Dr Mark Hamer
email: m.hamer@ucl.ac.uk

G David Batty

Michael G Marmot

Archana Singh-Manoux

Mika Kivimäki

Anti-depressant medication use and C-reactive protein: Results from two population-based studies

Keywords: Cardiovascular risk, depression, inflammation, selective serotonin reuptake inhibitors, tricyclic antidepressant

The use of anti-depressant medication has been linked to cardiovascular disease (CVD).

We examined the association between anti-depressant medication use and a marker of low grade systemic inflammation as a potential pathway linking anti-depressant use and CVD in two population based studies. Data were collected in a representative sample of 8,131 community dwelling adults (aged 47.4  15.9 yrs, 46.7% male) from the Scottish Health Surveys (SHS). The use of anti-depressant medication was coded according to the British National Formulary and blood was drawn for the measurement of C-reactive protein (CRP).

In a second study, we attempted to replicate our findings using longitudinal data from the Whitehall II study (n=4584, aged 55.5  5.9 yrs, mean follow-up 5.5 years).

Antidepressants were used in 5.6% of the SHS sample, with selective serotonin reuptake inhibitors (SSRIs) being the most common. There was a higher risk of elevated CRP (>3 mg/L) in users of tricyclic antidepressant (TCA) medication (multivariate adjusted odds ratio (OR) = 1.52, 95% CI, 1.07 -2.15), but not in SSRI users (multivariate adjusted OR = 1.07, 95% CI, 0.81 -1.42). A longitudinal association between any antidepressant use and subsequent CRP was confirmed in the Whitehall cohort. In summary, the use of antidepressants was associated with elevated levels of systemic inflammation independently from the symptoms of mental illness and cardiovascular co-morbidity. This might be a potential mechanism through which antidepressant medication increases CVD risk. Further data are required to explore the effects of dosage and duration of antidepressant treatment.

.

Introduction

As a result of the secular rises in mental health problems, antidepressant medication is currently commonly prescribed across Europe and the United States [START_REF] Olfson | National patterns in antidepressant medication treatment[END_REF][START_REF] Reid | Long term treatment of depression with selective serotonin reuptake inhibitors and newer antidepressants[END_REF]. The use of anti-depressant medication has been linked with a greater risk of weight gain [START_REF] Aronne | Weight gain in the treatment of mood disorders[END_REF], diabetes [START_REF] Rubin | Elevated depression symptoms, antidepressant medicine use, and risk of developing diabetes during the diabetes prevention program[END_REF][START_REF] Andersohn | Long-term use of antidepressants for depressive disorders and the risk of diabetes mellitus[END_REF][START_REF] Kivimäki | Antidepressant use before and after the diagnosis of type 2 diabetes: A longitudinal modeling study[END_REF], and with an increased risk of cardiovascular disease (CVD) events in most [START_REF] Cohen | Excess risk of myocardial infarction in patients treated with antidepressant medications: association with use of tricyclic agents[END_REF][START_REF] Tata | General population based study of the impact of tricyclic and selective serotonin reuptake inhibitor antidepressants on the risk of acute myocardial infarction[END_REF][START_REF] Chen | Risk of cerebrovascular events associated with antidepressant use in patients with depression: a population-based, nested case-control study[END_REF][START_REF] Fosbøl | Prognosis in heart failure and the value of {beta}-blockers are altered by the use of antidepressants and depend on the type of antidepressants used[END_REF][START_REF] Krantz | Psychotropic medication use and risk of adverse cardiovascular events in women with suspected coronary artery disease: outcomes from the Women's Ischemia Syndrome Evaluation (WISE) study[END_REF][START_REF] Smoller | Antidepressant use and risk of incident cardiovascular morbidity and mortality among postmenopausal women in the Women's Health Initiative study[END_REF], but not all [START_REF] Taylor | Effects of antidepressant medication on morbidity and mortality in depressed patients after myocardial infarction[END_REF][START_REF] Knol | No increased incidence of diabetes in antidepressant users[END_REF][START_REF] O'connor | Antidepressant use, depression, and survival in patients with heart failure[END_REF] studies. Low grade systemic inflammatory processes may be a key mechanism underlying some of the potential adverse effects of antidepressant medication, particularly raised CVD risk. While a series of studies suggest an association between depressive symptoms and higher levels of inflammatory biomarkers [START_REF] Dinan | Inflammatory markers in depression[END_REF], the link between anti-depressant use and inflammation has not been previously tested in a large sample of the general population.

The aim of the present study was therefore to examine the association between antidepressant medication use and C-reactive protein (CRP) in a representative sample of community dwelling adults. Additionally, we performed a replication study based on longitudinal data from the Whitehall II cohort.

Methods

Scottish Health Survey

The Scottish Health Survey is a cross-sectional study that is typically conducted serially every 3-5 years, that draws a different nationally representative sample of the general population living in households for each survey (see Scottish Health Survey publications).

For the present analysis we combined data from separate surveys sampled in 1998 (n=4649), 2003 (n=2824) and 2008 (n=658). Participants from the different survey years were comparable in terms of demographics and risk factors. Study participants gave full informed consent and ethical approval was obtained from the London Research Ethics Council. Data were collected during two household visits. During the first visit, trained interviewers collected data on a range of demographic and health related behaviours, including self-reported smoking (current/ex-smoker/never), alcohol intake, and participation in any form of physical activity (including walking for any purposes, heavy domestic activity, recreational sports and exercise). Psychological distress was measured using the 12 item version of the General Health Questionnaire (GHQ-12), a widely-utilized measure of psychological distress in population-based studies [START_REF] Goldberg | The validity of two versions of the GHQ in the WHO study of mental illness in general health care[END_REF]. The GHQ-12 enquires about symptoms in the last 4 weeks. We employed a GHQ-12 cut off score of 4 to denote psychological distress. This definition has been validated against standardised psychiatric interviews and has been strongly associated with depression and anxiety [START_REF] Goldberg | The validity of two versions of the GHQ in the WHO study of mental illness in general health care[END_REF].

During the second household visit, conducted within a few days of the first, trained nurses collected information about physician-diagnosed CVD (stroke, ischemic heart disease, angina symptoms), diabetes, and hypertension, recorded medication usage, measured weight and height to calculate body mass index [BMI; weight (kilograms)/height (meters) squared], and took blood samples from consenting participants. The use of antidepressant medication was coded according to the British National Formulary (http://bnf.org/bnf) (codes; 040301 to 040304). Peripheral blood was collected in serum tubes and spun at room temperature. All blood samples were frozen at -70 °C until assay.

The analysis of CRP levels was performed using the N Latex high sensitivity CRP mono immunoassay on the Behring Nephelometer II analyser. The limit of detection was 0.17 mg/L and the coefficient of variation was less than 6% for this assay. All analyses from each survey were carried out in the same laboratory according to Standard Operating Procedures by State Registered Medical Laboratory Scientific Officers.

Antidepressant medication use was split into four categories, [none, tricyclics (TCAs), selective serotonin reuptake inhibitor (SSRIs), other]. We used χ 2 and one-way ANOVA to examine differences in baseline characteristics between non-medicated and medicated groups. C-reactive protein was log transformed to normalise the data and we excluded any participants reporting an acute infection (bronchitis, influenza, etc) in the 3 weeks prior to the nurse visit since this may contribute to elevated levels of CRP.

Associations between anti-depressant medication use and CRP were examined using general linear models. We fitted several models that included a basic model adjusted for age and sex; and several mutually exclusive models including further adjustments for physical activity (<1 hr/wk; 1-3 hrs/wk; >3 hrs/wk), alcohol (never/ex drinker; trivial; moderate; heavy) and smoking (never; previous; current) (model 1); adjustment for body mass index (<18.5 kg/m 2 ; ≥18.5 <25 kg/m 2 ; ≥25 <30 kg/m 2 ; ≥30 kg/m 2 ) (model 2); adjustment for psychological distress (GHQ-12 score of zero; 1 -3; ≥4) (model 3). Finally we performed a fully adjusted model that contained all of the aforementioned variables plus physician diagnosed CVD, diabetes, hypertension. This modelling strategy allowed us to examine the relative importance of each co-variable in the model and crucially if the associations between anti-depressant use and CRP were independent of important confounding factors. We also used multivariate logistic regression to compute odds ratios (OR) with accompanying 95% confidence intervals (CI) for the association between medication use and CRP as a binary outcome (>3 mg/L). This cut point was based on previous clinical guidelines [START_REF] Pearson | Centers for Disease Control and Prevention; American Heart Association. Markers of inflammation and cardiovascular disease: application to clinical and public health practice: A statement for healthcare professionals from the Centers for Disease Control and Prevention and the American Heart Association[END_REF].

Whitehall II study

We performed a replication study based on data from the Whitehall II prospective cohort study of British civil servants. CRP was measured from 4584 participants in 1991-1993 (baseline) and again in 2003-2004 (follow-up). The analysis of CRP from serum stored at -80ºC was performed using a high-sensitivity immunonephelometric assay in a BN ProSpec nephelometer (Dade Behring, Milton Keynes, UK) [START_REF] Kivimäki | Does high C-reactive protein concentration increase atherosclerosis? The Whitehall II study[END_REF]. All other baseline characteristics were assessed in 1997-1999 and included age, sex, smoking (current vs other), physical activity (sedentary vs non-sedentary), BMI, prevalent CHD (history of myocardial infarction [MI] or definite angina), diabetes (according to World Health Organization definition) and hypertension (systolic/diastolic blood pressure <140/90 mm Hg or on antihypertensive treatment) and psychological distress (GHQ-12 caseness).

Use of antidepressants (yes/no) was requested at baseline in 1997-1999 and at follow-up in 2003-2004 (mean follow-up 5.5  0.5 years).

Replication analyses with longitudinal data from Whitehall II used linear and logistic regression analyses to examine the association of antidepressant use in [START_REF] Goldberg | The validity of two versions of the GHQ in the WHO study of mental illness in general health care[END_REF]-1999with CRP in 2003[START_REF] Danesh | C-reactive protein and other circulating markers of inflammation in the prediction of coronary heart disease[END_REF] after adjustment for baseline characteristics (i.e., smoking, physical activity, alcohol consumption, psychological distress, CHD, diabetes, hypertension, body mass index, and psychological distress in [START_REF] Goldberg | The validity of two versions of the GHQ in the WHO study of mental illness in general health care[END_REF]-1999and CRP in 1991-1993)). To assess dose-response associations we repeated this analysis with antidepressant use in 1997-1999 and 2003-2004 (0, 1 or 2 times) as the exposure variable. There were no clear differences in our results between men and women, so the data were pooled and sex-adjusted. Analyses were conducted using SPSS version 14 (Scottish Health Survey) and SAS version 9.2 (Whitehall II).

Results

Scottish Health Survey

The initial sample with available data on CRP consisted of 10,518 participants, although after exclusions (n=854 with infection in last 3 wks; n=863 missing BMI; n=670 missing demographics), the analytical sample comprised 8131 adults (aged 47.4  15.9 yrs, 3797 male, 4334 women). Excluded participants were older (52.3 vs. 47.4, p<0.001) and were somewhat more likely to be using anti-depressant medication (8.3% vs. 5.6%, p<0.001).

In this sample, SSRIs were the most commonly used type of anti-depressants. The medicated study participants were older, more likely to be female, smokers, ex-drinkers, more sedentary, obese, displayed higher levels of psychological distress and greater morbidity (Table 1). TCA users were on average older than SSRI users (54.8 vs 47.2 yrs, p<0.001); TCA users had lower levels of psychological distress (34.2 vs 42.7%, p<0.001); although levels of obesity (31.6 vs 25.4%, p=0.12) and smoking rates (41.3 vs 42.3%, p=0.88) did not significantly differ between TCA and SSRI users, respectively.

In linear analyses that examined the associations between anti-depressants and CRP, the use of TCAs remained consistently associated with CRP after a series of adjustments for potential confounders (Table 2). There was also an association between SSRI use and CRP, although this association was largely confounded by smoking. All of the covariates were strongly associated with CRP (p<0.001). We re-ran the models replacing BMI with waist/hip ratio as a marker of central adiposity but this did not impact on the results, and there remained an elevated level of CRP in TCA users (fully adjusted =0.35, 95% CI, 0.17 -0.53). Sex stratified analyses suggested that the associations of TCA anti-depressant use were somewhat stronger in women (fully adjusted =0.37, 95% CI, 0.17 -0.58) than in men (=0.30, 95% CI, -0.04 -0.64).

A similar pattern of results emerged using logistic regression, that showed a greater risk of elevated CRP (>3 mg/L) in users of TCA medication (multivariate adjusted OR = 1.52, 95% CI, 1.07 -2.15), but not in SSRI users (multivariate adjusted OR = 1.07, 95% CI, 0.81 -1.42). We performed a number of sensitivity analyses to examine potential effect modification by obesity, psychological distress, and smoking. The removal of obese participants (n=1842) or smokers (n=2281), or participants with history of CVD (n=518) essentially did not alter the results, while removal of participants with psychological distress (n=1113) strengthened the association between TCA use and CRP (multivariate adjusted OR = 1.73, 95% CI, 1.13 -2.64).

Whitehall II study

Clinical characteristics of the 3313 men and 1271 women at mean age of 55.5 years are presented in table 1. A lower proportion of the Whitehall II sample were taking medication for depression (2.6%) compared to the Scottish Health Survey (5.6%). Those on antidepressants at baseline displayed higher levels of psychological distress and were more likely to be female, obese, non-drinkers, and use lipid-lowering medication compared with those not using anti-depressants (Table 1).

Fifty-four participants (1.2%) were treated with antidepressants both at baseline and follow-up. Linear regression analyses on the longitudinal association showed a doseresponse relation between antidepressant use and subsequent CRP, which was strongest for participants that used antidepressants at both time points (Table 3). This association was largely robust to adjustment for confounders. The association between use of antidepressants only at baseline and CRP at follow up was not statistically significant.

Discussion

In this study of a representative sample of community dwelling adults we observed an association between antidepressant medication use and CRP, which was particularly strong for TCAs. A longitudinal association between any antidepressant use and subsequent CRP was replicated in the Whitehall study, although we were unable to investigate the role of medication type as this information had not been collected. These associations were independent of psychological distress, a measure of common mental disorder that is reflective of symptoms of anxiety and depression. In addition, antidepressant users were more likely to be obese and have cardiovascular disease co-morbidity at baseline. The use of TCA therapy in clinical practise has largely been replaced by newer antidepressant medication such as SSRIs, which demonstrated weaker associations with CRP in the present study and could be largely explained through confounding effects of lifestyle risk factors such as smoking. This is consistent with recent work that showed the association between depressive symptoms and CRP was largely explained through smoking, weight gain and physical inactivity [START_REF] Hamer | Persistent depressive symptomatology and inflammation: Do health behaviours and weight control mediate this relationship?[END_REF].

The association between antidepressant use and risk of CVD is controversial. In prospective studies of participants with existing CVD some studies have shown protective effects of SSRIs on recurrent MI and death [START_REF] Taylor | Effects of antidepressant medication on morbidity and mortality in depressed patients after myocardial infarction[END_REF] whilst other studies have shown increased risk in TCA and SSRI users [START_REF] Fosbøl | Prognosis in heart failure and the value of {beta}-blockers are altered by the use of antidepressants and depend on the type of antidepressants used[END_REF][START_REF] Krantz | Psychotropic medication use and risk of adverse cardiovascular events in women with suspected coronary artery disease: outcomes from the Women's Ischemia Syndrome Evaluation (WISE) study[END_REF]. In another study, the increased risk of mortality associated with antidepressants was not robust to adjustment for depressive symptoms, thus suggesting an effect of depression but not medication [START_REF] O'connor | Antidepressant use, depression, and survival in patients with heart failure[END_REF]. In an occupational cohort study of initially healthy participants, use of TCAs, but not SSRIs, was associated with increased risk of MI over 4.5 years follow up [START_REF] Cohen | Excess risk of myocardial infarction in patients treated with antidepressant medications: association with use of tricyclic agents[END_REF]. In the Women's Health Initiative study of postmenopausal women, TCAs and SSRIs were associated with increased risk of mortality, and SSRIs with greater risk of hemorrhagic and fatal stroke, although not with MI over 5.9 years of follow up [START_REF] Smoller | Antidepressant use and risk of incident cardiovascular morbidity and mortality among postmenopausal women in the Women's Health Initiative study[END_REF].

Short term trials of antidepressant medication in patients with major depression have produced inconsistent findings with regards to systemic markers of inflammation such as CRP [START_REF] O'brien | Antidepressant therapy and C-reactive protein levels[END_REF][START_REF] Dawood | Specific serotonin reuptake inhibition in major depressive disorder adversely affects novel markers of cardiac risk[END_REF][START_REF] Tousoulis | Antidepressive treatment as a modulator of inflammatory process in patients with heart failure: effects on proinflammatory cytokines and acute phase protein levels[END_REF]. However, these inconsistencies might reflect the different types of patients contained in each study, which included patients with coronary heart disease, heart failure, and patients with depression but otherwise healthy. Thus the underlying medical condition and other medications may have also impacted on levels of inflammation. The mechanisms that might link anti-depressants with low grade inflammation remain poorly understood. However, it is possible that factors such as weight gain and impaired glucose tolerance play a role. In addition antidepressants might impact on glucocorticoid receptors and the hypothalamic pituitary adrenal axis, which has both anti-and pro-inflammatory effects [START_REF] Pariante | Risk factors for development of depression and psychosis. Glucocorticoid receptors and pituitary implications for treatment with antidepressant and glucocorticoids[END_REF][START_REF] Pace | Cytokines and glucocorticoid receptor signaling. Relevance to major depression[END_REF].

Elevated levels of systemic inflammation might be a potential mechanism through which antidepressant medication increases CVD risk [START_REF] Danesh | C-reactive protein and other circulating markers of inflammation in the prediction of coronary heart disease[END_REF]. However, the role of CRP in CVD aetiology is highly controversial, since the evidence from Mendelian randomisation studies does not support a causal role [START_REF] Kivimäki | Does high C-reactive protein concentration increase atherosclerosis? The Whitehall II study[END_REF][START_REF] Zacho | Genetically elevated C-reactive protein and ischemic vascular disease[END_REF]Elliot et al., 2009) whereas there is supportive evidence of causality from experimental work in animals and humans [START_REF] Venugopal | Demonstration that Creactive protein decreases eNOS expression and bioactivity in human aortic endothelial cells[END_REF][START_REF] Verma | A self-fulfilling prophecy: C-reactive protein attenuates nitric oxide production and inhibits angiogenesis[END_REF][START_REF] Bisoendial | Activation of inflammation and coagulation after infusion of C-reactive protein in humans[END_REF]. Therefore, further studies are required to better understand the association between antidepressant use, low grade inflammation, and CVD risk. Other mechanisms might also be important. For example, tricyclic antidepressants may induce weight gain and have been shown to have a number of other cardio-toxic effects that might explain the increased risk of CVD, including orthostatic hypotension, reduced heart rate variability, QT interval prolongation, and greater risk of hypertension [START_REF] Roose | Comparison of paroxetine and nortriptyline in depressed patients with ischemic heart disease[END_REF][START_REF] Licht | Depression is associated with decreased blood pressure, but antidepressant use increases the risk for hypertension[END_REF][START_REF] Kemp | Impact of depression and antidepressant treatment on heart rate variability: A review and metaanalysis[END_REF].

Several limitations should be noted. We did not have information regarding dosage in either of our studies, which is a clear limitation. Furthermore, we did not have detailed information on the duration of anti-depressant treatment, although we attempted to examine this issue in the Whitehall II study by defining long term use as participants that reported taking antidepressants at baseline and follow up. The results showed that those participants taking antidepressants at baseline and follow up did demonstrate significantly elevated CRP, although use of anti-depressants at one time point only was not associated with CRP.

These data therefore suggest that chronic use might be an important factor, although further studies will be required to fully examine this issue. The associations of antidepressant use with CRP were independent of psychological distress, which reflects symptoms of depression and anxiety. However, we were unable to adjust specifically for a clinically diagnosed depression. Nevertheless, the strengths of our analyses include the sampling of a large, representative general population-based group, thus our results can be generalised to the wider community. In addition, the findings were replicated in a second study. The findings were generally more robust in women, who demonstrated higher levels of CRP, greater psychological distress and more use of anti-depressants compared with men.

However, these results should be treated cautiously since they might reflect a lack of statistical power in men.

In summary, previous evidence suggests an association between depressive symptoms and higher levels of inflammatory biomarkers. The present study suggests that the use of antidepressants was associated with elevated levels of systemic inflammation independently from symptoms of mental illness. This might be a potential mechanism through which antidepressant medication increases CVD risk.

Table 1 .

 1 Baseline characteristics of the study samples in Scottish Health Surveys (n=8131) and Whitehall II (n=4584)

		Scottish Health Surveys		Whitehall II
	Variable	Non-medicated	Medicated	p-value	Non-medicated	Medicated	p-value
		(n=7675)	(n=456)		(n=4470) †	(n=114) †
	Age (yrs)	47.216.0	50.514.1	<0.001	55.55.9	55.25.9	0.56
	Male (%)	48.0	24.8	<0.001	72.7	55.3	<0.001
	Smoking (%)
	Never/quit	72.7	57.9	<0.001	91.7	90.4	0.61
	Current	27.2	42.1		8.3	9.6
	Physical activity (%)
	None (<1 hr/wk)	27.6	39.1	<0.001	11.5	17.5	0.07
	Active	72.4	60.9		88.5	82.5
	Alcohol (%)*
	Never/ex-drinkers	4.9/3.8	5.1/11.4	<0.001	13.9	19.3	0.04
	Trivial	11.4	20.0		2.1	5.3
	Moderate	62.8	53.2		61.7	56.1
	Heavy	17.1	10.3		22.3	19.3
	BMI category (%)
	<18.5 kg/m 2	3.7	4.8	0.019	0.6	1.0	0.15
	≥18.5 <25 kg/m 2	34.0	28.3		43.0	42.0

Table 2 .

 2 Adjusted regression coefficients (95% CI) for the relation of antidepressant medication use with C-reactive protein in the Scottish

	Health Survey
	Drug class	N	Mean± SD Age & sex adjusted Model 1	Model 2	Model 3	Fully adjusted
			(mg\L)	β (95% CI)*	β (95% CI)*	β (95% CI)*	β (95% CI)*	β (95% CI)*
	None	7675 3.02±5.80	Reference	Reference	Reference	Reference	Reference
	TCAs	155	5.72±9.00	0.54 (0.35 to 0.73)	0.44 (0.25 to 0.62)	0.48 (0.30 to 0.66)	0.50 (0.31 to 0.69)	0.35 (0.17 to 0.52)
	SSRIs	260	3.56±5.19	0.20 (0.06 to 0.35)	0.13 (-0.02 to 0.27)	0.17 (0.03 to 0.31)	0.15 (0.01 to 0.30)	0.06 (-0.07 to 0.20)
	Other	41	4.02±4.39	0.24 (-0.12 to 0.60)	0.16 (-0.20 to 0.51)	0.28 (-0.06 to 0.62)	0.18 (-0.18 to 0.54)	0.16 (-0.18 to 0.49)

* Regression coefficients represent log transformed CRP values. Model 1; adjusted for age, sex, smoking, physical activity, alcohol Model 2; adjusted for age, sex, body mass index Model 3, adjusted for age, sex, psychological distress Fully adjusted; age, sex, smoking, physical activity, alcohol, psychological distress (GHQ-12≥4), physician diagnosed CVD, diabetes, hypertension, lipid lowering medication, body mass index

Table 3 .

 3 Adjusted regression coefficients (95% CI) for the relation of antidepressant medication use with C-reactive protein at follow up in Whitehall II. Regression coefficients represent log transformed CRP values. A category of "missing data" was added to measures of BMI and physical activity. Model 1; adjusted for age, sex, baseline CRP, smoking, physical activity, alcohol consumption. Model 2; adjusted for age, sex, baseline CRP, body mass index Model 3, adjusted for age, sex, baseline CRP, psychological distress (GHQ-12≥4) Fully adjusted; age, sex, baseline CRP, smoking, physical activity, alcohol consumption, psychological distress, CHD, diabetes, hypertension, body mass index, lipid lowering medication.

				Age, sex and baseline
			Mean± SD	CRP adjusted	Model 1	Model 2	Model 3	Fully adjusted
		N	(mg\L)	β (95% CI)*	β (95% CI)*	β (95% CI)*	β (95% CI)*	β (95% CI)*
	Antidepressant
	use at baseline
	No	4470 2.51±4.69	Reference	Reference	Reference	Reference	Reference
	Yes	114	2.77±3.71	0.15 (-0.03 to 0.32)	0.14 (-0.04 to 0.31)	0.14 (-0.03 to 0.31)	0.13 (-0.05 to 0.30)	0.11 (-0.07 to 0.28)
	Antidepressant
	use over 2 time
	points
	0	4382 2.51±4.68	Reference	Reference	Reference	Reference	Reference
	1	148	2.66±4.55	0.09 (-0.06 to 0.24)	0.08 (-0.07 to 0.23)	0.10 (-0.05 to 0.25)	0.07 (-0.09 to 0.22)	0.07 (-0.09 to 0.22)
	2	54	3.37±4.64	0.28 (0.03 to 0.54)	0.27 (0.02 to 0.52)	0.24 (-0.01 to 0.49)	0.27 (0.01 to 0.52)	0.21 (-0.04 to 0.45)

*

Sources of Funding:

The Scottish Health Survey is funded by the Scottish Executive. The funder played no role in the design and conduct of the study; collection, management, analysis, and interpretation of the data; and preparation, review, or approval of the manuscript. The Whitehall II study has been supported by grants from the British Medical


Aihua Fan
email: ai-hua.fan@u-picardie.fr

Lingmin Liao
email: lingmin.liao@univ-paris12.fr

ON MINIMAL DECOMPOSITION OF p-ADIC POLYNOMIAL DYNAMICAL SYSTEMS

Keywords: 2000 Mathematics Subject Classification. Primary 37E99; Secondary 11S85, 37A99 . p-adic dynamical system, minimal component, quadratic polynomial

A polynomial of degree ≥ 2 with coefficients in the ring of padic numbers Zp is studied as a dynamical system on Zp. It is proved that the dynamical behavior of such a system is totally described by its minimal subsystems. For an arbitrary quadratic polynomial on Z 2 , we exhibit all its minimal subsystems.

Introduction

Let Z p be the ring of p-adic integers (p being a prime number). Let f ∈ Z p [x] be a polynomial of coefficients in Z p and with degree deg f ≥ 2. It is simple fact that f : Z p → Z p is a 1-Lipschitz map. In this paper we study the topological dynamical system (Z p , f ). We refer to [START_REF] Walters | An Introduction to Ergodic Theory[END_REF] for dynamical terminology and [START_REF] Koblitz | p-adic Numbers, p-adic Analysis, and Zeta-Functions[END_REF][START_REF] Mahler | p-adic numbers and their functions[END_REF][START_REF] Schikhof | Ultrametric calculus, An introduction to p-adic analysis[END_REF][START_REF] Serre | A course in arithmetic[END_REF] for notions related to p-adic numbers.

Our first theorem is a general result which shows that a polynomial system admits at most countably many minimal subsystems. This describes to some extent the dynamical behavior of the system (see Theorem 4.2).

Theorem A. Let f ∈ Z p [x] with deg f ≥ 2. We have the following decomposition

Z p = A B C
where A is the finite set consisting of all periodic points of f , B = i B i is the union of all (at most countably many) clopen invariant sets such that each B i is a finite union of balls and each subsystem f : B i → B i is minimal, and each point in C lies in the attracting basin of a periodic orbit or of a minimal subsystem.

We will refer to the above decomposition as the minimal decomposition of the system f : Z p → Z p . A finite periodic orbit of f is by definition a minimal set. But for the convenience of the present paper, only the sets B i in the above decomposition are called minimal components.

Recently the theory of Non-Archimedean, in particular of p-adic, dynamical systems has been intensively developed ([6, 7, 8], [START_REF] Dremov | On the chaotic properties of quadratic, p-adic mathematical physics[END_REF], [START_REF] Dragovich | Linear fractional p-adic and adelic dynamical systems[END_REF], [START_REF] Fan | p-adic repellers in Qp are subshifts of finite type[END_REF], [START_REF] Gundlach | On ergodic behavior of p-adic dynamical systems[END_REF], [START_REF] Herman | Generalization of some theorem of small divisors to nonarchimedean fields[END_REF], [START_REF] Hsia | A weak Néron model with applications to p-adic dynamical systems[END_REF][START_REF] Hsia | Closure of periodic points over a non-Archimedean field[END_REF], [START_REF] Khrennikov | On the number of cycles of p-adic dynamical systems[END_REF], [START_REF] Kingsbery | Measurable dynamics of maps on profinite groups[END_REF], [START_REF] Lubin | Non-archimedean dynamical systems[END_REF], [START_REF] Rivera-Letelier | Dynamique des fonctions rationnelles sur des corps locaux, in Geometric methods in dynamics[END_REF], [START_REF] Thiran | p-adic dynamics[END_REF], [START_REF] Woodcock | p-adic chaos and random number generation[END_REF]). See also the monographes [START_REF] Anashin | Applied algebraic dynamics[END_REF], [START_REF] Khrennikov | P-adic deterministic and random dynamics[END_REF], [START_REF] Silverman | The arithmetic of dynamical systems[END_REF] and the bibliographies therein.

There were few works done on the minimal decomposition. Multiplications on Z p (p ≥ 3) were studied by Coelho and Parry [START_REF] Coelho | Ergodicity of p-adic multiplications and the distribution of Fibonacci numbers[END_REF] and general affine maps were studied by Fan, Li, Yao and Zhou [START_REF] Fan | Strict ergodicity of affine p-adic dynamical systems on Zp[END_REF]. The minimal decomposition of a polynomial system has been known in these cases and only in these cases. Quadratic polynomials will be studied at the end of the present paper.

One of interesting problems well studied in the literature is the minimality of the system f : Z p → Z p , which corresponds to the situation where A = C = ∅ and B consists of one minimal component ( [START_REF] Anashin | Uniformly distributed sequences of p-adic integers, (Russian)[END_REF][START_REF] Anashin | Uniformly distributed sequences in computer algebra or how to construct program generators of random numbers[END_REF][START_REF] Anashin | Uniformly distributed sequences of p-adic integers[END_REF][START_REF] Anashin | Ergodic transformations in the space of p-adic integers[END_REF][START_REF] Chabert | Minimal dynamical systems on a discrete valuation domain[END_REF][START_REF] Desjardins | Polynomial Mappings mod p n[END_REF][START_REF] Yu | p-adic quantum mechanics with p-adic valued functions[END_REF][START_REF] Yu | p-adic discrete dynamical systems and their applications in physics and cognitive sciences[END_REF][START_REF] Khrennikov | Ergodicity in the p-adic framework[END_REF][START_REF] Khrennikov | P-adic deterministic and random dynamics[END_REF][START_REF] Knuth | The art of computer programming[END_REF][START_REF] Larin | Transitive polynomial transformations of residue rings, (Russian. Russian summary) Diskret[END_REF]).

The above theorem shows that there are only a finite number of periodic orbits. The possible periods are shown in the following theorem (see Theorem 4.1). The statements 1)-3) were known to Pezda [START_REF] Pezda | Polynomial cycles in certain local domains[END_REF], the statements 1) and 2) are also found by Desjardins and Zieve [START_REF] Desjardins | Polynomial Mappings mod p n[END_REF] in a different way. The statement 4) is new.

Theorem B. Let f ∈ Z p [x].
1) If p ≥ 5, the periods of periodic orbits are of the form ab with a|(p -1) and 1 ≤ b ≤ p.

2) If p = 3, the periods of periodic orbits must be 1, 2, 3, 4, 6 or 9.

3) If p = 2, the periods of periodic orbits must be 1, 2 or 4. 4) Let p = 2. If there is 4-periodic orbit, then f (z mod 2) mod 2 should be a permutation on Z/2Z. There is no 4-periodic orbit for quadratic polynomials.

What kind of set can be a minimal component of a polynomial system? In a recent work, Chabert, Fan and Fares [START_REF] Chabert | Minimal dynamical systems on a discrete valuation domain[END_REF] showed that each minimal component B i must be a Legendre set and that in general, any Legendre set is a minimal component of some 1-Lipschtz system. We will show that for a polynomial system, the minimal components B i are Legendre sets of special forms. Let (p s ) s≥1 be a sequence of positive integers such that p s |p s+1 for every s ≥ 1. We denote by Z (ps) the inverse limit of Z/p s Z, which is called an odometer. The map x → x+1 is called the adding machine on Z (ps) . We will prove the following theorem (see Theorem 5.2). As we have already pointed out, the minimal decomposition is fully studied for affine maps. It seems much more difficult to study the minimal decomposition for higher order polynomials. In this paper, we try to attack the problem for quadratic polynomials. For an arbitrary 2-adic quadratic polynomial f (x) = ax 2 + bx + c on Z 2 , we find all its minimal components.

Theorem C. Let f ∈ Z p [x] with deg f ≥ 2. If E is a minimal clopen invariant set of f , then f : E → E is
As we shall see, such a quadratic system f : Z 2 → Z 2 is conjugate to one of the following quadratic polynomials

x 2 -λ, x 2 + bx, x 2 + x -d where λ ∈ Z 2 , b ≡ 1 (mod 2) and √ d ∈ Z 2 .
Our results are stated in Theorems 6.1-6.9. Let us state here some of these results.

Theorem D. Consider the polynomial x 2 -λ on Z 2 .
1) If λ ≡ 0 (mod 4), then there are two attracting fixed points, one in 4Z 2 with basin 2Z 2 , and the other one in 1 + 4Z 2 with basin 1 + 2Z 2 .

2) If λ ≡ 1 (mod 4), then the whole Z 2 is attracted into a periodic orbit of period 2 with one orbit point in 4Z 2 , and the other one in 3 + 4Z 2 .

3) If λ ≡ 2 (mod 4), then there are two attracting fixed points, one in 2 + 4Z 2 with basin 2Z 2 , and the other one in 3 + 4Z 2 with basin 1 + 2Z 2 . 4) If λ ≡ 3 (mod 4), then the whole Z 2 is attracted into a periodic orbit of period 2 with one orbit point in 1 + 4Z 2 , and the other one in 2 + 4Z 2 .

Theorem E. Consider the polynomial f (x) = x 2 + x on Z 2 [x].
There is one fixed point 0. We have f (1 + 2Z 2 ) ⊂ 2Z 2 and we can decompose 2Z 2 as

2Z 2 = {0}   n≥2 2 n-1 + 2 n Z 2   . Each 2 n-1 + 2 n Z 2 (n ≥ 2) consists of 2 n-2
pieces of minimal components:

2 n-1 + t2 n + 2 2n-2 Z 2 , t = 0, . . . , 2 n-2 -1. Theorem F. Consider the polynomial f (x) = x 2 + x -d with d ≡ 3 (mod 4). Then f (2Z 2 ) ⊂ 1 + 2Z 2 and 1 + 2Z 2 is the unique minimal component of f .
The main idea used in the paper comes from Desjardins and Zieve's work [START_REF] Desjardins | Polynomial Mappings mod p n[END_REF] and the Ph.D thesis of Zieve [START_REF] Zieve | Cycles of Polynomial mappings[END_REF]. Let E be an f -invariant compact set. It is now well known that the subsystem (E, f ) is minimal if and only if the induced map f n : E/p n Z → E/p n Z is minimal (transitive) for any n ≥ 1 (see [START_REF] Anashin | Ergodic transformations in the space of p-adic integers[END_REF][START_REF] Chabert | Minimal dynamical systems on a discrete valuation domain[END_REF]). The idea of Desjardins and Zieve is to establish relations between f n 's cycles and f n+1 's cycles, by linearizing the k-th iteration f k n+1 on a cycle of f n of length k. The paper is organized as follows. In Section 2, we give a full development of the idea in [START_REF] Desjardins | Polynomial Mappings mod p n[END_REF] by studying the induced dynamical systems f n on Z/p n Z when p ≥ 3. Section 3 is devoted to the case of p = 2 which was not treated in [START_REF] Desjardins | Polynomial Mappings mod p n[END_REF]. As we shall see, the situation in the case p = 2 is not exactly the same as in the case p ≥ 3. In Sections 4 and 5, we investigate how a minimal component is formed by analyzing the reduced maps f n (n ≥ 1) and we prove the decomposition theorem. In Section 5, we discuss the possible forms of minimal components. In Section 6, we give a detailed description of the minimal decomposition for an arbitrary quadratic polynomial system on Z 2 .

Induced dynamics on

Z/p n Z (p ≥ 3)
The main core of this section follows Desjardins and Zieve [START_REF] Desjardins | Polynomial Mappings mod p n[END_REF]. We shall give more details and rewrite some proofs for reader's convenience. The case p = 2, which is a little bit special, will be fully discussed in the next section.

Let p ≥ 3 be a prime (we may replace 3 by 2 in many places). Let n ≥ 1 be a positive integer. Denote by f n the induced mapping of f on Z/p n Z, i.e.,

f n (x mod p n ) = f (x) mod p n .
Many properties of the dynamics f are linked to those of f n . One is the following.

Theorem 2.1 ([4], [10]). Let f ∈ Z p [x] and E ⊂ Z p be a compact f -invariant set. Then f : E → E is minimal if and only if f n : E/p n Z p → E/p n Z p is minimal for each n ≥ 1. It is clear that if f n : E/p n Z p → E/p n Z p is minimal, then f m : E/p m Z p → E/p m Z p
is also minimal for each 1 ≤ m < n. So, the above theorem shows that it is important to investigate under what condition, the minimality of f n implies that of f n+1 .

Assume that

σ = (x 1 , • • • , x k ) ⊂ Z/p n Z is a cycle of f n of length k (also called k-cycle), i.e., f n (x 1 ) = x 2 , • • • , f n (x i ) = x i+1 , • • • , f n (x k ) = x 1 .
In this case we also say σ is at level n. Let

X := k i=1 X i where X i := {x i + p n t; t = 0, • • • , p -1} ⊂ Z/p n+1 Z. Then f n+1 (X i ) ⊂ X i+1 (1 ≤ i ≤ k -1) and f n+1 (X k ) ⊂ X 1 .
In the following we shall study the behavior of the finite dynamics f n+1 on the f n+1 -invariant set X and determine all cycles in X of f n+1 , which will be called lifts of σ (from level n to level n + 1). Remark that the length of any lift σ of σ is a multiple of k.

Let g := f k be the k-th iterate of f . Then, any point in σ is fixed by g n , the n-th induced map of g. For x ∈ σ, denote

a n (x) := g ′ (x) = k-1 j=0 f ′ (f j (x)) (2•1) b n (x) := g(x) -x p n = f k (x) -x p n . (2•2)
The values on the cycle σ = (x 1 , . . . , x k ) of the functions a n and b n are important for our purpose. They define, for each x, an affine map

Φ(x, t) = b n (x) + a n (x)t (x ∈ σ, t ∈ Z/pZ).
The 1-order Taylor expansion of g at x implies g(x + p n t) ≡ x + p n b n (x) + p n a n (x)t ≡ x + p n Φ(x, t) (mod p 2n ).

(2•3)

An important consequence of the last formula shows that g n+1 : X i → X i is conjugate to the linear map

Φ(x i , •) : Z/pZ → Z/pZ.
We could call it the linearization of g n+1 : X i → X i . For any x ∈ X (and even for any x ∈ Z p ), we can define the values of a n (x) and b n (x) by the formulas (2•1) and (2•2). As we shall see in the following lemma, the coefficient a n (x) (mod p) is always constant on X i and the coefficient b n (x) (mod p) is also constant on X i but under the condition a n (x) ≡ 1 (mod p).

Denote by v p (n) the p-valuation of n.

Lemma 2.2. Let n ≥ 1 and σ = (x 1 , • • • , x k ) be a k-cycle of f n . (i) For 1 ≤ i, j ≤ k, we have a n (x i ) ≡ a n (x j ) (mod p n ).
(ii) For for 1 ≤ i ≤ k and 0 ≤ t ≤ p -1, we have

a n (x i + p n t) ≡ a n (x i ) (mod p n ). (iii) For 1 ≤ i ≤ k and 0 ≤ t ≤ p -1, we have b n (x i + p n t) ≡ b n (x i ) (mod p A ),
where A := min{v p (a n (x i ) -1), n} = min{v p (a n (x j ) -1), n} for 1 ≤ i, j ≤ k.

(iv) For all 1 ≤ i, j ≤ k we have

min{v p (b n (x i )), A} = min{v p (b n (x j )), A}. Consequently, if a n (x i ) ≡ 1 (mod p n ), min{v p (b n (x i )), n} = min{v p (b n (x j )), n}.
Proof. Assertion (i) follows directly from the definition of a n (x i ) and the fact that

σ = (x i , f n (x i ), • • • , f k-1 n (x i )). The assertion (ii) is a direct consequence of a n (x i + p n t) ≡ k j=1 f ′ (f j (x i + p n t)) ≡ k j=1 f ′ (f j (x i )) (mod p n ).
The 1-order Taylor expansion of g at x i gives

g(x i + p n t) -(x i + p n t) ≡ p n g(x i ) -x i p n + p n t(g ′ (x i ) -1) (mod p 2n ). Hence b n (x i + p n t) ≡ b n (x i ) + t(a n (x i ) -1) (mod p n ).
Then (iii) follows. Write

g(f (x i )) -f (x i ) = f (f k (x i )) -f (x i ) = f (x i + p n b n (x i )) -f (x i ).
The 1-order Taylor expansion f at x i leads to

g(f (x i )) -f (x i ) ≡ p n b n (x i )f ′ (x i ) (mod p 2n ). Hence we have b n (f (x i )) ≡ b n (x i )f ′ (x i ) (mod p n ).
Since when A = 0, the result is obvious, we may suppose that A = 0. Then a n (x i ) ≡ 1 (mod p) (for 1 ≤ i ≤ k) which implies f ′ (x i ) ≡ 0 (mod p) for all 1 ≤ i ≤ k. Notice that f (x i ) ≡ x i+1 (mod p n ). Then by (iii), we obtain (iv).

According to Lemma 2.2 (i) and (ii), the value of a n (x) (mod p n ) does not depend on x ∈ X. According to Lemma 2.2 (iii) and (iv), whether b n (x) ≡ 0 (mod p) does not depend on x ∈ X if a n (x) ≡ 1 (mod p). For simplicity, sometimes we shall write a n and b n without mentioning x.

The above analysis allows us to distinguish the following four behaviors of f n+1 on X:

(a) If a n ≡ 1 (mod p) and b n ≡ 0 (mod p), then Φ preserves a single cycle of length p, so that f n+1 restricted to X preserves a single cycle of length pk. In this case we say σ grows.

(b) If a n ≡ 1 (mod p) and b n ≡ 0 (mod p), then Φ is the identity, so f n+1 restricted to X preserves p cycles of length k. In this case we say σ splits.

(c) If a n ≡ 0 (mod p), then Φ is constant, so f n+1 restricted to X preserves one cycle of length k and the remaining points of X are mapped into this cycle. In this case we say σ grows tails.

(d) If a n ≡ 0, 1 (mod p), then Φ is a permutation and the ℓ-th iterate of Φ reads

Φ ℓ (x, t) = b n (a ℓ n -1)/(a n -1) + a ℓ n t so that Φ ℓ (t) -t = (a ℓ n -1) t + b n a n -1 .
Thus, Φ admits a single fixed point t = -b n /(a n -1), and the remaining points lie on cycles of length d, where d is the order of a n in (Z/pZ) * . So, f n+1 restricted to X preserves one cycle of length k and p-1 d cycles of length kd. In this case we say σ partially splits. Now let us study the relation between (a n , b n ) and (a n+1 , b n+1 ). Our aim is to see the change of nature from a cycle to its lifts. Lemma 2.3. Let σ = (x 1 , . . . , x k ) be a k-cycle of f n and let σ be a lift of σ of length kr, where r ≥ 1 is an integer. We have

a n+1 (x i + p n t) ≡ a r n (x i ) (mod p n ), ( 1
≤ i ≤ k, 0 ≤ t ≤ p -1) (2•4) pb n+1 (x i + p n t) ≡t(a n (x i ) r -1) + b n (x i ) 1 + a n (x i ) + • • • + a n (x i ) r-1 (mod p n ). (2•5)
Proof. The formula (2•4) follows from

a n+1 ≡ (g r ) ′ (x i + p n t) ≡ (g r ) ′ (x i ) ≡ r-1 j=0 g ′ (g j (x i )) ≡ a r n (mod p n ).
By repeating r times of the linearization (2•3), we obtain

g r (x i + p n t) ≡ x i + Φ r (x i , t)p n (mod p 2n ),
where Φ r means the r-th composition of Φ as function of t, and

Φ r (x i , t) = ta n (x i ) r + b n (x i ) 1 + a n (x i ) + • • • + a n (x i ) r-1 .
Thus (2•5) follows from the definition of b n+1 and the above two expressions.

By Lemma 2.3, we obtain immediately the following proposition.

Proposition 2.4. Let n ≥ 1. Let σ be a k-cycle of f n and σ be a lift of σ. Then we have 1) if a n ≡ 1 (mod p), then a n+1 ≡ 1 (mod p); 2) if a n ≡ 0 (mod p), then a n+1 ≡ 0 (mod p); 3) if a n ≡ 0, 1 (mod p) and σ is of length k, then a n+1 ≡ 0, 1 (mod p); 4) if a n ≡ 0, 1 (mod p) and σ is of length kd where d ≥ 2 is the order of a n in (Z/pZ) * , then a n+1 ≡ 1 (mod p). This result is interpreted as follows in dynamical system language: 1) If σ grows or splits, then any lift σ grows or splits.

2) If σ grows tails, then the single lift σ also grows tails.

3) If σ partially splits, then the lift σ of the same length as σ partially splits, and the other lifts of length kd grow or split.

If σ = (x 1 , • • • , x k ) is a cycle of f n which grows tails, then f admits a k-periodic point x 0 in the clopen set X = k i=1 (x i + p n Z p ) and X is contained in the attracting basin of the periodic orbit x 0 , f (x 0 ), • • • , f k-1 (x 0 ).
With the preceding preparations, we are ready to prove the following Propositions 2.5-2.7 which predict the behavior of the lifts of a cycle σ by the properties of σ. We refer the reader to [START_REF] Desjardins | Polynomial Mappings mod p n[END_REF] for their proofs. Otherwise we can follow the similar proofs of Propositions 3.1-3.3 in the case p = 2. Proposition 2.5 ([12]). Let σ be a growing cycle of f n and σ be the unique lift of σ.

1) If p ≥ 3 and n ≥ 2 then σ grows.

2) If p > 3 and n ≥ 1 then σ grows.

3) If p = 3 and n = 1, then σ grows if and only if b 1 (x) ≡ g ′′ (x)/2 (mod p).

According to 1) and 2) of Proposition 2.5, in the cases p ≥ 3, n ≥ 2 and p >

3, n ≥ 1, if σ = (x 1 , • • • , x k )
grows then its lift also grows, and the lift of the lift will grow and so on. So, the clopen set

X = k i=1 (x i + p n Z p ) is a minimal set by Theorem 2.1. Let A n (x) := v p (a n (x) -1), B n (x) := v p (b n (x)). By Lemma 2.2, for a cycle σ = (x 1 , . . . , x k ), min{A n (x i ), n} does not depend on the choice of x i , 1 ≤ i ≤ k and if B n (x i ) < min{A n (x i ), n} then B n (x i ) does not depend on the choice of x i , 1 ≤ i ≤ k.
Sometimes, there is no difference when we choose x i or x j in the cycle. So, without misunderstanding, we will not mention x i in A n and B n (see the proof of Proposition 3.2 for the details corresponding to the case p = 2).

We say that a cycle σ at level n splits ℓ times if σ splits, and the lifts of σ at level n + 1 split and inductively all lifts at level n + j (2 ≤ j < ℓ) split. Similarly, one can imagine what we mean if we say a cycle grows ℓ times. That a cycle grows forever means that it grows infinite times.

Proposition 2.6 ([12]

). Let p ≥ 3 and n ≥ 1. Let σ be a splitting cycle of f n .

1) If min{A n , n} > B n , every lift splits B n -1 times then all lifts at level n + B n grow forever.

2) If A n ≤ B n and A n < n, there is one lift which behaves the same as σ (i.e., this lift splits and A n+1 ≤ B n+1 and A n+1 < n + 1) and other lifts split A n -1 times then all lifts at level n + A n grow forever..

3) If B n ≥ n and A n ≥ n, then all lifts split at least n -1 times.

Proposition 2.7 ([12]

). Let p ≥ 3 and n ≥ 1. Let σ be a partially splitting k-cycle of f n and σ be a lift of σ of length kd, where d is the order of a n in Z/pZ. 1) If A n+1 < nd, then σ splits A n+1 -1 times then all lifts at level n + A n+1 grow forever.

2) If A n+1 ≥ nd, then σ splits at least nd -1 times.

We remark that in the partially splitting case, min{A n+1 (x), nd} depends only on the lifting cycle of f n+1 of length kd but not on x (see [START_REF] Desjardins | Polynomial Mappings mod p n[END_REF], Corollary 3).

Induced dynamics on

Z/p n Z p (p = 2)
In this section we focus on the special case p = 2 which is not considered in [START_REF] Desjardins | Polynomial Mappings mod p n[END_REF]. The first part in the preceding section (where p ≥ 3 is not explicitly assumed) remains true for p = 2. Notice that when p = 2, there is no partially splitting cycle.

We only need to study how a cycle grow or split. We distinguish four cases. Let σ be a cycle of f n . We say σ strongly grows if a n ≡ 1 (mod 4) and b n ≡ 1 (mod 2), and σ weakly grows if a n ≡ 3 (mod 4) and b n ≡ 1 (mod 2). We say σ strongly splits if a n ≡ 1 (mod 4) and b n ≡ 0 (mod 2), and σ weakly splits if a n ≡ 3 (mod 4) and b n ≡ 0 (mod 2).

The following results hold true when p = 2. Their proofs are postponed and got together at the end of this section. Proposition 3.1. Let σ be a cycle of f n (n ≥ 2). If σ strongly grows then the lift of σ strongly grows. If σ weakly grows then the lift of σ strongly splits.

The first assertion of Proposition 3.

1 implies that if σ = (x 1 , • • • , x k ) is a strongly growing cycle of f n (n ≥ 2), then (x i + p n Z p ) is a minimal set. Recall that A n (x) = v 2 (a n (x) -1), B n (x) = v 2 (b n (x)).
In the following proposition, the x in A n (x), B n (x) can be chosen any

x i of the cycle σ = (x 1 , • • • , x k ) (see its proof).
Proposition 3.2. Let σ be a strongly splitting cycle of f n (n ≥ 2). 1) If min{A n , n} > B n , then all lifts strongly split B n -1 times, then all the lifts at level n + B n strongly grow.

2) If A n ≤ B n and A n < n, then one lift behaves the same as σ (i.e., this lift strongly splits and A n+1 ≤ B n+1 and A n+1 < n + 1). The other one splits A n -1 times, then all the lifts lifts at level n + A n strongly grow forever.

3) If B n ≥ n and A n ≥ n, then all lifts strongly split at least n -1 times.

Proposition 3.3. Let σ be a weakly splitting cycle of f n (n ≥ 2). Then one lift behaves the same as σ and the other one weakly grows and then strongly splits.

To prove these propositions, we need the following lemmas.

Lemma 3.4. Let σ be a growing cycle of f n (n ≥ 2). Then

a n+1 (x i ) ≡ 1 (mod 4), (3•1) 2b n+1 (x i + p n t) ≡ b n (x i )(1 + a n (x i )) (mod 4). (3•2)
Proof. Taking p = 2 and r = 2 in (2•4), we get

a n+1 (x i ) ≡ a 2 n (x i ) (mod 2 n ). Since n ≥ 2 and a n ≡ 1 (mod 2), we obtain (3•1).
Taking p = 2 and r = 2 in (2•5), we get

2b n+1 (x i + 2 n t) ≡ t(a n (x i ) 2 -1) + b n (x i )(1 + a n (x i )) (mod 2 n ).
Since n ≥ 2 and a n (x i ) ≡ 1 (mod 2), we obtain (3•2).

Lemma 3.5. Let σ be a splitting cycle of

f n . If A n < n, then A n+1 = A n and if A n ≥ n, then A n+1 ≥ n. Consequently, min{A n+1 , n} = min{A n , n}. (3•3)
Proof. We need only to notice that we have a n+1 ≡ a n (mod 2 n ) when σ splits.

Lemma 3.6. Let σ = (x 1 , . . . , x k ) be a splitting cycle of f n . Then for 1 ≤ i ≤ k and for t = 0 or 1, we have

2b n+1 (x i + 2 n t) ≡ b n (x i ) + t(a n (x i ) -1) (mod 2 n ). (3•4)
Consequently, we have

B n+1 (x i + 2 n t) = B n (x i ) -1 if B n (x i ) < min{A n (x i ), n}, (3•5)
Proof. Since σ splits, taking p = 2 and r = 1 in (2•5), we obtain the result.

The following lemma concerns an elementary property of polynomials on Z 2 .

Lemma 3.7. Let h ∈ Z 2 [x]. If a ≡ b (mod 2), then h ′ (a) ≡ h ′ (b) (mod 4). Furthermore, if h ′ (a) ≡ 1 (mod 2), then h ′ (a)h ′ (b) ≡ 1 (mod 4) .
Proof. It suffices to notice that the coefficient of x 2k+1 in h ′ (x) is equal to 0 (mod 2).

Lemma 3.8. Let σ be a growing k-cycle of f n (n ≥ 1). Then its lift strongly grows or strongly splits.

Proof. Let x 1 be a point in σ. What we have to show is a n+1 (x 1 ) ≡ 1 (mod 4). Since σ is a growing k-cycle, we have

f k (x 1 ) ≡ x 1 (mod 2 n ), a n (x 1 ) = (f k ) ′ (x 1 ) ≡ 1 (mod 2).
So, by Lemma 3.7, we have

a n+1 (x 1 ) = (f 2k ) ′ (x 1 ) = (f k ) ′ (x 1 )(f k ) ′ (f k (x 1 )) ≡ 1 (mod 4).
A direct consequence is the following result.

Corollary 3.9. If a cycle grows twice (maybe between the two growths, there are several splittings), then all the lifts will grow forever.

Proof. Let σ be the lift of the growing cycle σ. Assume that after several times of splitting, one of lifts of σ grows (then all the lifts at the same level grow). By Lemma 3.8, this growing lift at a level n ≥ 2 must strongly grow. Thus by Proposition 3.1, the lifts will grow forever.

We are now going to prove Propositions 3.1-3.3.

Proof of Proposition 3.1. If σ grows, then by (3•1), the lift of σ strongly grows or strongly splits. If σ strongly grows, then by (3•2), we have

2b n+1 (x i + p n t) ≡ 2b n (x i ) (mod 4). Thus b n+1 (x i + p n t) ≡ b n (x i ) ≡ 0 (mod 2)
. Hence the lift of σ strongly grows.

If σ weakly grows, then by (3•2), we have

2b n+1 (x i + p n t) ≡ 0 (mod 4). Thus b n+1 (x i + p n t) ≡ 0 (mod 2).
Hence the lift of σ strongly splits.

Proof of Proposition 3.2. First notice that if σ strongly splits then a n ≡ 1 (mod 4). Since n ≥ 2, by Lemma 2.2 we have a ℓ ≡ 1 (mod 4) for all ℓ > n. So, all the lifts strongly grow or strongly split. Proposition 3.2 contains three cases which are defined by some conditions on A n and B n . If such a condition is satisfied, we say σ or (A n , B n ) belongs to the corresponding case.

Case 1: min{A n , n} > B n . Recall that by Lemma 2.2, both min{A n (x i ), n} and min{A n (x i ), B n (x i ), n} are independent of x i . Thus in this case, we can simply write A n and B n . By (3•5), we have

B n+1 = B n -1. Thus by (3•3) min{A n+1 , n + 1} ≥ min{A n+1 , n} = min{A n , n} > B n > B n+1 .
Hence the lifts of σ still belong to Case 1. By induction, we know that after ℓ := B n times, B n+ℓ = 0 (i.e. b n+ℓ = 0 mod p). Since σ strongly splits, we have a n+ℓ ≡ 1 (mod 4). Thus the lifts at level n + ℓ strongly grow. That is to say all lifts of σ split B n -1 times, then all the lifts lifts at level n + B n strongly grow forever.

Case 2: A n ≤ B n and A n < n. Since A n (x i ) < n for some i, implies for all 1 ≤ i ≤ k, A n (x i ) = A n (x j ). We can also deduce that if A n (x i ) ≤ B n (x i ) for some i then for all i A n (x i ) ≤ B n (x i ). Otherwise, if A n (x j ) > B n (x j ) for some j, then by Lemma 2.2, B n (x i ) = B n (x j ) < A n (x j ) = A n (x i
) which leads to a contradiction. So in this case, we can choose any 1 ≤ i ≤ k and we simply write A n and B n . By Lemma 3.5, we have A n+1 = A n . Since B n ≥ A n , there exists one t such that b n + t(a n -1) ≡ 0 (mod 2 An+1 ), and the other one which we can write as 1t such that b n + t(a n -1) ≡ 0 (mod 2 An+1 ).

Hence by (3•4), for one lift of σ B n+1 ≥ A n and for the other one B n+1 = A n -1. Thus for one lift, A n+1 = A n ≤ B n+1 , and A n+1 = A n < n + 1. Therefore, this lift belongs to Case 2. For the other one, B n+1 = A n -1 = A n+1 -1 < A n+1 , and B n+1 = A n -1 < n + 1. Thus this lift belongs to Case 1. By induction, we know that one lift of σ behaves the same as σ (i.e., strongly splits and satisfies the condition of Case 2 at level n + 1) and the other one splits A n -1 times, then the lifts strongly grow.

Case 3: B n ≥ n and A n ≥ n. First we notice that by Lemma 2.2, if for some 1 ≤ i ≤ k, we have B n (x i ) ≥ n and A n (x i ) ≥ n, then for all 1 ≤ i ≤ k, the same property established. The following statement will be the same if we choose another i. So we still simply write A n and B n . By the definition of b n , if the cycle splits, the order of b n deceases at most one when the level goes up one step. Since B n ≥ n, we have B n+1 ≥ n -1, and if n ≥ 2, the lifts of σ still strongly split. Thus by induction, the lifts of σ split at least n -2 times. But after that we can not give any more information.

Proof of Proposition 3.3. Since σ weakly splits, a n+1 ≡ a n ≡ 3 (mod 4). Thus A n+1 = A n = 1 < n and B n ≥ 1 = A n . Thus (A n , B n ) belongs to Case 2 in Proposition 3.2. By the proof of Proposition 3.2, we know that for one lift of σ, B n+1 ≥ A n and then A n+1 = A n ≤ B n+1 . Thus this lift behaves the same as σ.

For the other lift, B n+1 = A n -1 = 0. Hence this second lift weakly grows, and then its lift strongly splits by Proposition 3.1. Therefore, we complete the proof.

Minimal decomposition

If a cycle always grows (grows forever) then it will produce a minimal component of f . If a cycle always splits (splits infinite times) then it will produce a periodic orbit of f . If a cycle grows tails, it will produce an attracting periodic orbit with an attracting basin. We shall describe this more precisely.

Let σ = (x 1 , . . . , x k ) be a cycle of f n . Recall that in this case σ is called a k-cycle at level n. Let

X := k i=1 (x i + p n Z p ).
There are four special situations for the dynamical system f : X → X.

(S1) Suppose σ grows tails. Then f admits a k-periodic orbit with one periodic point in each ball x i + p n Z p (1 ≤ i ≤ k), and all other points in X are attracted into this orbit. In this situation, if x is a point in the k-periodic orbit, then

|(f k ) ′ (x)| p < 1 since (f k ) ′ (x) = a m (x) ≡ 0 (mod p m ) for all m ≥ n. The periodic orbit (x, f (x), • • • , f k-1 (x)) is then attractive.
(S2) Suppose σ grows and its lifts always grow. Then f is transitive (minimal) on each X/p m Z p m ≥ n. Thus, by Theorem 2.1, f is minimal on X. In this case, we say that σ is a starting growing cycle at level n.

(S3) Suppose σ splits and there is a splitting lift at each level larger than n. Then there is a k-periodic orbit with one periodic point in each

x i + p n Z p (1 ≤ i ≤ k).
We say that σ is a starting splitting cycle at level n. In this situation, if x is a point in the k-periodic orbit, then (f k ) ′ (x) = 1 since (f k ) ′ (x) = a m (x) ≡ 1 (mod p m ) for all m ≥ n. Thus the periodic orbit (x, f (x),

• • • , f k-1 (x)) is indifferent.
(S4) Suppose σ = (x 1 , . . . , x k ) partially splits (p ≥ 3). Then by Proposition 2.7, there is one lift of length k which still partially splits like σ. Thus there is a k-periodic orbit with one periodic point in each

x i + p n Z p (1 ≤ i ≤ k). In this situation, if x is a point in the k-periodic orbit formed above, then |(f k ) ′ (x)| p = 1 since (f k ) ′ (x) = a m (x) ≡ 0, 1 (mod p m ) for all m ≥ n. Hence, the periodic orbit (x, f (x), • • • , f k-1 (x)) is indifferent.

Now we can deduce all possible periods of the polynomial systems on

Z p . Theorem 4.1. Let f ∈ Z p with deg f ≥ 2.
1) If p ≥ 5, the lengths of periodic orbits are of the form ab with a|(p -1) and 1 ≤ b ≤ p;

2) If p = 3, the lengths of periodic orbits must be 1, 2, 3, 4, 6 or 9;

3) If p = 2, the lengths of periodic orbits must be 1, 2 or 4. 4) Let p = 2. If there is 4-periodic orbit, then f 1 should be a permutation on Z/2Z. There is no 4-periodic orbit for quadratic polynomials. Remark 1. The statements 1)-3) were due to Pezda ([32]). By using the idea we explained in the preceding sections which is quite different from that of Pezda, Desjardins and Zieve ([12]) also gave 1) and 2). The statement 4) is new.

Proof. We only show 3) and 4), because the proofs of 1) and 2) are similar and can be found in [START_REF] Desjardins | Polynomial Mappings mod p n[END_REF] and [START_REF] Pezda | Polynomial cycles in certain local domains[END_REF].

Notice that any periodic orbit comes from an infinite sequence of splitting of some cycle, and that the length of the periodic orbit is the length of the starting splitting cycle. So, what we want to study are all possible lengths of starting splitting cycles.

The possible lengths of cycles at the first level (i.e. the cycles of f 1 on Z/2Z) are 1 and 2. Notice that the growth of length must be multiplied 2, according to our discussion in the preceding sections. So, the possible lengths of cycles are 2 k (k ≥ 0). However, by Corollary 3.9, if a cycle grows twice it will grow forever. There, any cycle of length 2 k (k ≥ 3), which must have grown twice, can not be a starting splitting cycle. Hence the lengths of starting splitting cycles can only be 1, 2, 4. This completes the proof of 3).

If there is a periodic orbit of length 4, there must be a starting splitting cycle of length 4. This is possible only in the following case: at the first level, f 1 admit a 2-cycle. Otherwise it needs to grow twice and then its lifts will grow forever. This will produce a clopen minimal set not a periodic orbit. This is the first part of 4). For the second part of 4), one can see from our study on the quadratic polynomials in Section 6 (Theorems 6.1-6.9).

Theorem 4.2. Let f ∈ Z p [x] with deg f ≥ 2.
We have the following decomposition

Z p = A B C
where A is the finite set consisting of all periodic points of f , B = i B i is the union of all (at most countably many) clopen invariant sets such that each B i is a finite union of balls and each subsystem f : B i → B i is minimal, and each point in C lies in the attracting basin of a periodic orbit or of a minimal subsystem.

Proof. We first explain that there are only finitely many periodic points. In fact, by Theorem 4.1, there are only finitely many possible lengths of periods. Periodic points are solutions of the equations f qi (x) = x with {q i } being one of possible lengths of periods. Since deg f ≥ 2, each equation admits a finite number of solutions. So, there is only a finite number of periodic points.

We start from the second level. Decompose Z p into p 2 balls with radius p -2 . Each ball is identified with a point in Z/p 2 Z. The induced map f 2 admits some cycles. The points outside any cycle are mapped into the cycles. The ball corresponding to such a point will be put into the third part C. From now on, we really start our analysis with cycles at level n ≥ 2. Let σ = (x 1 , . . . , x k ) be a cycle at level n ≥ 2. Let

X = k i=1 (x i + p n Z p ).
Suppose p ≥ 3. We distinguish four cases. (P1) σ grows tails. Then by (S1), the clopen set X consists of a k-periodic orbit and other points are attracted by this periodic orbit. So, X contributes to the first part A and the third part C.

(P2) σ grows. Then by Proposition 2.5, σ is in the situation (S2). Therefore X is a minimal component. So, X ⊂ B.

(P3) σ splits. Then we shall apply Proposition 2.6.

• If σ belongs to Case 1 described by Proposition 2.6, then after finitely many times of splitting, the lifts will grow forever and so they are in the situation of (S2). Therefore we get a finite number of minimal components, all belonging to B. • If σ belongs to Case 2, then there is one lift of σ sharing the property (S3), and other lifts different from the cycle containing the periodic orbit (at any level m ≥ n + 1) find themselves in the situation (S2) after finitely many times of lifting. Therefore, we get a periodic orbit and countable infinite minimal components. • If σ belongs to Case 3, then σ splits into p n cycles at level 2n. These cycles at level 2n may continue this procedure of analysis of (P3). But this procedure can not continue infinitely, because there is only a finite number of periodic points. So, all these cycle may continue to split but they must end with their lifts belonging either to Case 1 or Case 2 in Proposition 2.6. So, X contributes to both A and B. (P4) σ partially splits. Then σ is in the situation (S4). Thus there comes out a periodic orbit. Suppose σ m is the lift of σ containing the periodic orbit at level m ≥ n + 1. If σ m belongs to Case 1 in Proposition 2.7, then the other lifts different from σ m+1 , will be in the situation (S1) after finite times. If σ m belongs to Case 2 in Proposition 2.7, then each of other lifts different from σ m+1 , split to be p nd-1 cycles at level nd. We then go to (P3) for these cycles at level nd.

Suppose p = 2. We distinguish five cases. (Q1) σ grows tails. Then σ is in the situation (S1). We have the same conclusion as (P1) above.

(Q2) σ strongly grows. Then by Proposition 3.1, σ is in the situation (S2). We have the same conclusion as (P2) above.

(Q3) σ strongly splits. By Proposition 3.2, the arguments are the same as (P3): The procedures will be ended if the condition 1) or 2) in Proposition 3.2 is satisfied. If the condition 3) in Proposition 3.2 is satisfied, we repeat the analysis of (Q3) for the lifts of σ. But the procedures will be eventually ended with the condition 1) or 2), because there is only a finite number of periodic points.

(Q4) σ weakly grows. Then by Proposition 3.1, the lift of σ strongly splits. We are then in the case (Q3).

(Q5) σ weakly splits. By Proposition 3.3, then one lift is in the situation (S3) which produces a periodic orbit, and the other lifts different from the cycle containing the periodic orbit, at any level m ≥ n + 1, will weakly grow. Then we are in the case (Q4).

All the above procedures will stop. So, we get the decomposition in finite steps.

We have excluded the affine polynomials from the theorem. Exactly speaking, the conclusion is false for affine polynomials. For example, every points Z p are fixed by f (x) := x. Anyway, affine polynomials have been fully studied in [START_REF] Fan | Strict ergodicity of affine p-adic dynamical systems on Zp[END_REF].

Corollary 4.3. Let f ∈ Z p [x] with deg f ≥ 2.
If f admits an indifferent fixed point or a periodic orbit, then there exists a sequence of minimal components with their diameters and their distances from the fixed point or the periodic orbit tending to zero.

Proof. Suppose (x 1 , . . . , x k ) is an indifferent periodic orbit. Let x

(n) j ∈ Z/p n Z and x (n) j ≡ x j (mod p n ) for 1 ≤ j ≤ k. Then σ n = (x (n) 1 , . . . , x (n)
k ) is a splitting or partially splitting cycle at level n. By the procedures of the decomposition, the cycle σ n should be in the situation (S3) or (S4). That is to say σ n splits for all n or σ n partially splits for all n.

Since there are only finite number of periodic orbits, for any ǫ > 0 small enough, there is no other periodic orbits in the ǫ neighborhood of the orbit (x 1 . . . , x k ). Take n such that p -n < ǫ. Then the lifts of σ n which are different to σ n+1 will never split infinitely. Hence they will grow after finite times. Then all the lifts of σ n which are different to σ n+1 , considered as union of balls, consist of finite number of minimal components. Since these balls are contained in x j + p n Z p for each j respectively. Thus there is a minimal component such that the diameter and the distance to the orbit (x 1 , . . . , x k ) are all less than p -n . The result is obtained if we consider infinitely n and find one minimal component for each n.

Conjugacy classes of Minimal subsystems

Recently, Chabert, Fan and Fares [START_REF] Chabert | Minimal dynamical systems on a discrete valuation domain[END_REF] proved that minimal sets of a 1-Lipschitz map are Legendre sets. We shall prove that minimal sets of a polynomial are some special Legendre sets. A set E ⊂ Z p is a Legendre set if for any s ≥ 1 and any x ∈ E/p s Z p , the number

q s := Card y ∈ E/p s+1 Z p : y ≡ x mod p s is independent of x ∈ E/p s Z p . Let p s := q 1 q 2 • • • q s (∀s ≥ 1).
It is clear that p s = Card E/p s Z p . We call (p s ) s≥1 the structure sequence of E.

Consider the inverse limit

Z (ps) := lim ← Z/p s Z.
This is a profinite group, usually called an odometer, and the map τ : x → x + 1 is called the adding machine on Z (ps) .

Theorem 5.1 ([10]). Let E be a clopen set in Z p and f : E → E be a 1-Lipschitz map. If the dynamical system (E, f ) is minimal, then f is an isometry, E is a Legendre set and the system (E, f ) is conjugate to the adding machine (Z (ps) , τ ) where (p s ) is the structure sequence of E. On the other hand, on any Legendre set there exists at least one minimal map.

We improve the above result in the case of polynomials by giving more information on the structure sequence.

Theorem 5.2. Let f ∈ Z p [x] with deg f ≥ 2. If E is a minimal clopen invariant set of f , then f : E → E is conjugate
to the adding machine on an odometer Z (ps) , where

(p s ) = (k, kd, kdp, kdp 2 , • • • )
with some 1 ≤ k ∈ N, k ≤ p and d|(p -1).

Proof. By our previous discussion on the cycles of f n on Z/p n Z, a clopen minimal set E is formed when a cycle grows forever. If n is the starting level for the cycle to grow, then E is a union of some balls with radius p -n . Therefore, for s ≥ n, every nonempty intersection of E with a ball of radius p -s contains p balls of radius p -(s+1) . That is to say q s = p. From the cycle at the first level to the starting growing cycle at level n, the of cycle length is multiplied by 1, p or some d satisfying d|(p -1). That is to say for 1 ≤ s < n, every nonempty intersection of E with a ball of radius p -s contains the same number (1, p or d) of balls of radius p -(s+1) . Thus E is a Legendre set. To determine p s for 1 ≤ s < n, we distinguish three cases: p ≥ 5, p = 3, p = 2. Case p ≥ 5. In this case, when a cycle grows, its lift grows forever. A cycle at level 1 may start with growing, several times of splitting or several times of partially splitting and then the lifts grow forever. Therefore, there are three ways to form a minimal set. We show the three ways by the growth of cycle length as follows (k being the length of the cycle σ at the level 1).

Case 1. σ grows:

(k, kp, kp 2 , . . . ), Case 2. σ splits:

(k, k, . . . , k, kp, kp 2 , . . . ), Case 3. σ partially splits:

(k, kd, . . . , kd, kdp, kdp

The above three cases correspond to three kinds of adding machines. However, by the result of Buescu and Stewart [START_REF] Buescu | Liapunov stability and adding machines[END_REF], the adding machines in both Case 1 and Case 2 are conjugate to (Z (ps) , τ ) where p s = (k, kp, kp 2 , . . . ). In Case 3, the adding machines are all conjugate to (Z (ps) , τ ) where p s = (k, kd, kdp, kdp 2 , . . . ) and d|(p -1), d ≥ 2.

Case p = 3. We distinguish four cases. Case 1. σ grows and its lift also grows:

(k, kp, kp 2 , . . . ), Case 2. σ grows but its lift splits:

(k, kp, . . . , kp, kp 2 , . . . ), Case 3. σ splits:

(k, k, . . . , k, kp, kp 2 , . . . ), Case 4. σ partial splits:

(k, kd, . . . , kd, kdp, kdp 2 , . . . ), d|(p -1), d ≥ 2.

Then (E, f ) is conjugate to (Z (ps) , τ ) where p s = (k, kd, kdp, kdp 2 , . . . ) with 1 ≤ k ≤ p, and d|(p -1). Case p = 2. We distinguish twelve cases.

(1, 1, 1, . . .

1 strongly split , 2, 2 2 , 2 3 , . . . ), (1, 1 strongly grows
, 2, 2 2 , 2 3 , . . . ),

, 2 3 , . . . ), (2, 2 2 , . . . , 2 2 strongly split , 2 3 , . . . ), (2, 2 2 strongly grows , 2 3 , . . . ).
In any of these cases, the system (E, f ) is conjugate to (Z 2 , x + 1).

2-adic Quadratic Polynomials

In this section, we undertake a full investigation on the minimal decomposition of 2-adic quadratic polynomial systems on Z 2 of the form:

f (x) := ax 2 + bx + c (a, b, c ∈ Z 2 , a = 0).
As we shall see, the system f (x) = ax 2 +bx+c is conjugate to one of the following quadratic polynomials

x 2 -λ, x 2 + bx, x 2 + x -d where λ ∈ Z 2 , b ≡ 1 (mod 2) and √ d ∈ Z 2 .
Let us state our results on the minimal decomposition of (Z p , f ). The proofs are postponed at the end of this section. By the way, we shall discuss the behavior of f on the field Q p .

If a ≡ 1 (mod 2), then lim n→∞ |f n (x)| = ∞ for any x ∈ Z 2 . An elementary calculation shows that ax 2 + bx + c on Z 2 is conjugate to x 2 + bx + c on Z 2 through the conjugacy x → ax. If a ≡ 0 (mod 2), then lim n→∞ |f n (x)| = ∞ for any x ∈ 1 a Z 2 and ax 2 + bx + c on 1 a Z 2 is conjugate to x 2 + ax + ac on Z 2 through the conjugacy x → ax. Thus without loss of generality, we need only to consider the quadratic polynomials of the form

x 2 + bx + c (b, c ∈ Z 2 ).
We distinguish two cases according to b ≡ 0 (mod 2) or b ≡ 1 (mod 2)

If b ≡ 0 (mod 2), x 2 + bx + c is conjugate to

x 2 -λ with λ = b 2 -4c-2b 4
, through the conjugacy x → x + B 2 . Theorem 6.1. Consider the polynomial f (x) = x 2λ on Z 2 .

1) If λ ≡ 0 (mod 4), then f admits two attracting fixed points, one in 4Z 2 with 2Z 2 as its attraction basin, and the other one in 1+4Z 2 with 1+2Z 2 as its attraction basin.

2) If λ ≡ 1 (mod 4), then the whole Z 2 is attracted into a periodic orbit of period 2 with one orbit point in 4Z 2 and the other one in 3 + 4Z 2 .

3) If λ ≡ 2 (mod 4), then f admits two attracting fixed points, one in 2+4Z 2 2Z 2 as its attraction basin, and the other one in 3 + 4Z 2 with 1 + 2Z 2 as its attraction basin.

4) If λ ≡ 3 (mod 4), then the whole Z 2 is attracted into a periodic orbit of period 2 with one orbit point in 1 + 4Z 2 and the other one in 2 + 4Z 2 .

If b ≡ 1 (mod 2), then x 2 + bx + c is conjugate to

x 2 + x -d where d = (b-1) 2 -4c 4 ∈ Z 2 , through x → x + b-1 2 . It is clear that x 2 + x -d admits fixed points if and if only √ d ∈ Z 2 .
Thus we need to study the case

x 2 + x -d with √ d ∈ Z 2 and the case x 2 + x -d with d ∈ Z 2 but √ d ∈ Z 2 . If √ d ∈ Z 2 (i.e.
2Z 2 = {0}   n≥2 2 n-1 + 2 n Z 2   . Each 2 n-1 + 2 n Z 2 (n ≥ 2) consists of 2 n-2
pieces of minimal components:

2 n-1 + t2 n + 2 2n-2 Z 2 , t = 0, . . . , 2 n-2 -1. Denote N * = N \ {0}. If b ≡ 1 (mod 2) but b = 1, we distinguish four subcases: • b = 1 -4m, m ∈ Z 2 \ {0}; • b = -1 -4m, m ∈ Z 2 with v 2 (m) ∈ 1 + 2N; • b = -1 -4m, m ∈ Z 2 with v 2 (m) ∈ 2N * ; • b = -1 -4m, m ∈ Z 2 with v 2 (m) = 0. If f (x) = x 2 + (-1 -4m)x with v 2 (m) = 0, then f is conjugate to g(x) = x 2 + (-1 -4(-m -1))x with v 2 (-m -1) = v 2 (m + 1) ≥ 1 through x → x -4m -2.
Thus the last case is reduces to the second and the third case. So we need only consider the first three cases.

Before the statement of the following results, we would like to give some terminology to simplify our statements.

We say a 1-cycle (x) at level n is of type I-[k] if it splits k times then its lifts grow forever. In this case, the ball x + p n Z p is decomposed into p k pieces of minimal components. Such a component is a ball of radius p -n-k . Sometimes the ball x + p n Z p is said to be of type I- [k].

We say the a 2-cycle (x, y) at level n is of type II-[k] if it splits k times then its lifts grow forever. In this case, the union of two balls (x + p n Z p ) ∪ (y + p n Z p ) is decomposed into p k pieces of minimal components. Such a component is a union of two balls of radius p -n-k . The union (x + p n Z p ) ∪ (y + p n Z p ) is sometimes said to be of type II- [k]. Remark that the union (x + p n Z p ) ∪ (y + p n Z p ) may be a ball of radius p -n+1 .

If an invariant subset E ⊂ Z p is a union of invariant subsets F n ⊂ Z p , n ∈ J ⊂ N where each F n is of type I-[k], we will denote it as

E = n∈J F n -{I-[k]}.
Similarly, if each F n ⊂ Z p , n ∈ J ⊂ N where each F n is a union of two balls of type II-[k], we will denote it as

E = n∈J F n -{II-[k]}.
Now we are ready to state the following theorems. Theorem 6.3. Consider f (x) = x 2 + (1 -4m)x with m ∈ Z 2 \ {0}. Then f admits two fixed points 0 and 4m, and f (1 + 2Z 2 ) ⊂ 2Z 2 . We can decompose 2Z 2 as

2Z 2 = {0, 4m} E 1 E 2 E 3 ,
where

E 1 = 2≤n<v2(m)+3 2 n-1 + 2 n Z 2 -{I-[n -2]}, E 2 = n>v2(m)+3 2 n-1 + 2 n Z 2 -{I-[v 2 (m) + 1]}, E 3 = n>v(m)+3 4m + 2 n-1 + 2 n Z 2 -{I-[v 2 (m) + 1]}. Theorem 6.4. Consider f (x) = x 2 + (-1 -4m)x with v 2 (m) ∈ 1 + 2N.
Then f admits two fixed points 0 and 4m + 2, and f (1 + 2Z 2 ) ⊂ 2Z 2 . We can decompose 2Z 2 as

2Z 2 = {0, 4m + 2} E 1 E 2 E 3 ,
where

E 1 = n≥4 4m + 2 + 2 n-2 + 2 n-1 Z 2 -{II-[1]}, E 2 = 4≤n≤⌊v2(m)/2⌋+3 2 n-2 + 2 n-1 Z 2 -{II-[2n -5]}, E 3 = n>⌊v2(m)/2⌋+3 2 n-2 + 2 n-1 Z 2 -{II-[v 2 (m) + 1]}. Theorem 6.5. Consider f (x) = x 2 + (-1 -4m)x with v 2 (m) ∈ 2N * .
Then f admits fixed points 0 and 4m + 2, and f (1 + 2Z 2 ) ⊂ 2Z 2 . The invariant set 2Z 2 admits the following form

2Z 2 = {0, 4m + 2} E 1 E 2 E 3 2 v2(m)/2+1 + 2 v2(m)/2+2 Z 2 ,
where

E 1 = n≥4 4m + 2 + 2 n-2 + 2 n-1 Z 2 -{II-[1]}, E 2 = 4≤n<v2(m)/2+3 2 n-2 + 2 n-1 Z 2 -{II-[2n -5]}, E 3 = n>v2(m)/2+3 2 n-2 + 2 n-1 Z 2 -{II-[v 2 (m) + 1]}. Denote E = 2 v2(m)/2+1 + 2 v2(m)/2+2 Z 2 . (1) If v 2 (m) = 2 and v 2 (m -4) = 3, then E is of type II-[4]. (2) If v 2 (m) = 2 and v 2 (m -4) ≥ 5, then E is of type II-[5]. ( 3
) If v 2 (m) = 2 and v 2 (m-4) = 4
, then there exists a 2-periodic orbit with one point x 1 ∈ 4 + 16Z 2 and the other x 2 ∈ 12 + 16Z 2 ; and we can decompose

E as E = {x 1 , x 2 } E 4
, where

E 4 = k≥5 (x 1 + 2 k-1 + 2 k Z 2 ) ∪ (x 2 + 2 k-1 + 2 k Z 2 ) -{II-[5]}. (4) If v 2 (m) ≥ 4 and v 2 (m -2 v2(m) ) < v 2 (m) + 3, then E is of type II-[v 2 (m - 2 v2(m) ) + 1]. (5) If v 2 (m) ≥ 4 and v 2 (m -2 v2(m) ) ≥ v 2 (m) + 3,
then there exists a 2periodic orbit with one point x ′ 1 ∈ 2 v2(m)/2+1 + 2 v2(m)/2+3 Z 2 and the other x ′ 2 ∈ 2 v2(m)/2+1 + 2 v2(m)/2+2 + 2 v2(m)/2+3 Z 2 ; and we can decompose E as

E = {x ′ 1 , x ′ 2 } E ′ 4 , where E ′ 4 = k≥v2(m)/2+4 (x ′ 1 + 2 k-1 + 2 k Z 2 ) ∪ (x ′ 2 + 2 k-1 + 2 k Z 2 ) -{II-[v 2 (m) + 1]}.
Now we are left to study the polynomials f

(x) = x 2 + x -d with d ∈ Z 2 but √ d ∈ Z 2 .
We distinguish four cases.

Theorem 6.6. Consider f (x) = x 2 + x-d with d ≡ 0 (mod 4) and

√ d ∈ Z 2 . Then f (1 + 2Z 2 ) ⊂ 2Z 2 and 2Z 2 is decomposed as finite number of minimal components. Let n 0 = ⌊v 2 (d)/2⌋ + 1. (1) If v 2 (d) = 2 and v 2 (d -4) = 3, then 2Z 2 consists of three minimal compo- nents: 4Z 2 , 2 + 8Z 2 and 6 + 8Z 2 . (2) If v 2 (d) = 2 and v 2 (d -4) = 4, then 2Z 2 consists of five minimal compo- nents: 4Z 2 , 2 + 16Z 2 , 6 + 16Z 2 , 10 + 16Z 2 , and 14 + 16Z 2 . (3) If v 2 (d) ≥ 3 and v 2 (d) is odd, then 2Z 2 = E 1 E 2 , where E 1 = 2≤n≤n0 2 n-1 + 2 n Z 2 -{I-[n -2]}, E 2 = 2 n0 Z 2 -{I-[n 0 -1]}. (4) If v 2 (d) ≥ 3 and v 2 (d) is even, then 2Z 2 = E ′ 1 E ′ 2 E ′ 3 , where E ′ 1 = 2≤n≤n0-1 2 n-1 + 2 n Z 2 -{I-[n -2]}, E ′ 2 = 2 n0 Z 2 -{I-[n 0 -2]}, E ′ 3 = 2 n0-1 + 2 n0 Z 2 -{I-[v 2 (d -2 v2(d) ) -n 0 ]}. Theorem 6.7. Consider f (x) = x 2 + x -d with d ≡ 1 (mod 4) and √ d ∈ Z 2 . Then f (2Z 2 ) ⊂ 1 + 2Z 2 and 3 + 4Z 2 is of type II-[1]. Let d = 5 + 8t with t ∈ Z 2 . If v 2 (t) ≤ 1, then 1 + 4Z 2 is of type II-[v 2 (t) + 2]. If v 2 (t) ≥ 2, then 1 + 4Z 2 = {x 1 , x 2 } E 1 E 2 E 3 ,
with the form

E 1 = (a + 2 4 Z 2 ) ∪ (f (a) + 2 4 Z 2 ) -{II-[3]}, E 2 = (b + 2 5 Z 2 ) ∪ (f (b) + 2 5 Z 2 ) -{II-[3]}, E 3 = n≥6 (x 1 + 2 n Z 2 ) ∪ (x 2 + 2 n Z 2 ) -{II-[3]}, and x 1 , x 2 is a 2-periodic orbit such that x 1 ∈ c + 2 5 Z 2 and x 2 ∈ f (c) + 2 5 Z 2 . Precisely, (1) If v 2 (t) = 2 and v 2 (t -4) = 3, then a = 1, b = 25, c = 9. (2) If v 2 (t) = 2 and v 2 (t -4) ≥ 4, then a = 1, b = 9, c = 25. (3) If v 2 (t) = 3, then a = 9, b = 1, c = 17. (4) If v 2 (t) ≥ 4, then a = 9, b = 17, c = 1. Theorem 6.8. Consider f (x) = x 2 + x -d with d ≡ 2 (mod 4) and √ d ∈ Z 2 . Then f (1 + 2Z 2 ) ⊂ 2Z 2 .
(

) If v 2 (d -2) = 2, then 2Z 2 is of type II-[1]. (2) If v 2 (d -2) = 3, then 8Z 2 ∪ (f (0) + 8Z 2 ) is of type II-[1], (4 + 8Z 2 ) ∪ (f (4) + 1
2 ) consists of a 2-periodic orbit with one point x 1 ∈ 4 + 8Z 2 and the other x 2 ∈ f (4) + 8Z 2 , and for each n ≥ 4, (

x 1 + 2 n Z 2 ) ∪ (x 2 + 2 n Z 2 ) is of type II-[2]; (3) If v 2 (d-2) ≥ 4, then 4+8Z 2 ∪(f (4)+8Z 2 ) is of type II-[1], 8Z 2 ∪(f (0)+8Z 2 )
consists of a 2-periodic orbit with one point x 1 ∈ 8Z 2 and the other x 2 ∈ f (0) + 8Z 2 , and for each n ≥ 4, (

x 1 + 2 n Z 2 ) ∪ (x 2 + 2 n Z 2 ) is of type II-[2].
Theorem 6.9. For f (x) = x 2 + xd with d ≡ 3 (mod 4), the ball 2Z 2 is mapped into the ball 1 + 2Z 2 which is the unique minimal component.

We prove Theorems 6.1-6.5. The proofs of Theorems 6.6-6.9 will be omitted since they are similar to those of Theorems 6.1-6.5.

Proof of Theorem 6.1. Let f (x) = x 2λ. Then f ′ (x) = 2x and (f 2 ) ′ (x) = 4x 3 -4λx.

1) If λ ≡ 0 (mod 4), then 2 + 4Z 2 and 3 + 4Z 2 are mapped into 4Z 2 and 1 + 4Z 2 respectively, and 4Z 2 and 1+4Z 2 are mapped into themselves respectively. Consider the cycles (0) and (1) of f 2 . We have a 2 (0) = f ′ (0) ≡ 0 (mod 2) and a 2 (1) = f ′ (1) ≡ 0 (mod 2).

Thus cycles (0) and (1) grow tails, hence there will form two attracting fixed points, one in 4Z 2 with basin 2Z 2 , and the other one in 1 + 4Z 2 with basin 1 + 2Z 2 .

2) If λ ≡ 1 (mod 4), then 1 + 4Z 2 and 2 + 4Z 2 are mapped into 4Z 2 and 3 + 4Z 2 respectively, and 4Z 2 and 3 + 4Z 2 are mapped into 3 + 4Z 2 and 4Z 2 respectively. Consider the cycle (0, 3) of f 2 . We have a 2 (0) = (f 2 ) ′ (0) ≡ 0 (mod 2).

Thus cycle the cycle (0, 3) grows tails, hence there will form an attracting 2-periodic orbit, with one periodic point in 4Z 2 , and the other one in 3 + 4Z 2 . We also see that the attracting basin is the whole Z 2 .

The proofs of 3) and 4) are similar to the proofs of 1) and 2).

Proof of Theorem 6.2. Let f (x) = x 2 + x. We will use a diagram to show the structure of the dynamics of f . At level n, the "→" stands for the transformation of the elements of Z/p n Z under f n . Thus the diagram shows that f 1 (1) = 0, f 1 (0) = 0, i.e., f (1 + 2Z 2 ) ⊂ 2Z 2 and f (2Z 2 ) ⊂ 2Z 2 . and f 2 (0) = 0, f 2 (2) = 2, i.e., f (4Z 2 ) ⊂ 4Z 2 and f (2 + 4Z 2 ) ⊂ 2 + 4Z 2 .

Since f (1 + 2Z 2 ) ⊂ 2Z 2 and f -1 (1 + 2Z 2 ) = ∅, we need only to consider 2Z 2 . From the diagram, we also see that (0) is the only cycle of f 1 with length 1, and (0), [START_REF] Anashin | Uniformly distributed sequences in computer algebra or how to construct program generators of random numbers[END_REF] are two lifts of (0). will start our examination from the level 2. Since Thus the cycle (2) strongly grows which implies that the lift of (2) still grows, and so on. Hence 2 + 4Z 2 is a minimal component. By induction we know that for all n ≥ 2

A n (0) = ∞ and B n (0) = ∞.

Thus the cycle (0) of f n-1 always splits to be two cycles (0) and (2 n-1 ) of f n , and the number 0 should be a fixed point. Now for n ≥ 2, let us consider the cycle (2 n-1 ) of f n . With the same calculations, a n (2 n-1 ) = 2 n + 1 and b n (2 n-1 ) = 2 n-2 .

Thus a n (2 n-1 ) ≡ 1 (mod 4) and

A n (2 n-1 ) = n and B n (2 n-1 ) = n -2.

Hence, the cycle (2 n-1 ) strongly splits and B n < min{A n , n}. By Proposition 3.2, the lift of (2 n-1 ) splits B n -1 = n -3 times then all lifts strongly grow. Thus there are 2 n-2 pieces of minimal components which constitute 2 n-1 + 2 n Z 2 . They are 2 n-1 + t2 n + 2 2n-2 Z 2 , t = 0, . . . , 2 n-2 -1.

This concludes the proof of Theorem 6.2.

Proof of Theorem 6.3. Let f (x) = x 2 + (1 -4m)x. We see that there are two fixed points 0 and 4m, and 1 + 2Z 2 is mapped into 2Z 2 . We are concerned with the invariant subset 2Z 2 .

Consider 2 n-1 + 2 n Z 2 (n ≥ 2). We study the cycle (2 n-1 ) at level n. We have a n (2 n-1 ) = 2 n -4m + 1 b n (2 n-1 ) = 2 n-2 -2m, thus a n (2 n-1 ) ≡ 1 (mod 4) and if 2 ≤ n < v 2 (m) + 3,

A n (2 n-1 ) ≥ n B n (2 n-1 ) = n -2.

If n = 2, then the cycle (2 n-1 ) strongly grows. If n > 2, then the cycle (2 n-1 ) strongly splits and B n < min{A n , n}. By Proposition 3.2, the lift of (2 n-1 ) strongly splits B n -1 = n -3 times then all lifts strongly grow. Thus we will obtain the part E 1 in Theorem 6.3.

If n > v 2 (m) + 3,

A n (2 n-1 ) = v 2 (m) + 2 B n (2 n-1 ) = v 2 (m) + 1.
Hence, the cycle (2 n-1 ) strongly splits and B n < min{A n , n}. By Proposition 3.2, the lift of (2 n-1 ) strongly splits B n -1 = v 2 (m) times then all lifts strongly grow.

Hence we have the part E 2 . Consider 4m + 2 n-1 + 2 n Z 2 (n > v 2 (m) + 3). Let s n ≡ 4m + 2 n-1 (mod 2 n ) and 0 ≤ s n < 2 n . We study the cycle (s n ) at level n. We have

a n (s n ) = 2s n -4m + 1 b n (s n ) = s n (s n -4m) 2 n ,
thus a n (s n ) ≡ 1 (mod 4) and

A n (s n ) = v 2 (m) + 2 B n (s n ) = v 2 (m) + 1.

  conjugate to the adding machine on an odometer Z (ps) , where (p s ) = (k, kd, kdp, kdp 2 , • • • ) with integers k and d such that 1 ≤ k ≤ p and d|(p -1).

x 2 +Theorem 6 . 2 .

 262 xd has a fixed point), then x 2 + xd conjugates tox 2 + bx with b = 1 -2 √ d, through x → x + √ d. If b = 1, the minimal decomposition of x 2 + x is as follows. Consider the polynomial f (x) = x 2 + x on Z 2 [x].There is one fixed point 0. We have f (1 + 2Z 2 ) ⊂ 2Z 2 and we can decompose 2Z 2 into

a 2 (have a 2 ( 2 )

 222 0) = f ′ (0) = 1 and b 2 (0) = f (0) -0 2 2 = 0,we have a 2 (0) ≡ 1 (mod 4) andA 2 (0) = ∞ and B 2 (0) = ∞.Thus the cycle (0) strongly splits. Sincea 2 (2) = f ′ (2) =5 and b 2 (≡ 1 (mod 4) and A 2 (2) = 2 and B 2 (0) = 0.

On the minimal decomposition of p-adic polynomial dynamical systems

Hence, B n < min{A n , n}. By Proposition 3.2, the cycle (s n ) strongly splits and the lift of (s n ) strongly splits B n -1 = v 2 (m) times then all lifts strongly grow. Therefore, we have the part E 3 . This completes the proof.

Proof of Theorem 6.4. Let f (x) = x 2 + (-1 -4m)x with v 2 (m) ∈ 1 + 2N. We see that there are two fixed points 0 and 4m + 2, and 1 + 2Z 2 is mapped into 2Z 2 .

Since 0 and 4m + 2 are two fixed points, there are cycles (0) and (t n ) at each level, where t n ≡ 4m + 2 (mod 2 n ) and 0 ≤ t n < 2 n . Consider the cycles (0) and (t n-2 ) at level n -2. By studying the a n-2 , b n-2 of these two cycles, we know that they weakly split. By Proposition 3.3, after splitting, half of lifts weakly grow. Thus we will obtain two 2-cycles: (2 n-2 , 2 n-2 + 2 n-1 ) and (s n , s n + 2 n-1 ) at level n, where s n ≡ 4m + 2 + 2 n-2 (mod 2 n-1 ) and 0 ≤ s n < 2 n-1 .

For each n ≥ 4, we study the cycle (s n , s n + 2 n-1 ) at level n. We have

thus a n (s n ) ≡ 1 (mod 4) and

Hence, the cycle (s n , s n + 2 n-1 ) strongly splits and B n < min{A n , n}. Therefore, by Proposition 3.2, the lift of (s n , s n + 2 n-1 ) strongly splits B n -1 = 1 -1 = 0 times then all lifts strongly grow. Thus we obtain E 1 in Theorem 6.4. Now we study the cycle (2 n-2 , 2 n-2 + 2 n-1 ) at level n ≥ 4. We have

Thus a n (2 n-2 ) ≡ 1 (mod 4) and for each n > ⌊ v2(m)

Hence, the cycle (2 n-2 , 2 n-2 + 2 n-1 ) strongly splits and B n < min{A n , n}. Therefore, by Proposition 3.2, the lift of (2 n-2 , 2 n-2 + 2 n-1 ) strongly splits

times then all lifts strongly grow. Thus we have

) strongly splits and

Therefore, the lift of (2 n-2 , 2 n-2 + 2 n-1 ) strongly splits at least n -1 times. But except this we do not obtain any further more information. Thus Proposition 3.2 is not sufficient for us. Now we do some calculations directly. For any point 2

where

Thus the cycles grow at level 3n -5. By Corollary 3.9, the cycles grow always.

Therefore we obtain the part E 2 which completes the proof.

Proof of Theorem 6.5. Let f (x) = x 2 + (-1 -4m)x with v 2 (m) ∈ 2N * . We see that there are two fixed points 0 and 4m + 2, and 1 + 2Z 2 is mapped into 2Z 2 .

As the proof of Theorem 6.4, we study two 2-cycles: (2 n-2 , 2 n-2 + 2 n-1 ) and (s n , s n +2 n-1 ) at level n, where s n ≡ 4m+2+2 n-2 (mod 2 n-1 ) and 0 ≤ s n < 2 n-1 . The existence of E 1 , E 2 , E 3 are the same as that of Theorem 6.4.

Consider 2 n-2 + 2 n-1 Z 2 with n = v2(m) 2 + 3. We are going to study the cycle (2 n-2 , 2 n-2 + 2 n-1 ) at level n. We study the points

With the same calculation in the proof of Theorem 6.4, we have the same equation (6•1). To continue the proof, we will distinguish two cases:

) -(2 n-2 + t2 n-1 ) ≡ 0 (mod 2 9 ) ≡ 0 (mod 2 10 ).

Hence we will obtain (1) and (2) of Theorem 6.5. Since f 2 (x)x = x(x -4m -2)(x 2 -4mx -4m), f has 2-periodic orbit if and only if x 2 -4mx -4m = 0 has solutions different to 0 and 4m + 2 in Z 2 . But x 2 -4mx -4mx = 0 has solution 0 or 4m + 2 only if m = 0 or m = -1. Thus for the case v 2 (m) ∈ N * , f has 2-periodic orbit if and if only △:= 16m 2 + 16m has square roots in Z 2 . By the standard argument in number theory (see [START_REF] Serre | A course in arithmetic[END_REF], p.18), this is equivalent to 2 -v2(m) m(m + 1) ≡ 1 (mod 8). By some basic calculations it is then equivalent to v 2 (m -4) = 4. This is nothing but the rest case we need to study. Thus for v 2 (m -4) = 4, there exists a 2-periodic orbit.

From the equation x 2 -4mx -4m = 0, the periodic point can be written as

Recall that we are concerned with 2 n-2 + 2 n-1 Z 2 (n = 4) which is the union of two balls 2 n-2 + 2 n Z 2 and 2 n-2 + 2 n-1 + 2 n Z 2 , and we are studying the cycle (2 n-2 , 2 n-2 + 2 n-1 ) at level n = 4. Thus we have x 1 ≡ 4 (mod 16) and x 2 ≡ 12 (mod 16).

For each k ≥ 5, we consider the union of the two balls (

We study the cycle (s 1 , s 2 ) where

where

Here we have used the property that x 1 is a solution of the equation x 2 -4mx -4m = 0.

Since v 2 (m) = 2 and v 2 (x 1 ) = 2, we get v 2 (Φ) = k + 2. Thus

).

Hence we have (3). Now we are left to treat the case v 2 (m) ≥ 4. In this case the equation x 2 -4mx -4m = 0 admits solutions if and only if

Then we will obtain [START_REF] Anashin | Ergodic transformations in the space of p-adic integers[END_REF].

m -2 -v2(m) m(m + 1) .

For each k ≥ v2(m) 2 + 4, we consider (x ′ 1 + 2 k-1 + 2 k Z 2 ) ∪ (x ′ 2 + 2 k-1 + 2 k Z 2 ). For every point

1 + 2 k-1 + t2 k ) ≡ 0 (mod 2 v2(m)+k+1 ) ≡ 0 (mod 2 v2(m)+k+2 ).

Hence we have [START_REF] Anashin | Applied algebraic dynamics[END_REF]. This completes the proof.