\},$
$\displaystyle\Sigma_{\bar{0}}=\left\\{(w_{i})_{i}\in\Sigma_{2}:\text{ there
exists }k\in{\mathbb{N}}\text{ s.t. for every }i\geq k,w_{i}=0\right\\}.$

Also consider $\Sigma_{2,0}=\Sigma_{2}\setminus\Sigma_{\bar{1}}$ and endow it
with the induced topology.

######  Theorem 2.4.

There exits an homemorphism $\eta:\Sigma_{2,0}\mapsto\Sigma_{\geq 0}$ such
that

$O_{0}\circ\eta=\eta\circ O.$

###### Proof.

In order to prove this result it will be convenient to recode the space
$\Sigma_{2,0}$. For every integer $k\geq 0$ we the define the block

$\displaystyle b_{k}=\begin{cases}0&\text{ if }k=0,\\\
\underbrace{1,\dots,1}_{k\textrm{-times}},0&\text{ if }k\geq 1.\end{cases}$
(2.1)

Every sequence $(w_{i})_{i}\in\Sigma_{2,0}$ can be uniquely written as an
infinite sequence in the alphabet $\\{b_{k}:k\geq 0\\}$. For example,

$(0,0,1,1,1,0,1,0,1,1,1,1,1,0,1,0,\dots)=(b_{0},b_{0},b_{3},b_{1},b_{5},b_{1},\dots).$

Note that the map $\eta:\Sigma_{2,0}\mapsto\Sigma_{\geq 0}$ defined by

$\eta((w_{1},w_{2},\dots))=\eta(b_{k_{1}},b_{k_{2}},\dots)=(k_{1},k_{2},\dots),$

is an homeomorphism. The following holds

$(\eta\circ
O\circ\eta^{-1})(k_{1},k_{2},k_{3},\dots)=(\underbrace{0,\dots,0,}_{k_{1}\textrm{-times}}(k_{2}+1),k_{3},\dots).$
(2.2)

Indeed, we have
$\eta^{-1}(k_{1},k_{2},\dots)=(\underbrace{1,\dots,1,}_{k_{1}\textrm{-times}}0,b_{k_{2}},\dots)$.
Thus,

$(\eta\circ
O\circ\eta^{-1})\left(\underbrace{1,\dots,1,}_{k_{1}\textrm{-times}}0,b_{k_{2}},\dots\right)=\left(\underbrace{0,\dots,0}_{k_{1}\textrm{-times}},1,b_{k_{2}},\dots\right),$

and the conclusion follows from the fact that $1,b_{k}=b_{k+1}$. That is,

$O_{0}=(\eta\circ O\circ\eta^{-1}).$

∎

######  Remark 2.5.

For every $k\in{\mathbb{N}}$, let ${\mathbb{N}}_{k}:=\\{n\in{\mathbb{N}}:n\geq
k\\}$ and $\Sigma_{\geq k}={\mathbb{N}}_{k}^{{\mathbb{N}}}$ endowed with the
product topology. The map $O_{k}:\Sigma_{\geq k}\mapsto\Sigma_{\geq k}$
defined by

$O_{k}((w_{1},w_{2},w_{3},\dots))=(\underbrace{k,\dots,k,}_{(w_{1}-k)\textrm{-times}}(w_{2}+1),w_{3},\dots).$

is topologically conjugated to $O_{0}$

The following _odometric_ property follows from Theorem 2.4 and Proposition
2.2.

######  Remark 2.6.

Given an infinite word $\omega=(w_{1},w_{2},w_{3},\dots)\in\Sigma_{\geq 0}$
and any finite word $(\bar{w_{1}},\bar{w_{2}},\dots,\bar{w_{k}})$, with
$\bar{w}_{j}\in{\mathbb{N}}$, there exists $n>0$ such that
$O_{0}^{n}(\omega)=(\bar{w_{1}}\,\bar{w_{2}},\dots,\bar{w_{k}},\dots)$. That
is, given a finite word, any infinite word has an iterate that starts exactly
as the given finite word.

######  Definition 2.7.

For every $k\in{\mathbb{N}}$, the _shift_ map is the transformation
$\sigma:\Sigma_{\geq k}\mapsto\Sigma_{\geq k}$ defined by

$\sigma((w_{1},w_{2},w_{3}\dots))=(w_{2},w_{3}\dots).$

This map is somehow at the opposite dynamical end of the odometer. Indeed, it
has infinite entropy, it is topologically mixing, and for every
$n\in{\mathbb{N}}$ it has countable many periodic orbits of prime period $n$.

###  2.1. Renormalization Property

The odometer and the shift map satisfy a renormalization property analogous to
that satisfied by the geodesic and horocycle flows in the unit tangent bundle
of a hyperbolic surface (see [Da, Property 3.3 p.119] or [SaS]). Indeed,
denote by $\sigma_{2}:\Sigma_{2}\to\Sigma_{2}$ the shift map on $\Sigma_{2}$.
For every $m,n\in{\mathbb{N}}\cup\\{0\\}$ and $w\in\Sigma_{2}$ we have

$\left(O^{m}\circ\sigma_{2}^{n}\right)(w)=\left(\sigma_{2}^{n}\circ
O^{m2^{n}}\right)(w).$ (2.3)

The number $2$ in the exponent of $O$ in equation (2.3) is related to the fact
that the topological entropy of $\sigma_{2}$ is $\log 2$. In our next result
we establish a formula analogous to that in equation (2.3) for the odometer
$O_{0}$ and the full-shift on a countable alphabet $\sigma:\Sigma_{\geq
0}\to\Sigma_{\geq 0}$. We stress that the later system has infinite entropy
and that, in this case, the relation depends upon the point where it is
evaluated.

######  Proposition 2.8.

For every $w=(k_{1},k_{2},\dots)\in\Sigma_{\geq 0}$ and
$m,n\in{\mathbb{N}}\cup\\{0\\}$, we have

$\left(O_{0}^{m}\circ\sigma^{n}\right)(w)=\left(\sigma^{n}\circ
O_{0}^{m2^{n}2^{\sum_{i=1}^{n}k_{i}}}\right)(w).$ (2.4)

###### Proof.

Recall that, restricted to the appropriate non-compact subspace, we have
$O=\eta^{-1}\circ O_{0}\circ\eta$. Replacing in equation (2.3) we obtain, for
every $(k_{1},k_{2},\dots)\in\Sigma_{\geq 0}$ and
$m,n\in{\mathbb{N}}\cup\\{0\\}$, that

$O_{0}\circ(\eta\circ\sigma_{2}^{n}\circ\eta^{-1})=(\eta\circ\sigma_{2}^{n}\circ\eta^{-1})\circ
O_{0}^{m2^{n}}.$ (2.5)

The map $\sigma$ can be thought of as the _accelerated_ version of
$\sigma_{2}$ in the following sense: let
$w=(k_{1},k_{2},k_{3},\dots)\in\Sigma_{\geq 0}$, then

$(k_{2},k_{3},\dots)=\sigma(w)=(\eta\circ\sigma_{2}^{k_{1}+1}\circ\eta^{-1})(w).$
(2.6)

Combining equations (2.3), (2.5) and (2.6), we obtain the desired result. ∎

##  3\. Counting finite words

In this section we extend the action of the odometer to finite words. We prove
that arranging them in appropriate binary tree, the extension of the odometer
runs through the tree from left-right and top-down. Identifying finite words
with rational numbers by means of certain continued fractions, we recover well
known binary trees made out of rational numbers, such as the Kepler tree or
the Calkin and Wilf tree. Moreover, this extension of the odometer to finite
words allow us to recover, in a unified way, several counting results for the
rational numbers.

###  3.1. Odometric binary tree

We will represent the set of finite words, that we denote by
$\Omega_{1}=\bigcup_{j\geq 1}{\mathbb{N}}_{1}^{j}$, as a binary tree.

######  Definition 3.1.

The odometric binary tree, denoted by $\mathcal{T}_{1}$, is a tree defined by
the following rules:

  1. (a)

The leafs of $\mathcal{T}_{1}$ are the elements in $\Omega_{1}$.

  2. (b)

Every leaf $\omega=(w_{1},w_{2},\dots,w_{j})\in\Omega_{1}$ has two sons
displayed in the following order:

$\omega^{1}=(1,w_{1},w_{2},\dots,w_{j})\quad\textrm{and}\quad\omega^{+1}=(w_{1}+1,w_{2},\dots,w_{j}).$

  3. (c)

The root of the tree is $(1)\in\Omega_{1}$.

Note that, apart from $(1)$, every leaf is the son of exactly one leaf. The
level $1$ of the tree, denoted by $\mathcal{T}^{1}_{1}$, contains $(1)$ as its
only element. Inductively, for $l\in{\mathbb{N}}$, we say that sons of a leaf
at level $\mathcal{T}_{1}^{l}$ are at the level $\mathcal{T}_{1}^{l+1}$.

The first four levels of the odometric binary tree are displayed in Figure 1.

[.(1) [.(11) [.(111) [.(1111) ] [.(211) ] ] [.(21) [.(121) ] [.(31) ] ] ]
[.(2) [.(12) [.(112) ] [.(22) ] ] [.(3) [.(13) ] [.(4) ] ] ]]

Figure 1. The first four levels of the odometric binary tree or the
$\mathcal{T}_{1}$ tree.

######  Definition 3.2.

For a finite word $\omega=(w_{1},w_{2},\dots,w_{j})$, we define its _sum_ ,
that we denote $s(\omega)$, by

$s(\omega)=w_{1}+w_{2}+\dots+w_{j}.$

For $l\in{\mathbb{N}}$, denote by
$\Omega_{1}^{l}=\\{\omega\in\Omega_{1}:s(\omega)=l\\}$.

######  Lemma 3.3.

The odometric binary tree verifies:

  1. (a)

Every element $\omega\in\Omega_{1}$ appears exactly once as a leaf in the
odometric binary tree.

  2. (b)

For $l\geq 1$, we have $\mathcal{T}_{1}^{l}=\Omega_{1}^{l}$.

###### Proof.

We first prove (b). Note that, for $l\geq 1$, the set
$\Omega_{1}^{l}=\\{\bar{\omega}\in\Omega_{1}:s(\bar{\omega})=l\\}$ has
cardinality $2^{l-1}$. Indeed, any element with sum $l$ can be obtained from a
list $1\ 1\dots 1\ 1$ of length $l$, by intercalating a $"+"$ symbol or a
colon $","$ symbol at each of the $(l-1)$ places between the ones, and
performing the corresponding sums. Due to the sons rule, given
$\omega\in\mathcal{T}_{1}^{l}$, we have
$s(\omega^{1})=s(\omega^{+1})=s(\omega)+1$. Hence,
$\mathcal{T}^{l}_{1}\subset\Omega_{1}^{l}$. Since the tree is binary, and
$\mathcal{T}_{1}^{1}$ has $1$ element, the cardinality of
$\mathcal{T}_{1}^{l}$ is $2^{l-1}$, which implies
$\mathcal{T}_{1}^{l}=\Omega_{1}^{l}$, concluding (b). For
$\omega\in\Omega_{1}$, part (b) implies that
$\omega\in\mathcal{T}_{1}^{s(\omega)}$. This fact, together with a cardinality
argument, allow us to obtain (a). ∎

######  Definition 3.4.

The reverse lexicographic order $<_{\Omega_{1}}$ on $\Omega_{1}$ is defined
by:

  1. (a)

For $\omega,\bar{\omega}\in\Omega_{1}$, if $s(\omega)<s(\bar{\omega})$ then
$\omega<_{\Omega_{1}}{\bar{\omega}}$.

  2. (b)

If $s(w_{1},w_{2},\dots,w_{j})=s(\bar{w}_{1},\bar{w}_{2},\dots,\bar{w}_{r})$
then $\omega<_{\Omega_{1}}{\bar{\omega}}$ provided that

$w_{j}=\bar{w}_{r},\ w_{j-1}=\bar{w}_{r-1}\ ,\dots,w_{j-t-1}=\bar{w}_{r-t-1},\
\textrm{and }w_{j-t}<\bar{w}_{r-t},$

for some $t\in{\mathbb{N}}$ with $j>t$ and $r>t$.

Note that this is well defined, the strict inequality in (b) occurs, otherwise
both sums should be different. An example of the order is
$(1,1,3)<_{\Omega_{1}}(4,2)$ and $(2,2,2,1)<_{\Omega_{1}}(4,2,1)$. Note that
the reverse lexicographic order $<_{\Omega_{1}}$ is a total order on
$\Omega_{1}$.

Given an integer $k\geq 0$, we define the block
$\hat{b}_{k}=0\underbrace{1\dots 1}_{k-\textrm{times}}$. Note that
$\hat{b}_{k}$ has length $k+1$ and that is the complementary version of the
block $b_{k}$, defined in (2.1). To every $\omega\in\Omega_{1}$ we associate a
positive integer $n_{\omega}$, which in binary notation corresponds to:

$n_{\omega}=\hat{b}_{w_{j}-1}\hat{b}_{w_{j-1}-1}\dots\hat{b}_{w_{1}-1}.$

For instance, to $\omega=(4,2,1)$ we associate $n_{\omega}=0\ 01\ 0111$, which
in decimal notation is $n_{\omega}=23$. Note that $n_{(s)}=2^{s-1}-1$ and
$n_{(1,\dots,1)}=0$.

######  Lemma 3.5.

Let $\omega,\bar{\omega}\in\Omega_{1}$. We have that,
$\omega<_{\Omega_{1}}\bar{\omega}$ if and only if
$2^{s(\omega)-1}+n_{\omega}<2^{s(\bar{\omega})-1}+n_{\bar{\omega}}$.

###### Proof.

Note that $n_{\omega}\in\\{0,1,\dots,2^{s(\omega)-1}-1\\}$, which implies
$2^{s(\omega)-1}+n_{\omega}<2^{r}$ for any $r\geq s(\omega)$. When
$s(\omega)=s(\bar{\omega})$, the definitions allow us to conclude. ∎

We can now produce a more refined version of Lemma 3.3.

######  Proposition 3.6.

The odometric binary tree verifies:

  1. (a)

The leafs in the level $\mathcal{T}_{1}^{l}$ appears from left-rightin the
order induced by $<_{\Omega_{1}}$ on $\Omega_{1}^{l}$. In particular, the
leftmost leaf is $(\underbrace{1,\dots,1}_{l-\textrm{times}})$ and the
rightmost leaf is $(l)$.

  2. (b)

The finite word $\omega=(w_{1},w_{2},\dots,w_{j})$ corresponds to the leaf in
the $n_{\omega}$ position of the level $\mathcal{T}_{1}^{s(\omega)}$.

###### Proof.

We proceed by induction on $l$. Assume that (a) holds for $l\geq 1$. Take
$\omega=(w_{1},\dots,w_{j}),\bar{\omega}=(\bar{w}_{1},\dots,\bar{w}_{r})$,
such that $\omega<_{\Omega_{1}}\bar{\omega}$ and $s(\omega)=s(\bar{\omega})=l$
(that is, $\omega,\bar{\omega}\in\mathcal{T}_{1}^{l}$). It is easy to see that
$\omega^{1}<_{\Omega_{1}}\omega^{+1}$. Since $s(\omega)=s(\bar{\omega})$ there
exists $t>1$ such that $w_{t}<\bar{w}_{t}$, that leads to
$\omega^{+1}<_{\Omega_{1}}\bar{\omega}^{1}$, concluding the inductive step,
and hence (a). Part (b) follows from (a) and Lemma 3.3. ∎

######  Remark 3.7.

Note that Proposition 3.6.(b) implies that the odometric binary tree is a
Cartesian Tree, as introduced by Vuillemin [Vui].

###  3.2. Odometric action on finite words.

In this section we extend the action of the odometer $O_{1}$ to finite words.
There is a natural way to define $O_{1}$ for words of length strictly greater
than one in a way that the action is consistent with the order induced by
$<_{\Omega_{1}}$. For words of length one there is a great deal of freedom. We
will concentrate on two possible choices that will be useful later.

######  Definition 3.8.

Let $(w_{1},w_{2},\dots,w_{j})\in\Omega_{1}$. If $j>1$ we define the action of
$O_{1}$ by

$O_{1}(w_{1},w_{2},\dots,w_{j})=(\underbrace{1,\dots,1,}_{w_{1}-1\text{
times}}w_{2}+1,w_{3},\dots,w_{j}).$ (3.1)

If $j=1$ the action $O_{1}(w_{1})$ can be defined arbitrarily. That is, it can
be any finite word. The two main examples we consider are:

$O_{1}(w_{1})=(\underbrace{1,1,\dots,1}_{w_{1}-\textrm{times}})\quad\text{and}\quad
O_{1}(w_{1})=(\underbrace{1,1,\dots,1}_{(w_{1}+1)-\textrm{times}}).$

We call the first example cyclic odometer while the former top-down odometer.

######  Lemma 3.9.

Let $\omega=(w_{1},w_{2},\dots,w_{j})$ be a finite word, with $j>1$. Then,
when regarded as leafs of the odometric binary tree, the leaf $O_{1}(\omega)$
is the leaf located one position to the right of $\omega$ at the same level.

###### Proof.

From equation (3.1) we have that $s(O_{1}(\omega))=s(\omega)$. We will proceed
by induction on $l=s(\omega)$. For $l=2$ we have $O_{1}(11)=(2)$. Assume the
claim holds for $l\geq 2$. From the inductive hypothesis, $O_{1}(\omega)$ is
the next leaf to the right from $\omega$ at the level $l$. From equation (3.1)
we see that $O_{1}(\omega^{1})=\omega^{+1}$. Therefore,

$\displaystyle O_{1}(\omega^{+1})=O_{1}((w_{1}+1),w_{2},\dots,w_{j})$
$\displaystyle=$
$\displaystyle(\underbrace{1,\dots,1}_{w_{1}-\textrm{times}},w_{2}+1,\dots,w_{j})$
$\displaystyle=$
$\displaystyle(1,\underbrace{1,\dots,1}_{(w_{1}-1)-\textrm{times}},w_{2}+1,\dots,w_{j})$
$\displaystyle=$ $\displaystyle O_{1}(\omega)^{1},$

concluding the proof. ∎

######  Lemma 3.10.

Let $\omega=(w_{1},w_{2},\dots,w_{j})$ be a finite word, with $j>1$. Then,

$n_{O_{1}(\omega)}=n_{\omega}+1.$ (3.2)

That is, $O_{1}(\omega)$ is the next finite word for the complete order
$<_{\Omega_{1}}$, provided that the leaf $\omega$ is not a rightmost leaf in
the tree.

The following result, which describes the orbits of the top-down odometer on
finite words, is a consequence Lemmas (3.9) and (3.10).

######  Theorem 3.11 (Counting Finite Words).

For every $w_{1}\geq 1$ define

$O_{1}(w_{1})=(\underbrace{1,\dots,1}_{(w_{1}+1)-\textrm{times}}).$

Then, the map $n\mapsto O^{n}(1)$ is a bijection between
${\mathbb{N}}\cup\\{0\\}$ and $\Omega_{1}$. That is, the orbit of $(1)$ by
$O_{1}$ passes trough every finite word in $\Omega_{1}$ once and only once.
Moreover, the orbit follows the $<_{\Omega_{1}}$ order of $\Omega_{1}$.

The next result is a manifestation of the fact that odometers are $+1$ adding
machines.

######  Corollary 3.12.

Let $O_{1}$ be defined as in Theorem 3.11, then

$2^{s(O_{1}(\omega))-1}+n_{O_{1}(\omega)}=2^{s(\omega)-1}+n_{\omega}+1.$

###  3.3. Odometric tree and odometric action for $\Omega_{k}$

The results for $\mathcal{T}_{1}$ and $O_{1}$ can be extended so as to
consider finite words having letters larger than an arbitrary number. Let
$k\geq 0$, the set of finite words with symbols starting with $k$ will be
denoted by $\Omega_{k}=\bigcup_{j\geq 1}{\mathbb{N}}_{k}^{j}$.

######  Definition 3.13.

Denote by $(\cdot)_{k}:\Omega_{k}\to\Omega_{1}$ the bijection between
$\Omega_{k}$ and $\Omega_{1}$ defined for
$\omega=(w_{1},\dots,w_{j})\in\Omega_{k}$ by

$(\omega)_{k}=(w_{1},\dots,w_{j})_{k}=(w_{1}-(k-1),\dots,w_{j}-(k-1)).$

The $k$-odometric binary tree, denoted by $\mathcal{T}_{k}$, is obtained from
$\mathcal{T}_{1}$ by applying $(\cdot)_{k}^{-1}$ to every leaf of
$\mathcal{T}_{1}$. The son-rule for $\mathcal{T}_{k}$ can be expressed by: a
leaf $\omega=(w_{1},w_{2},\dots,w_{j})\in\Omega_{k}$ has two sons

$\omega^{1}=(k,w_{1},w_{2},\dots,w_{j})\quad\textrm{and}\quad\omega^{+1}=(w_{1}+1,w_{2},\dots,w_{j}).$

######  Definition 3.14.

The $k$-sum $s_{k}$ of a finite word $\omega=(w_{1},\dots,w_{j})\in\Omega_{k}$
is defined by:

$s_{k}(\omega)=s((\omega)_{k})=\sum_{1\geq r\geq j}w_{r}-j(k-1).$

The $k$-reverse lexicographic order on $\Omega_{k}$, denoted by
$<_{\Omega_{k}}$, is defined by:

$\omega<_{\Omega_{k}}\bar{\omega}\quad\textrm{if and only
if}\quad(\omega)_{k}<_{\Omega_{1}}(\bar{\omega})_{k}.$

Let $\Omega_{k}^{l}=\\{\omega\in\Omega_{k}\ |\ s_{k}(\omega)=l\\}$.

######  Proposition 3.15.

The $k$-odometric binary tree verifies:

  1. (a)

The leafs in the level $\mathcal{T}_{k}^{l}$ appears from left-right in the
order induced by $<_{\Omega_{k}}$ on $\Omega_{k}^{l}$. In particular, the
leftmost leaf is $(\underbrace{k,\dots,k}_{l-\textrm{times}})$ and the
rightmost leaf is $(l)$.

  2. (b)

The finite word $\omega=(w_{1},w_{2},\dots,w_{j})$ is located at the leaf in
the $n_{(\omega)_{k}}$ position of the level
$\mathcal{T}_{k}^{s_{k}(\omega)}$.

######  Definition 3.16.

We define the $k$-odometric action $O_{k}:\Omega_{k}\to\Omega_{k}$ by
$O_{k}(\omega)=(O_{1}((\omega)_{k}))_{k}^{-1}$. In other words, for
$(w_{1},w_{2},\dots,w_{j})\in\Omega_{k}$, with $j>1$ we have

$O_{k}(w_{1},\dots,w_{j})=(\underbrace{k,\dots,k,}_{(w_{1}-k)-\textrm{times}}(w_{2}+1),\dots,w_{j}).$

As in the case for $k=1$, the action $O_{k}$ at words of the form $(l)$ can be
defined in an arbitrary fashion.

######  Lemma 3.17.

If $\omega=(w_{1},\dots,w_{j})\in\Omega_{k}$, with $j>1$, then $O_{k}(\omega)$
is the next finite word for the complete order $<_{\Omega_{k}}$. Hence,
provided that $\omega$ is not a rightmost leaf in $\mathcal{T}_{k}$, then
$O_{k}(\omega)$ is the next leaf to the right of $\omega$, at the level
$\mathcal{T}_{k}^{s_{k}(\omega)}$.

###  3.4. Examples

There exists different numeration systems with countably many symbols. Finite
words can be interpreted using them. For example, the continued fraction and
the backward continued fraction systems. In these cases, finite words
correspond to rational numbers. By means of these identifications we will
construct, or recover, both the Kepler and the Calkin and Wilf trees.
Moreover, the odometric action on these trees allow us to recover different
ways of counting rational numbers in a unified way. We begin, though, with the
more standard binary expansion and its relation to the usual dyadic odometer.

####  3.4.1. Binary expansion and Dyadic Tree.

Dyadic numbers in $(0,1)\subset\mathbb{R}$ are rationals that can be written
as a finite binary expansion $x=\sum_{1\leq n\leq j}\frac{a_{n}}{2^{n}}$, with
$a_{n}\in\\{0,1\\}$ and $a_{j}=1$. Therefore, to every dyadic number
$x\in(0,1)$ we can associate a finite word $(a_{1},\dots,a_{j})$ with
$a_{i}\in\\{0,1\\}$. As in the proof of Theorem 2.4, we can recode this finite
word in $\Omega_{0}$ using the blocks
$b_{m}=\underbrace{1,\dots,1}_{m-\textrm{times}},0$. The final block in the
form $\underbrace{1,\dots,1}_{m-\textrm{times}}$ will be coded by $b_{m}$. For
example, $\frac{19}{32}=0,10011\mapsto(1,0,2)\in\Omega_{0}$. The $0$-odometric
binary tree $\mathcal{T}_{0}$ has finite words ending by $0$. This produces
repetitions in the coding of dyadic numbers. Nevertheless, the sub-tree
$\mathcal{T}_{0,(1)}$, composed by the descendants of $(1)$ never ends by $0$,
and contains a unique coded version of every dyadic number in
$(0,1)\subset{\mathbb{R}}$. The leftmost leaf of the level $l$ is
$(\underbrace{0,\dots,0}_{(l-1)-\textrm{times}},1)$ and the rightmost leaf is
$(l)$. The following are the son-rules for $\mathcal{T}_{0,(1)}$ in terms of
dyadic numbers: the number $x=0,b_{w_{1}}b_{w_{2}}\dots b_{w_{j}}$ has two
sons, in the following order

$0,b_{0}b_{w_{1}}b_{w_{2}}\dots b_{w_{j}}=\frac{x}{2}\quad\textrm{and}\quad
0,b_{w_{1}+1}b_{w_{2}}\dots b_{w_{j}}=\frac{1+x}{2}.$

[.1/2 [.1/4 [.1/8 [.1/16 ] [.9/16 ] ] [.5/8 [.5/16 ] [.13/16 ] ] ] [.3/4 [.3/8
[.3/16 ] [.11/16 ] ] [.7/8 [.7/16 ] [.15/16 ] ] ]]

Figure 2. The first four levels of the dyadic tree or the
$\mathcal{T}_{0,(1)}$ tree.

See Figure 2 for the first four levels of this tree. The odometric action
$O_{0}$ defined so that
$O_{0}(w_{1})=(\underbrace{0,\dots,0}_{w_{1}-\textrm{times}},1)$ (that is, the
top-down definition restricted to $\mathcal{T}_{0,(1)}$), coincides with the
action of the the von Newmann Kakutani odometer restricted to dyadic numbers
(see Section 5.1).

######  Remark 3.18.

It is a direct consequence of Theorem 3.11 that the map $n\mapsto
O_{0}^{n}(1)$ is a bijection between the sets of finite words with symbols in
$\\{0,1\\}$ and ${\mathbb{N}}\cup\\{0\\}$.

####  3.4.2. Continued fractions and the Kepler Tree.

Every real number $x\in(0,1)$ can be written as a continued fraction of the
form

$x=\textrm{ }\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{a_{3}+\dots}}}=\textrm{
}[a_{1}\ a_{2}\ a_{3}\ \dots],$

where $a_{i}\in\mathbb{N}$. As it is well known [HW, Theorem 170], arithmetic
properties of the number are captured by the expansion. In particular, a
number $x\in(0,1)$ is irrational if and only if its continued fraction
expansion has infinitely many terms. Moreover, irrational numbers have a
unique continued fraction expansion. On the other hand, rational numbers have
two different expansions, both of them finite and one of them having last
digit equal to $1$. In fact $[a_{1}\ a_{2}\ a_{3}\ \dots\ a_{n}\ 1]=[a_{1}\
a_{2}\ a_{3}\ \dots\ (a_{n}+1)]$. Hence, every rational in $(0,1)$ has two
representatives as leafs of the tree $\mathcal{T}_{1}$. If we restrict
ourselves to the sub-tree of the descendants of $(2)$, denoted by
$\mathcal{T}_{1,(2)}$, we obtain that every rational number in $(0,1)$ appears
exactly once, see Figure 3. The son-rule can be computed in terms of continued
fractions. Indeed, if $\frac{p}{q}=[a_{1}\ a_{2}\ \dots\ a_{n}]$ then

$[1\ a_{1}\ \dots\
a_{n}]=\cfrac{1}{1+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\dots}}}=\cfrac{1}{1+\cfrac{p}{q}}=\frac{q}{p+q},$

and

$[(a_{1}+1)\ \dots\
a_{n}]=\cfrac{1}{(a_{1}+1)+\cfrac{1}{a_{2}+\dots}}=\cfrac{1}{1+\cfrac{q}{p}}=\frac{p}{p+q}.$

[.1/2 [.1/3 [.1/4 [.1/5 ] [.4/5 ] ] [.3/4 [.3/7 ] [.4/7 ] ] ] [.2/3 [.2/5
[.2/7 ] [.5/7 ] ] [.3/5 [.3/8 ] [.5/8 ] ] ]]

Figure 3. The first four levels of the Kepler tree or the
$\mathcal{T}_{1,(2)}$ tree.

Hence, the tree ${\mathcal{T}_{1,(2)}}$ coincides with the well known Kepler
tree (see [K, Oeis] or [B, p.383]). In particular, every rational number in
$(0,1)$ appears once in this tree, and the exact position in the tree can be
computed from its continued fraction.

######  Remark 3.19.

The top-down odometric action on the Kepler tree induces a bijection between
the rational numbers in $(0,1)$ and ${\mathbb{N}}\cup\\{0\\}$. Indeed, it
suffices to consider the map $n\mapsto O_{1}^{n}((2))$ and to identify each
finite word $O_{1}^{n}((2))$ with the corresponding rational.

In section 5.2 we give a complete account of the dynamics of the odometric
action $O_{1}$ on the interval and, in particular, on the Kepler tree.

####  3.4.3. Backward continued fractions and the Calkin and Wilf tree.

Every real number $x\in[0,1)$ can be written as backward continued fraction of
the form

$x=1-\textrm{
}\cfrac{1}{a_{1}-\cfrac{1}{a_{2}-\cfrac{1}{a_{3}-\dots}}}=\textrm{ }[a_{1}\
a_{2}\ a_{3}\ \dots]_{B},$

where $a_{i}\geq 2$. Irrational numbers have a unique backward continued
fraction expansion which is infinite. Rational numbers have two different
expansions: a finite one and an infinite one of that ends with a tail of $2$’s
(see [Ka, Theorem 1.2]). In fact,

$[a_{1}\ a_{2}\ a_{3}\ \dots\ a_{n}]_{B}=[a_{1}\ a_{2}\ a_{3}\ \dots\
(a_{n}+1)\ 2\ 2\ 2\ \dots]_{B}.$ (3.3)

Hence, every rational in $(0,1)$ has a unique representative as a leaf of the
tree $\mathcal{T}_{2}$. The son rule can be computed in terms of the backward
continued fraction. If $\frac{p}{q}=[a_{1}\ a_{2}\ a_{3}\ \dots\ a_{n}]_{B}$
then

$[2\ a_{1}\ \dots\
a_{n}]_{B}=1-\cfrac{1}{2-\cfrac{1}{a_{1}-\cfrac{1}{a_{2}-\dots}}}=1-\cfrac{1}{1+\cfrac{p}{q}}=\frac{q}{p+q},$

and

$[(a_{1}+1)\ \dots\
a_{n}]_{B}=1-\cfrac{1}{(a_{1}+1)-\cfrac{1}{a_{2}-\dots}}=1-\cfrac{1}{1+\cfrac{q}{q-p}}=\frac{q}{2q-p}.$

The Calkin and Wilf tree, see [CW, p.361], has the following son-rule: every
leaf $\frac{a}{b}$ has two sons, $\frac{a}{a+b}$ and $\frac{a+b}{b}$. Thus,
leafs alternate rationals in $(0,1)$ and rationals larger than $1$. The odd
terms (that is, the leafs that are in $(0,1)\subset\mathbb{R}$) form a sub-
tree with the following son-rule: the leaf $\frac{a}{a+b}$ has two sons that
appear in the following order: $\frac{1}{2a+b}$ and $\frac{a+b}{a+2b}$, see
Figure 4.

[.1/2 [.1/3 [.1/4 [.1/5 ] [.4/7 ] ] [.3/5 [.3/8 ] [.5/7 ] ] ] [.2/3 [.2/5
[.2/7 ] [.5/8 ] ] [.3/4 [.3/7 ] [.4/5 ] ] ]] [.$\frac{a}{b}$ [.$\frac{a}{a+b}$
[.$\frac{a}{2a+b}$ ] [.$\frac{2a+b}{a+b}$ ] ] [.$\frac{a+b}{b}$
[.$\frac{a+b}{a+2b}$ ] [.$\frac{a+2b}{a+b}$ ] ]]

Figure 4. The first four levels of the Calkin and Wilf sub-tree or the
$\mathcal{T}_{2,(2)}$ tree and the son-rule.

Replacing $a=p$ and $q=a+b$, we see that the tree $\mathcal{T}_{2,(2)}$
coincides with the sub tree of the Calkin and Wilf tree of rationals in
$(0,1)$ (which, in turn, coincides with the odd leafs at every level), see
Figure 4. In particular, every rational in $(0,1)\subset\mathbb{R}$ appears
once and only once in this tree. Moreover, its exact position can be computed
from the backward continued fraction. Since every rational has an expansion
with infinitely many digits, the odometric action $O_{2}$ has a natural
definition on this tree. It turns out that the odometric action runs through
every leaf from left-right to top-down. More details in section 5.3.

######  Remark 3.20.

It is a direct consequence of Theorem 3.11 and that $O_{2}$ is uniquely
defined for rational numbers (by means of their infinite expansion) that the
map $n\mapsto O_{2}^{n}(1)$ is a bijection between the sets of rational
numbers in $(0,1)$ and ${\mathbb{N}}\cup\\{0\\}$. Indeed, we only need to
identify $O_{2}^{n}(1)$ with the corresponding rational.

##  4\. Odometers in non-compact Polish spaces

Using universality properties of the Baire space it is possible to extend the
results in section 2 to a broad range of spaces. Recall that a separable,
completely metrizable space is called _Polish space_. A topological space is
_zero dimensional_ if it is Hausdorff and has a basis consisting of clopen
sets. The odometer defied in the non-compact space $\Sigma_{\geq 1}$ can be
used to define odometers in several spaces.

######  Theorem 4.1.

Let $X$ be nonempty Polish zero-dimensional space, for which all compact
subsets have empty interior. Then, there exists a continuous map $O_{X}:X\to
X$ topologically conjugated to $O_{1}:\Sigma_{\geq 1}\to\Sigma_{\geq 1}$.

###### Proof.

Alexandrov and Urysohn [Ke, Theorem 7.7 p.37] proved that the Baire space is
the unique, up to homeomorphism, nonempty Polish zero-dimensional space, for
which all compact subsets have empty interior. Therefore, there exists an
homemorphism $g:\Sigma_{\geq 1}\to X$. It suffices to define $O_{X}:X\to X$ by

$O_{X}=g\circ O_{0}\circ g^{-1}.$

This readily yields the desired transformation. ∎

This result can be applied in the setting of countable Markov shifts. Let $M$
be a ${\mathbb{N}}\times{\mathbb{N}}$ matrix with entries $0$ or $1$. The
_symbolic space_ associated to $M$ with alphabet ${\mathbb{N}}$ is defined by

$\Sigma_{M}:=\left\\{(x_{1},x_{2},\dots)\in\Sigma_{\geq
1}:M(x_{i},x_{i+1})=1\text{ for every }i\in{\mathbb{N}}\right\\},$

As before, we endow ${\mathbb{N}}$ with the discrete topology and
${\mathbb{N}}^{{\mathbb{N}}}$ with the product topology. On $\Sigma_{M}$ we
consider the induced topology given by the natural inclusion
$\Sigma\subset{\mathbb{N}}^{{\mathbb{N}}}$. We stress that, in general, this
is a non-compact space. The space $\Sigma_{M}$ is locally compact if and only
if for every $i\in{\mathbb{N}}$ we have $\sum_{j\in{\mathbb{N}}}M(i,j)<\infty$
(see [Ki, Observation 7.2.3]). The space $\Sigma_{M}$ is metrizable. The
following result is direct consequence of Theorem 4.1.

######  Corollary 4.2.

Let $\Sigma$ be a non-locally compact symbolic space, then there exists a
continuous map $O_{\Sigma}:\Sigma\to\Sigma$ topologically conjugated to
$O_{1}:\Sigma_{\geq 1}\to\Sigma_{\geq 1}$.

Moore generally, for any Polish space with no isolated points we can induce,
by semi-conjugacy, an odometer. Indeed, it suffices to proceed as in Theorem
4.1 noticing that Sierpinski [Ke, 7.15 p.40] proved that if $X$ is a non empty
Polish space with no isolated points then there exists a continuous bijection
$g:\Sigma_{\geq 1}\to X$. We stress that, in this case, the map $g^{-1}$ need
not to be continuous.

##  5\. Odometers in the interval

Numeration systems have been related to odometers, at least, since the work of
von Neumann and that of Kakutani. In this section, by means of the results in
section 2, we extend this relation to a wide range of numeration systems using
infinite alphabets. Before moving to the countable alphabet setting we recall
the interval map realization of the odometer $O$. Let $(I_{n})_{n}$ be the
partition of the unit interval, where
$I_{n}=[\frac{2^{n-1}-1}{2^{n-1}},\frac{2^{n}-1}{2^{n}})$, for
$n\in{\mathbb{N}}$. The interval realization of the _dyadic odometer_ is the
interval map $\tilde{O}:[0,1)\to[0,1)$ defined by

$\tilde{O}(x)=x+\frac{3}{2^{n}}-1,$ (5.1)

for $x\in I_{n}$. We have that $\tilde{O}:[0,1]\setminus\\{{\text{dyadic
rationals}}\\}\to[0,1]\setminus\\{{\text{dyadic rationals}}\\}$ is
topologically conjugated to the odometer $O$ defined in the space $\Sigma_{2}$
minus the sequences that end with a tail of ones or with a tail of zeros.
Moreover, the action of $\tilde{O}$ can be extended to the dyadic rationals.
Thus, we obtain an interval map realization of Remark 3.18.

######  Remark 5.1.

The map $n\mapsto\tilde{O}(1/2)$ is a bijection between the sets of dyadic
numbers in $(0,1)$ and ${\mathbb{N}}\cup\\{0\\}$. That is, the orbit of $1/2$
by $\tilde{O}$ passes through every dyadic number in $(0,1)$ once and only
once.

###  5.1. Odometers obtained from interval maps

Given an interval map with countably many full branches, by means of the
results in section 2, we construct an associated odometer. Denote by
$\text{dom}(T)$ the domain of the function $T$. We now define the class of
interval maps we consider.

######  Definition 5.2.

Let $(I_{n})_{n}$ be a collection of intervals in $[0,1]$ having disjoint
interiors. A map $T:\bigcup_{n\geq 1}I_{n}\mapsto[0,1]$ is a Countable-Markov-
Interval map (CMI map) if the following holds:

  1. (a)

The map $T$ is full-branched, that is, for every $n\in{\mathbb{N}}$ we have
$(0,1)\subset T(I_{n})$.

  2. (b)

There exists $\alpha>0$ such that the map is pieecewise of class
$C^{1+\alpha}$.

  3. (c)

For every $x\in(0,1)$ we have that $|T^{\prime}(x)|>1$.

  4. (d)

If $x\in\text{dom}(T)\cap\\{0,1\\}$ then $|T^{\prime}(x)|\geq 1$.

For a CMI-map $T$ its corresponding _repeller_ is the set

$\Lambda_{T}=\left\\{x\in\text{dom}(T):T^{n}(x)\text{ is well defined for
every }n\in{\mathbb{N}}\right\\}.$

The set of points with _finite orbit_ for $T$ is defined by
$\mathcal{F}=\text{dom}(T)\setminus\Lambda_{T}$. We stress that for some maps
this set may be empty. For an interval $I$ let $\partial I$ be the boundary
points of $I$. Denote the set of _boundary_ points of $T$ by:

$\mathcal{B}_{T}=\bigcup_{n\geq 0}T^{-n}\left(\\{0,1\\}\cup\bigcup_{n}\partial
I_{n}\right).$

The following holds,

######  Lemma 5.3.

For every CMI-map $T$ there exist a continuous _coding map_
$\pi_{T}:\Sigma_{\geq 1}\to\Lambda_{T}$ such that

$\pi_{T}\circ T=\sigma\circ\pi_{T}.$

###### Proof.

The proof of this results is well known, see for example [Sa, Proposition
1.2], for points not belonging to the boundary of the partition
$(0,1)\setminus\mathcal{B}_{T}$. Since all iterates of the map $T$ are well
defined in $\Lambda_{T}$, the map $\pi_{T}$ can actually be defined be defined
in $\text{dom}(T)\setminus\mathcal{F}=\Lambda_{T}$. ∎

######  Remark 5.4.

The map $\pi_{T}:\Sigma_{\geq
1}\setminus\pi_{T}^{-1}\left(\mathcal{B}_{T}\right)\to\Lambda_{T}\setminus\mathcal{B}_{T}$
is an homemorphism.

######  Remark 5.5.

Note that we do not require that $(0,1)\subset\bigcup_{n}I_{n}$. Also, the
closure of the points in the boundary of the partition can be very large. It
can be, for example, a Cantor set.

To every CMI-map $T$ we can associate an odometer in the following way.

######  Definition 5.6.

Let $O_{T}:\Lambda_{T}\to[0,1]$ be the map defined by,

$O_{T}(x)=\left(\pi_{T}\circ O_{1}\circ\pi_{T}^{-1}\right)(x).$

It follows from the definition of $O_{T}$, Remark 5.4 and Theorem 2.4 that,
away from the boundary points, the map $O_{T}$ is topologically conjugated to
the odometer in a countable alphabet. If $\Lambda_{T}=[0,1)$ or
$\Lambda_{T}=(0,1]$ the map $O_{T}$ can be thought of as an infinite interval
exchange transformation.

######  Theorem 5.7.

Let $T$ be an CMI-map and $O_{T}$ the associated odometer. Then, the map
$O_{T}:\text{dom}(T)\setminus\mathcal{B}_{T}:\to[0,1]$ is topologically
conjugated to the $O_{0}$ odometer.

######  Corollary 5.8.

The map $O_{T}:\text{dom}(T)\setminus\mathcal{B}_{T}:\to[0,1]$ satisfies the
following dynamical properties:

  1. (a)

The map $O_{T}$ is minimal and uniquely ergodic.

  2. (b)

The map $O_{T}$ has zero topological entropy.

  3. (c)

The map $O_{T}$ has purely discrete spectrum and it is equicontinuous.

The behavior of $O_{T}$ on points with finite orbits $\mathcal{F}$ will depend
on how we define $T$ on the boundary points $\partial I$. In general, we are
interested in continuous extensions of $T$ at least from one-side at each
point in $\partial I$. In what follows we provide explicit examples of maps
$O_{T}$.

###  5.2. The odometer associated to the continued fraction expansion

Recall that, see section 3.4.2, every number in $(0,1)$ can be written as a
continued fraction expansion. There exists CMI-map closely related to this
expansion. Indeed, the _Gauss_ map $G:(0,1]\to(0,1]$, is the interval map
defined by

$G(x)=\frac{1}{x}-\left[\frac{1}{x}\right],$

where $[\cdot]$ is the integer part. If $x=[a_{1}a_{2}a_{3}\dots]$ then
$G(x)=[a_{2}a_{3}\dots]$. The repeller for $G$ is
$\Lambda_{G}=[0,1]\setminus{\mathbb{Q}}$ and the set with finite orbits is
$\mathcal{F}={\mathbb{Q}}$. We will assume that the continued fraction
expansion of rationals does no ends with a one. Denote by
$O_{G}:(0,1]\to(0,1]$ the transformation obtained from $G$ as given in
Definition 5.6 . The action of $O_{G}$ over the irrational numbers is, by
definition, given by:

######  Lemma 5.9.

If $x=[a_{1},a_{2},a_{3},\dots]\in[0,1)$ then

$O_{G}(x)=[\underbrace{1,1,\dots,1}_{(a_{1}-1)\textrm{-times}},a_{2}+1,a_{3},a_{4},\dots].$

We have two natural choices to define $O_{G}$ at the boundary points $[n]$. In
the first case we define $O_{G}$ to be continuous from the right, that is

$O_{G}[n]=\lim_{m\to\infty}O_{G}[(n-1)\ 1\
m]=\lim_{m\to\infty}[\underbrace{1\dots 1}_{n-2}\ 2\ m]=[\underbrace{1\dots
1}_{n-2}\ 2].$

In that case, $[n]$ is periodic for $O_{G}$, with period $2^{n-2}$. With this
choice, every rational number $[a_{1}a_{2}\dots a_{j}]$ is periodic, with
period $2^{\sum_{i}{a_{i}}-2}$. This corresponds to the cyclic odometer, see
Definition 3.8.

On the other hand, we could define $O_{G}$ to be continuous from the left. In
this case we obtain,

$O_{G}[n]=\lim_{m\to\infty}O_{G}[n\ m]=\lim_{m\to\infty}[\underbrace{1\dots
1}_{n-1}\ (m+1)]=[\underbrace{1\dots 1}_{n-1}].$

With this choice, the orbit of every rational number goes up the Kepler Tree
until it reaches $[1]$. Hence, every rational number in $(0,1)$ reaches $[1]$
in finite time. In what follows we will consider the first choice, that is, we
assume $O_{G}$ to be continuous from the right. Recall that the Fibonacci
Sequence is defined by,

$f_{0}=0,\ f_{1}=1,\ f_{n+1}=f_{n}+f_{n-1}.$

The following result explicitly describes the odometer $O_{G}$ (when it is
continuous from the right).

######  Proposition 5.10.

For $x\in\left[\frac{1}{n+1},\frac{1}{n}\right)\cap{\mathbb{Q}}^{c}$ we have
that

$O_{G}(x)=\frac{x(f_{n-1}-nf_{n})+f_{n}}{x(f_{n}-nf_{n+1})+f_{n+1}}.$

If $x=[a_{1},\dots,a_{n}]\in{\mathbb{Q}}$ then the point $x$ is periodic for
$O_{G}$ with period $2^{-2+\sum_{i=1}^{n}a_{i}}$.

###### Proof.

Any $x\in\left[\frac{1}{n+1},\frac{1}{n}\right)$ can be uniquely written as

$x=[n\ a]=\frac{1}{n+\frac{1}{a}},$ (5.2)

with $a\geq 1$ a real number. The action of the odometer is

$O_{G}[n\ a]=[\underbrace{1\dots 1}_{n-1}\ (a+1)].$ (5.3)

We claim that (5.3) leads to

$O_{G}[n\ a]=\frac{f_{n-1}a+f_{n}}{f_{n}a+f_{n+1}}.$ (5.4)

Indeed, reasoning by induction on $n$, we have the base case

$\displaystyle O_{G}[1\ a]=[a+1]=\frac{1}{a+1}.$

Consider now the case $(n+1)$, assuming that (5.4) holds

$\displaystyle O_{G}[(n+1)\ a]$ $\displaystyle=$
$\displaystyle[\underbrace{1\dots 1}_{n}\ (a+1)]$ $\displaystyle=$
$\displaystyle\frac{1}{1+[\underbrace{1\dots 1}_{n-1}\ (a+1)]}$
$\displaystyle=$
$\displaystyle\frac{1}{1+\frac{f_{n-1}a+f_{n}}{f_{n}a+f_{n+1}}}$
$\displaystyle=$
$\displaystyle\frac{f_{n}a+f_{n+1}}{a(f_{n}+f_{n-1})+f_{n+1}+f_{n}}=\frac{f_{n}a+f_{n+1}}{f_{n+1}a+f_{n+2}}.$

The equation (5.2) leads to

$a=\frac{x}{1-nx},$

and then

$\displaystyle O_{G}(x)$ $\displaystyle=$
$\displaystyle\frac{f_{n-1}\left(\frac{x}{1-nx}\right)+f_{n}}{f_{n}\left(\frac{x}{1-nx}\right)+f_{n+1}}$
$\displaystyle=$
$\displaystyle\frac{x(f_{n-1}-nf_{n})+f_{n}}{x(f_{n}-nf_{n+1})+f_{n+1}}.$

∎

The map $O_{G}$ has one fully supported ergodic measure which corresponds to
the projection of the unique invariant measure for $O_{1}$. Moreover, since
the rational points are periodic, the map also preserves countably many
ergodic measures supported on periodic orbits. There are no other ergodic
invariant measures. We therefore have,

######  Lemma 5.11.

The space of invariant probability measures for $O_{G}$ is a Choquet simplex
with countably many extreme points.

######  Remark 5.12.

The function $O_{G}$, or its inverse, has occurred previously in the
literature. For example, in [VPB] Viader, Paradis and Bibiloni proved that the
question mark function, introduced by Minkowski, corresponds to the asymptotic
distribution of the enumeration of the rational numbers in $(0,1)$ given by
the Kepler tree, see section 3.4.2. Note, however, that they do not explicitly
mention the Kepler tree and that their construction of the order is obtained
in a different, although equivalent, way than the one induced by the
ododmetric action. In [VPB, Section 2.1], computing the successor in the order
of the Kepler tree, they obtain an explicit formula for the inverse of
$O_{G}$. Also, Panti [Pa] constructed Kakutani von-Neumann odometers in higher
dimensional simplexes. The one dimensional version of his construction is
related to the inverse of the map $O_{G}$.

######  Remark 5.13.

We provide a flow version of the example described on this section that may be
of interest. Let $\tau:(0,1]\to{\mathbb{R}}$ be a continuous positive map
bounded away from zero. Denote by
$Y=\left\\{(x,t)\in(0,1)\times{\mathbb{R}}:0\leq t\leq\tau(x)\right\\}$.
Consider the equivalence relation $(x,\tau(x))\sim(O_{G}(x),0)$ and the
quotient space $Y/\sim$. The flow $\Phi=(\varphi_{t})_{t}$ defined on the
quotient space by

$\varphi_{t}(x,s)=(x,s+t)\quad\text{ whenever }\quad s+t\in[0,\tau(x)].$

This flow satisfies the following properties: it has one, and only one, fully
supported measure and countably many measures supported on periodic orbits.
Every point not in a periodic orbit is generic for the fully supported
measure. A renormalization formula for the flow $\Phi$ and the the suspension
flow with base the shift on countably many symbols and roof function $\tau$
can be obtain from the corresponding property established in section 2.1.

###  5.3. The odometer associated to the backward continued fraction expansion

Recall that, see section 3.4.3, every number in $(0,1)$ can be written as a
backward continued fraction expansion. There exists CMI-map closely related to
this expansion. Indeed, the Renyi map, $R:[0,1)\to[0,1)$, is defined by

$R(x)=\frac{1}{1-x}-\left[\frac{1}{1-x}\right].$

This map acts as the shift on the backward continued fraction (see [DK,
Chapter 11]). That is, if $x=[a_{1}a_{2}a_{3}\dots]_{B}$ then
$R(x)=[a_{2}a_{3}\dots]_{B}$. Note that $R(0)=0$ and that $R^{\prime}(0)=1$.
In particular, the Renyi map is not uniformly expanding. Renyi [R] showed that
there exists an infinite $\sigma-$finite invariant measure, $\mu_{R},$
absolutely continuous with respect to the Lebesgue measure. It is defined by

$\mu_{R}(A)=\int_{A}\dfrac{1}{x}\ dx,$

where $A\subset[0,1)$ is a Borel set. There is no finite invariant measure
absolutely continuous with respect to the Lebesgue measure. Denote by
$O_{R}:[0,1)\to[0,1)$ the transformation obtained from $R$, as given in
Definition 5.6. In [IP] we proved, using Minkowski’s question mark function,
that:

######  Proposition 5.14.

For every $x\in[0,1)$ we have that,

$O_{R}(x)=\frac{1}{2\left[\frac{1}{1-x}\right]+1-\frac{1}{1-x}}.$

Moreover, we established the following result which, in this setting, follows
directly from the definition of $O_{R}$:

######  Lemma 5.15.

If $x=[a_{1},a_{2},a_{3},\dots]_{B}\in[0,1)$ then

$O_{R}(x)=[\underbrace{2,2,\dots,2}_{(a_{1}-2)\textrm{-times}},a_{2}+1,a_{3},a_{4},\dots]_{B}.$

Interest in the map $O_{R}$ arises from the fact that the orbit of $x=0$
induces an explicit bijection between the sets of natural numbers and the
rational numbers in $(0,1)$. Indeed, the map $n\mapsto O_{R}^{n}(0)$ is the
required bijection. Of course, this readily follows Remark 3.20 and the
general ideas developed in this article. However, this fact was first observed
by Newman, while solving a problem posed by Knuth [Kn], although the way
$O_{R}$ is written is formally different. The proof original given of this
fact is based on work by Calkin and Wilf [CW] and can be found in [AZ, Chapter
17]. They considered the sequence $q(n)$ defined as the number of ways of
writing the integer $n$ as a sum of powers of $2$, each power being used at
most twice. They go on to prove that the map $n\to b(n)/b(n+1)$ is a bijection
between the natural and the rational numbers. It turns out that the numbers
$q(n)$ appear in the Calkin and Wilf tree (also known as the Euclid tree).
Newman constructed a map for which the orbit of zero realizes the Calkin and
Wilf bijection. Bonanno and Isola [BI, Theorem 2.3] proved that the map
$O_{R}$ is topologically conjugated to the interval map realization of dyadic
odometer. Their proof uses, instead of backward continued fractions, the usual
continued fractions. Again, the formal expressions for $O_{R}$ are different.
As we already mentioned, this counting property is a manifestation of Theorem
3.11 and of its version described in Remark 3.20.

Since every finite sequence can be realized as in infinite sequence ending by
a tail of $2$’s (see equation (3.3)), for the map $R$ the set $\mathcal{F}$ is
empty. Or, if we prefer, the odometric action on finite words corresponds to
the top-down odometer, see Definition 3.8. Therefore,

######  Proposition 5.16.

The map $O_{R}:[0,1)\to[0,1)$ is uniquely ergodic.

It turns out that the only invariant measure is the projection of the Lebesgue
measure through the question mark function (see [BI, IP]). In particular, the
invariant measure is singular with respect to the Lebesgue measure.

###  5.4. The odometer on numbers with partial quotients larger than $k$

In the previous examples the repeller together with the points with finite
orbits form an interval. In this section we provide an example in which this
is not the case. Let $k\in{\mathbb{N}}$ be a positive integer larger or equal
than $2$. Consider the set,

$\Lambda_{k}=\left\\{x=[a_{1},a_{2},\dots]\in(0,1):a_{i}\geq k\text{ for every
}i\in{\mathbb{N}}\right\\}.$

This is a totally disconnected set having positive Hausdorff dimension
strictly less than $1$. Let $G_{k}$ be the restriction of the Gauss map to the
set $\Lambda_{k}$. Denote by $O_{G_{k}}:(0,1/k]\to(0,1]$ the transformation
obtained from $G_{k}$, as given in Definition 5.6. Consider the following two
$k-$Fibonacci Sequences defined by,

$\displaystyle b_{0}=0,\ b_{1}=1,\ b_{n+1}=kb_{n}+b_{n-1},$ $\displaystyle
d_{0}=1,\ d_{1}=1,\ d_{n+1}=kd_{n}+d_{n-1}.$

An argument similar to that in Proposition 5.10 leads to the following:

######  Proposition 5.17.

Let $n\in{\mathbb{N}}\cup\\{0\\}$. If
$x\in\left[\frac{1}{n+1+k},\frac{1}{n+k}\right)\cap{\mathbb{Q}}^{c}\cap\Lambda_{k}$
then

$O_{G_{k}}(x)=\frac{x(b_{n}-nd_{n+1})+d_{n+1}}{x(b_{n+1}-nd_{n+2})+d_{n+2}}.$

######  Remark 5.18.

Note that for $n\in{\mathbb{N}}$, if $x\in(1/(n+1+k),1/(n+k))$ then

$\lim_{x\to\frac{1}{n+1+k}}O_{G_{k}}(x)=\frac{b_{n}}{b_{n+1}}.$

Moreover,

$\lim_{n\to\infty}\frac{b_{n}}{b_{n+1}}=\frac{1}{\varphi_{k}},$

where

$\varphi_{k}=\frac{k+\sqrt{k^{2}+4}}{2},$

is the $k-$golden mean.

The map $O_{G_{k}}$ has one fully supported ergodic measure which corresponds
to the projection of the unique invariant measure for $O_{k}$.

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