ON THE MUTUAL MULTIFRACTAL ANALYSIS FOR SOME NON-REGULAR MORAN MEASURES Текст научной статьи по специальности «Математика»

46 Probl. Anal. Issues Anal. Vol. 12 (30), No 1, 2023, pp. 46-71

DOI: 10.15393/j3.art.2023.12310

UDC 517.518.1

B. Selmi, N. Yu. Svetova

ON THE MUTUAL MULTIFRACTAL ANALYSIS FOR SOME NON-REGULAR MORAN MEASURES

Abstract. In this paper, we study the mutual multifractal Hausdorff dimension and the packing dimension of level sets K(a, ft) for some non-regular Moran measures satisfying the so-called Strong Separation Condition. We obtain sufficient conditions for the valid multifractal formalisms of such measures and discuss examples.

Key words: fractal/multifractal dimensions, Moran sets, non-regular Moran measures

2020 Mathematical Subject Classification: 28A20, 28A75, 28A78, 28A80, 49Q15

1. Introduction. Recently, the issue of using the methods of fractal geometry to compare the distributions of various probability measures has been actively discussed (see for example [17], [18], [21], [23]). However, in practical applications, the comparison of distributions of measures can be difficult. Often, different distributions of measures can give subtle or indistinguishable differences in the spectra. To solve this problem, various new methods for direct comparison of distributions are proposed. One such method is the mixed or mutual multifractal analysis [17], [18], [21], [23], which allows to better understand the local geometry of fractal measures and the simultaneous scale behavior of multiple measures.

The mixed multifractal analysis is a natural extension of the multi-fractal analysis of single objects, such as measures, functions, statistical data, distributions, etc. It has been developed quite recently from a purely mathematical point of view. In physics, and statistics, it was appearing in different forms, but not really, and strongly linked to the mathematical theory (see [12]). In some applications, such as clustering topics, each attribute in a data sample may be described by more than one type of measure. This leads researchers to apply measures well adapted for mixed-type data as [12]. Mixed multifractal analysis has been applied in

© Petrozavodsk State University, 2023

explaining joint movements in volatility for asset markets, such as joint multifractal Markov-switching models.

The mixed multifractal analysis is not really new in financial series processing. It has been, in contrast, merged under the name of multivariate multifractal analysis, where many situations in financial markets and their volatility have been described. Multivariate models have been also applied for long memory with mixture distributions [14], [15]. Multifractal analysis of measures for the so-called mixed logical dynamical models to the classification of signals, especially network traffic, is developed in [13]. These models are widely applied in the control of hybrid systems, such as multiserver ones.

Many authors were interested in studying the properties of mutual multifractal dimensions and spectra and establishing connections between them. More about the use of mixed multifractal analysis/formalism of measures, as well as functions and time series or images is developed in [1], [3], [4], [5], [6], [8], [9], [22], [24], [25], [26] and the references therein. The inverse problem of the mixed multifractal formalism and interesting examples are developed in [19].

From a mathematical point of view, Moran sets are generalizations of the classical self-similar objects, characterized by arbitrary basic sets in each step of construction. Moreover, the associated similarities at each step have also different rations from their predecessors. Moran's construction permits zero value for the lower limit of the contraction ratios (see [28]). Moran structures are also met in geography. Spatial autocorrelation in geographic information systems is based on the degree to which one object is similar to other nearby objects. Geographers call this concept Moran Index measures for spatial autocorrelation. Although the concept is defined independently in geography, it uses the similar idea of subdividing maps into Moran sets based on the axiom stating that everything is related to everything else, but near things are more related than distant things, see for example [2] and the references therein. More backgrounds and information on the applications may be found in [11], [20].

For given two compactly supported Borel probability measures ^ and v on Mra and for a, ft ^ 0, we consider the set

K(a, ft) = p E ; lim l0g ^^ - a and lim l0g (B'= ft}, l r-^0 log r r^0 log r J

where E = supp ^ x supp v and Br (x) is the closed ball with center x and radius r. That is, we will be interested in the set of points for which

the local dimensions of ^(Br(x)) and v(Br(x)) simultaneously describe the power-law behavior of the measures at a small radius r. The main problem is to estimate the size of this set. For some a, ft, the mutual Hausdorff and packing dimensions of these sets K(a, ft) have a close connection with the Legendre transform of some function r(q,t) associated with measures ^ and v. The purpose of this work is to obtain conditions for the valid and non-valid multifractal formalism for some non-regular Moran measures.

2. Preliminaries. Before detailing our results, let us recall the mutual multifractal formalism introduced by Svetova [25]. Let ^ and v be two Borel probability measures on Rra with the same compact supports supp ^ = supp v. For (q, t, s) e R3 and 5 > 0, we introduce

(E) = inf {I v(Bn (Xi))q V (Bn (xt))t(2rl)s}, i

where the infimum is taken over all centered ^-coverings of E c Rra. The mutual Hausdorff measure is defined as follows:

) = sup^(E), and ) = sup U^Q(F).

&>0 F cE

We make the dual definitions

^^(E) " sup (I v(Bn(Xi))qu(Bn(Xi))t(2Ti)s}, i

where the supremum is taken over all the centered ^-packings of E c Rra. We define the mutual packing as follows:

It holds, as for the case of the multifractal analysis of a single measure, that each of the measures 'H^3 and Vassigns a multifractal dimension to each subset E of Rra. They are respectively denoted by

dim*4,(E) = sup (s: H^;S(E) = 8} = inf (a: H^;S(E) = 0} ,

and

Dim*;,(E) = sup {s: V^S(E) = 8= inf : V^S(E) = 0}

Now, we define the multifractal function , : R2 ^ [—8, +8] as follows:

(Q,t) = dim^ (supp y), and (q,t) = Dim^ (supp

We denote by V(Rn) the set of Borel probability measures on Rn. A measure ^ e V(Rn) with support supp ^ is said to satisfy the doubling condition if

Pf \ v (

F(a,^) = nmsup I sup —-— , I < 8,

r\0 \xesuppn (x)j J

for all a > 1. Denote by Vd (Rn) the family of Borel probability measures

on Rn that satisfy the doubling condition.

Let {nkbe a sequence of positive integers. Define D0 = 0, for

any integer k ^ 1, set Dm,k = {(im'im+1 ...ik);1 ^ ij ^ nj, m ^ j ^ k},

and Dk = D1,k. Define D = U Dk. If a = (a1 ...ak) e Dk, and

k^0

t = (n ...rm)e Dm, then a * t = (ai... ak ri.. .rm) e Dk+m. If I ^ k, then a\l = (o1.. .ai). Suppose J is a closed interval of length 1.

Definition 1. [30], [31] The collection Q = [Ja ,a e D} of closed subin-tervals of J is called having the Moran structure, if it satisfies the following conditions:

(i) Jh = J;

(ii) For all k ^ 0 and o e Dk, J<r*i,..., are subintervals of Ja, and satisfy int(Ja*i) x int (Jo-*j) = 0 for i ^ j, where int(A) denotes the interior of the set A;

(iii) For any k ^ 1 and a e Dk_1, 1 ^ j ^ nk, \. <J*j. \ = ckj, where \A\

v'o \

denotes the diameter of A.

Suppose that Q is a collection of closed subintervals of J having the Moran structure, set

Ek = U , and E = Ek.

aeDk k'^0

It is clear that E is a nonempty set. The set E = E(Q) i s called the

Moran set associated with the collection Q.

Let Qk = {Ja; a e Dk}; obviously, Q = [J Qk. The elements of Qk are

k^0

called the basic elements of order k of the Moran set E, and the elements

of Q are called the basic elements of the Moran set E. Further we will assume that lim sup \Ja \ = 0.

aeDk

Suppose, for any integer k ^ 1, any a e Dk, and for 1 ^ j ^ nk+1, the (k + 1)-order basic element Ja*j c Ja. Let

dist(J,7^j, .

m * A ■

for all i ^ j, where dist(A,5) := inf dist(x,y) for any two sets A and

xeA, yeB

B, and dist(x,y) is the Euclidean distance between the points x and y.

Denote A := inf A k.

k^i

Definition 2. We say that the Moran set E satisfies the Strong Separation Condition (SSC) if A > 0.

Now, we define two probability measures on the Moran set E. For k ^ 1, let p = [pkj}™t 1, p = [pk j}™" 1 be two positive probability vectors,

nk nk

Pkj > 0, Pkj > 0, Yjpki " 1, and YjPkj = 1.

For a e Dk, k ^ 1, we define ^(Ja) = piaip2a2 ...Pkak and v(Ja) = ,p1a1 'P2<t2 .. .'Pk<rk. It is obvious that supp ^ = supp v. The measures ^ and v are defined on one set E = supp ^ = supp v. Denote

Pmin = min{'Pi j}, Pmax = max^-} for 1 ^ j ^ nk, 1 ^ i ^ k,

Pmin = min{pi j}, Pmax = max^} for 1 ^ j ^ Uk, 1 ^ i ^ k,

Cmin = min{cij}, Cmax = max^} for 1 ^ j ^ Uk, 1 ^ i ^ k.

3. The main result. Let ^, v be two compactly supported Borel probability measures on Rn. For a, ft ^ 0, let

* («J) " L; lim l0g (X)» " „, and - ft I

I log r log r )

We are interested in the estimation of the Hausdorff and packing dimension of K(a, ft). Let us mention that in the last decade there has been a great interest for the multifractal analysis and positive results have been obtained for various situations (see, for example, [16], [29], [31]). The

authors in [8], [9] prove the result of Theorem 3 in [25] under less restrictive assumptions, as follows:

Theorem 1. Let be two compactly supported Borel probability measures on Rra. Suppose that is differentiable at {q,t) and set a = - dB^q*q and ft = - dB^q,tq. Assume that

Wq*,B»'vPqq (supp ^ x supp v) > 0.

Then we have

dimH{K{a, ft)) = dimp{K{a, ft)) = B*tV{a, ft) = {a, ft),

where f *{a,ft) = inf (aq + ftt + f (a, ft)) denotes the Legendre transform

of the function f. Here dim h and dimp denote the Hausdorff and packing dimensions (see [16] for the definitions), and in this case we say that the mutual multifractal formalism is valid.

Let us define the function rk (q,t) as the only solution to the equation

2 KJ*)q■ v{J*Y-\J*ITk{q,t) = 1. (1)

aeDk

Using (1), the theorem of implicit differentiation shows that rk is partially

differentiable with respect to all variables. It is clear that if ^{J*) ^ \Ja \Sk

and v{J*) ^ \Ja\Sk, where sk satisfies c*k = 1, then rk{q,t) is strictly

aeDk

convex for {q,t) e R2. Define now the following functions: r{q,t) = liminf Tk{q,t), r{q,t) = limsuprk{q,t), r{q,t) = lim rk{q,t).

fc^ro

We can now state the main result of the paper. Explicit formulas for the mutual multifractal dimensions of the level sets K{a, ft), for which the classical formalism does not hold, are given by obtaining some new sufficient conditions (which are different from the ones in Theorem 1) for the non-validity of the mutual multifractal formalism of non-regular Moran measures, i.e., the case for which the multifractal functions v and B^,y do not necessarily coincide (see Figure 2). Our main theorem generalizes the main results of [7], [29], [30], [31] (by taking (q = 0 or t = 0) and q = t = 0).

Theorem 2. Assume that A > 0, and let {q,t) e R2. Let be two non-regular Moran measures on the Moran set E.

1) Suppose that , ^q^ exists, and

liminf V v (J* Y " (J* ) \J* IT{q 't] > 0, for all (q, t) e E2.

k—>m

<reDk

If (a, ft ) = - ( ^, ^ ), then dimH (K (a, ft)) = b^ (a, ft) = r* (a, ft). 2) Suppose that ^ ^Bq ^, BTpq'^ ) exists, and

limsup ^ P (J* )q " (J*) \J* I TBq 't] > 0, for all (q,t) e E2.

*eDk

If (a, ft ) = -( ^, , then, dimp (K (a, ft)) = B*>v (a,ft) = r*(a, ft).

3) Suppose that t(q,t) and ^8tpqq'^, drBqt'^ exist, and inequality r(q,t) < k + (k(q,t) is tru) for some constant c > 0, and all k ^ 1. If (a, ft)" - (^, ^), then,

dimH(K(a, ft)) = dimp(K(a, ft)) = (a, ft) = B*tv(a, ft) = r*(a,ft). Remark.

1) Our results hold naturally when replacing the interval J by a compact subset (denoted also J) of Era with int(J) = J.

2) If there exists a family of similitudes {Skj: k ^ 1,1 ^ j ^ nk} with J* = S* (J) = Shai oS2'*2 0• • -°Sk,uk (j), for a = (a\, ...,ak) e Dk,

then the corresponding Moran set E = p| [J J* is called a general-

k *

ized self-similar set, which is a generalization of the self-similar sets, and the Moran measure is an extension of the self-similar measure. This implies that our main results hold for the self-similar sets and measures.

3) For j ^ 0, we consider the following fractal set:

log

log (u(Br(x))

of \ i i- log (v(Br (X))) |

£= \x e suppv X supp V; -( , =

l r^o log v[HAx) >

Denote R+ x R+ := [0, + <x>[x]0, + <x>[. It is clear that

U K{a, ft) c £

*

(a,/3)eR+xR*

P

The union is composed of an uncountable number of pairwise disjoint nonempty sets. Theorems 1 and 2 show that, surprisingly, the Hausdorff and packing dimensions of £ {j) are fully carried by some subset K{a, ft). Also, our main results give an optimal lower bound of the Hausdorff and packing dimensions of £{j).

4. Examples. In this section, we illustrate our main results with two examples.

Example 1. Let I = [0,1], nk = 2, and ck = 5, for all k ^ 1. The set E is the middle-5 Cantor set, and ^ and v are two Bernoulli measures, such that p = v, with p := Pf = P? and p := Pf = P22. Then, for {q,t) e R2, we have:

h ( n ( (A log {pq + ' + Pq+*) {q,t) = Bp,„ {q,t) = t {q,t) = -^-,

and

dr {q,t) dr {q,t) pq+t logp + pq+t logp dq dt {pq+t + %>q+t) log 5

Now, it follows from Theorem 2 that

dimn{K{a, ft)) = dimp{K{a, ft)) = r*{a,ft),

for {a, ft) = — (^jq^, ^jf^j . Figure 1 shows the plots of the multifractal functions r and r*.

Example 2. Let {Sk)k be a sequence of integers, such that

Si = 1, S2 = 3, and Sk+i = 2Sk, @ k ^ 2. Define the family of parameters ni, Ci, and pi,m as follows:

3, if S2k-1 ^ i < S2k,

C

n1 = 2, ni =

2, if S2 k ^ i < S2 k+1,

and

1 f1, if S2k-1 ^ a < S2k,

Ci = Ci = \ 7

5 1, if ^2k ^ i<s2

5 , if ^2 k ^ I < 02k+1. Let {pa,m)'^l"1 and {pb,m)fri"1 be two probability vectors. We define

Pl,m = Pa,m, for all 1 ^ m ^ 2,

and

u

¡Pb,m, if S2k-1 ^ i < S2k, 1 ^ m ^ 3,

Pi,m

[Pa,m, if S2k ^ i < S2k+1, 1 ^ m ^ 2.

Let Nk be the number of integers i ^ k, such that pi,m = pa,m\ then

r . fNk 1 . r Nk 2 lim ini —— = — and lim sup —— = -.

k^+8 k 3 k^+8 k 3

Now, let ^ and v be two Moran measures, with ^ = v, which are generated by {Pi,m). For {q,t) e R2, we get

k

■log( 2 pqa+m) + (1 — f) lo^ z cm)

l vm=1 7 v 7 vm=1 7

Tk{q,t) =-E-(--

f log 5 + (1 — f) log 7

Let

3 log (ee <m) +1 log (ee c)

1 log5 + 1 log7

Figure 2: The plots of the functions t and r.

and

2 x /3

2

3 iog( 2 pI+£) +1 iog( 2 Pliï)

Q,t) =

3 log 5 + 3 log 7

Observing that the assumptions of Theorem 2 are satisfied, we obtain

r(q,t) = min j<fr(q,t), 4>(q,t)^ and f{q,t) = maxj<fr(q,t), 4>(q,t)^ . Figure 2 shows the plots of the functions r and f.

4. Proof of the main result. Suppose that the set E is a Moran set associated with the collection Q. Let ^ and v be two non-regular Moran measures supported by the set E. We start with the following interesting intermediate results.

Proposition 1. Let A > 0, 0 < r < A, and a ^ 1. For x e Ja, a e D, we find k,l e N, such that \Ja\k \ ^ r < \Ja\k-1\ and A\Ja\i\ ^ ar < A\Ja\i-1 \. Then 0 ^ k — I ^ M, where M is a positive constant.

Proof. Since 0 < A ^ 1 and a ^ 1, we have: -¡¿^ < 1. Then

j^f < f ^ 1. Therefore \Ja\k\ ^ \Ja\i\, which implies k ^ I. As k ^ I, the inequality r < \Ja\k-1\ can be rewritten as

r < \ Ja\l-1 *(l,1 +1,...,k-1)\ ^ \ Ja\l-l\ ■ C^na^1.

We have

r I f i I 1^1 rk-l-1 „k-l-1

I , \Ja\k-1\ \^a\l \ ■ ^max ^ cmax

^ \»v j- i w \o i max ^

^ ^ A\JaV\ ^ A\Ja\-1\ ^ A

1 ck--1 log4

Then, - « , and k — I < 6 ° + 1 = M. □

a A log cmax

Proposition 2. [10] If A > 0, then e VD (E).

Proposition 3. For any x e E, and small r > 0, we can find a e Dn, k,l e N. There are some constants Ai(q, t), A2(q, t), B1(q, t), and B2(q, t) for (q, t) e R2, such that

A^q, t)^(Ja]l )qu(Ja\i ) « /i(Br (x))qu(Br (x)) « A2(q, t)^(Ja]l )qu(Ja\i ),

and

Bi (q, t)ll(Jalk )qy(Ja\k ) « l(Br (x))qu(Br (x)) « B2(q, t)l(Ja\k )qv(Ja\k Y.

Proof. Fix x e E and r > 0. We can find a e Dn, such that x e Ja, and integers k, I e N, such that \Ja\k | « r < \Ja\k-1\, and A| Ja\i\ « r < A\ Ja\i-1 \. Notice that Ja\k Ç Br(x) and E x Br(x) Ç Ja\l. Let us estimate i(Br(x))qu(Br(x)) for different q and t.

1) Suppose q,t < 0. By Proposition 1, we have, for M ^ k — I:

l(Ja\l )Qv(Ja\l ) « l(Br (x))qu(Br (x)) « l(Ja\k )q»(Ja\k ) =

= (P 1aiP2a2 . . . Pkak Y(P1ai f>2a2 . . . Pkak Y = = (PiaiP2a2 . .. PlaiPl+1al+i . . . Pkak Y (PiaiP>2a2 . .. PlalPl+1al+i . . . Pkak Y « « P^-!)q^]tl(Ja\lYv(Ja\lY « P™nP^nl(Ja\lY^(Ja\lY.

2) For q,t ^ 0:

«ww " iBm^11' {x)ym)< "{Br(x)) «

« 4J4 "-rrrABr(x)XBr(x))' «

l(Ja\k)q v(Ja\k)

« k-Dl~{k-i)tl(Br(x))qv(Br(x)Y «

Pmin Pmin

« -M^mlB(x))q"(Br(x)Y « -u^lW^MiY.

i'mini'min i'mini'min

3) For q< 0, t^ 0:

MMl)MJ^) ^ (x))

Jo\l Y

HBr (x)Y

u(Br(x)) ^

w

Y 1

^ -±TTtMBr(x)yu(Br(x)) ^ -~Mt»(Br(x))qu(Br(x)) ^

U( J (r\k ) Pmin

Mq Mq

^ M^M)q"(Br(x)) ^ fiW^MJ^.

Pmin Pmin

4) For q ^ 0, t < 0, similarly to case 3, we have

1 fiMt

vWM < -Mn№r(x)Yu(Br(x)) ^ .

Pmin Pmin

Combining all these cases, we get

Ai(q,t)fi(.Ja\x)qv(Ja\) ^ MBr(x))qp(Br(x)) ^ A2(q,t)»(Ja\x)qv(Ja\),

where

Ai(q, t) =

i,

, < 0,

Mq ~m a 0

fmin-Pmin, y,1 ^ 0,

M , min

A2(q, t) =

M , Pmin,

Mq ~Mt + <■- 0

-Pmin-Pmin, Q,^ < 0,

i,

M min

M , min

q,t ^ 0, q< 0,t ^ 0, q^ 0,t < 0.

q< 0,t ^ 0 q^ 0,t < 0,

Similar arguments give

Bi(g,t)KJa\k)qv(J*\k) ^ (x))qu(Br(x)) ^ B2(q,t)fi(Ja\k)qv(Ja\k) where

Bite, t) =

i

M M min min

M min

M min

, < 0, q,t ^ 0, q< 0,t ^ 0

q^ 0,t < 0

B2(q, t) =

i,

M M min min

M min

M min

, < 0, q,t ^ 0,

q< 0,t ^ 0,

q^ 0,t < 0.

□

i

i

i

i

i

i

For (q, t, s) e R3, we denote

h = liminf V i(Ja)qu(Ja)t\Ja\s, and h = limsup V i(Ja)qu(Ja)t\Ja\s

n^rCT, ¿—I „ . „„ ¿—I

n^<x>

aeD.

Proposition 4. If A > 0, there exists a constant c > 0, such that, for any (q,t, s) e R3,

c- h ^ n^ (E) ^ h.

Proof. It is not hard to see that Wqf^s(E) ^ h. If h e (0, +8), there is a sufficiently large number n, such that

h

2 l(Jafy(Ja)Va\' > -

ae D2

Let 8 > 0 and {Bri(xi)} be a centered ^-covering of E. For any i e N, we choose a(i) e Dn, n ^ 1, such that xi e Japiq. For any i e N, let ki,li e N be such that

\ Ja(i) \ ki \ « n < \ Ja(i) \ ki-1 \, and A\ Ja(i) \ k \ « ^ < A\ Ja(i) \ k-1 \. If s ^ 0, then \Ja(t)\h\s « (*)s.

"•O I J /.M, I n . T /AI, n . ^ ,

4) . Thus

Let S < 0. Since \Ja(i)\li \ ^ CminJa(i)\k-1 ^ Cmin £, then \Ja(i)\k\S « CSrmn ( ï Y . Thus

(t■ \s ) 1 s ^ 0

A), where Ko H *<0'

There exists a probability measure Xg,t,s supported on E, such that

V(Ja{i) )qV(Ja{i) )t\Ja{i)\S

Xq,t,s(Ja(i)) =

2 l(Ja(i))q^(Ja(i))t\Ja(^)\S'

aeD\a(i)\

Obviously, Xq,t,s(E ) = Y^Xq,t,s (Ja(i)) = -

For a Moran set E and (q,t,s) e R3, we have, by using Proposition 3,

Xq,t,s (E ) «J]Xq,t,s (Bri (xi)) «J]Xq,t,s (Ja(i)\k ) =

n

S

KJ*(i)\l i Y^(J*(i)\k y\Ja{i)\l i I"

aeDii

< h li )qij(Ja{i)\h y\Ja(i)\li\s ^

11

. - . . v

K

2MBri (x,))qu(Br i (xi)y(2r i)s)

where K = -————-. Let c= —, hence, (2A)*Ai(q,t) K '

ch = chxq,t,s(E) ^ V^J(£) ^ V^(E) ^ -(E).

Let h = +8. For any e > 0, there is a sufficiently large n, such that

S KJ*)9u(J*y\Ja\S > 1.

*eDn

Then, for 6 = 6(n) > 0 and a centered ^-covering {Bri(xi)} of the Moran set E, we have

Xq,t,s (E) ^YjXq,t,s(Bri (xi)) ^ 2 Xq,t,s (J*(i)\k ) = i i,\o(i)\h\=n

V s ^(J*)q"(J*y\J*\s " 2 * (l)) (ri(1 (

*eD„

Then

22

Vff (E) ^ n%(E) ^ V^(E) ^ -^Xq^E) " JK.

Therefore, (E) = +8. □

Proposition 5. Let A > 0.

1) If 0 < h < +8, there are some constants A,B, such that Ah ^ (E) ^ Bh.

2) If h = 0, then, (E) = 0.

3) If h = +8, then, (E) = +8. Proof.

1) Since h > 0, we can find n, such that

S mj.* )qv(J* y\J*\s > h.

*eDn

Let 5 > 0, for a e Dn, take r e Dm, such that \J**T\ < S, and J**T c J*. Let x e E x J**T. Let 0 < r < 1, such that J* contains a ball Br* (x) of radius r* = r\J**T\/2. The collection of balls {Br* (x)} is a ^-packing of E. Thus,

(Cmin)m \ J* \ ^ \ J**r \ ^ (C max )m\J*\,

(Pmin)mMJ*) ^ »(J**r) ^ (Pmax)m»(J*(2) {(Pmin)mV(J*) ^ V(J**r) ^ (Pmax)mV(J*). For any (q,t,s) e E3, we get

M(J**r )q > ki»(J* )q, v(J**r ) ^ k2 u(J* ), \J**T\s > k3\J* \S.

Then

S v(J**r)qv(J**TY\J**r\s > hk2k3 S v(J*)qv(J*)V*\s > Ah,

*eD„ *eD„

where A = kik2k3/2. This implies that (E) ^ Ah. On the other hand, since h > 0, there is a sufficiently large n, such that

S M(J*)qv(J*)V*\s < 2h.

*eD„

Let 5 > 0, and {Br.(xi)} be a centered ^-packing of the Moran set E. For i e N, we can choose a(i) e Dn, such that xi e J*.. We can found ki, U e N, such that

\Ja(i)\ki \ ^ ri < \Ja(í)\ki-l\, and \J*(i)\k \ ^ ri < \J*(i)\U-l\.

If s ^ 0, then (2ri)s ^ 2s\J*(i)\ki\s, if s > 0. Hence,

2

(2ri)s ^ 2S\J*(i)\ki-i\s ^ \J*(i)\ki\s.

min

Therefore,

(2s s < 0

(2rty < C(s)\Jami\s, where C(s) = ,^ < (3)

V ^ ^min '

There exists a probability measure xg,t,s supported on E, for which,

Xg,t,s {J<r(i)) =

gp( ^ VI ^.n|s

2 ^{Ja(i))gV

gt r\Vi T r\\s

alt)) I'-1alt)\

apD\<r(i)|

By Proposition 3 and (3), for {q,t,s) e E3, we get 2(xt)gu(Bn(xi)t{2ri)s <

< B2{q,t)C (S) 2 ^{Ja{i)\ki )g v(Ja{i)\ki )t\Ja{i)\ki\S =

aeD„

_ rw ,\ni V{Ja(i)\ki) gV {Ja(i)\ki f\Ja{i)\ki\S V1 I T \g

= B2{q,t)C{s^--s 2j »{Ja) V {Ja) \J.a \ <

i /-I ^{Ja) ^ {.Ja) \ -J a \ aeDk.

aeDk.

< 2B2{q,t)C(s)h2Xg,t,s{Jad)\ki) < 2B2{q,t)C(s)h2Xg,t,s(Bn(xt)) = = 2B2.(Q,t)C(s)hYgt.J\ \B

{q,t)C{s)hXg,t,s( U BnPxi)) < 2B2{q,t)C{s)h.

Hence, it follows that Vq^s(E) < Vj^0(E) < Bh, where B = 2B2{q,t)C(s). 2) Let h = 0. Then, for any e > 0, there is a sufficiently large n, such that

2 V(Ja)gV(Ja)Va\S <

aeD„

Let 8 > 0, and {Br.{xt)} be a centered ^-packing of E. Then, for i e N, we can choose a(i) e Dn, xt e Japt), and ki,li e N, such that

\Ja(i)\ki \ < Ti < \ J a pi)\ki-l\, and A\Japt)\k \ < Ti < A\JaPi)\k-l\-

Then, Japi)\ki Q Bri(Xi), and E x Bn(xi) Q Japi)\ii. If s < 0, then (2r-i)s < 2s\Japi)\ki\s. If s > 0, then

2s

(2ri)S < 2S\Japi)\ki-l\S < \Japi)\ki\S-

^min

Therefore, (2rl)s ^ C(s)\J*mz\s. By Proposition 3,

»(Bri(xl))qp(Bri(xz)Y ^ B2(q,tMJ*(i)\ki)qv(J*(i)\ki)*.

Then,

S»(Bn(x%))qv(Bn(xl))t(2r) ^

^ C(s)B2(q,t)J]^(J*(i)\ki)qHJ*(i)\kiY\J*(i)\ki\s = c(s)B2(q,t)S ^ki)MTJ*qm;\]\Jafi^ S v(J*y-(J*)\J*\s ^

i Zj ^J* ) U(Ja ) \J* \ *PDk.

*eDki

^ £ C (s )B2(q, t) Sx,,t (J*(i)\ki) ^ £C (s)B2(q, t) Sxq,t (Bn (xt)) ^

<

eC (s )B2(q, t)xg,t (UBri ^)) ^ eC (s)B2(q, t).

Since e is small enough, then (E) = 0.

3) Since h = +8, then, for any e > 0, we can find an infinite number of integers n with

S » (J*YHJ*)t\J*\s> 1. (4)

*eD„

Now, take any open set U intersecting E; it contains a basic interval, say J*0 for a0 e Dn. For any a e Dk, re Dk+in, it follows from the definitions of » and v that

» (J**r) » (J*o*T) V (J**T) V (J*0*T) and \J**T \ \J*o*r \

V (J*) V (J*0) , V(J*) V(J*0) , \J* \ \J*0 \ .

Combining with (4), we get

^ ] » (Ja0*T) V(J*0*T) \J*0*T \ =

*eDk+i

,n

V (J*0 ) v (J*0 ) V (J**T) V (J**T ) \J*o\S\J**r\S

» (j* )qu(J* )\J*\\

y 1 ß {Jap Y " (Jap ) \Jyp\S y 1 ß {Jap Y V {Jap ) \J°P \

e ß (Ja Y V (Ja Y\Ja\S ^ ß (Ja Y " (Ja Y \Ja \

aeDk

By using a similar argument as in the proof of the assertion above, and (2), we learn that there are three positive constants k\, k2 and k3, such that, as e ^ 0

T,q,t,s(-p^ t \ ^ hfahv {J<rp Y V {J°p Y\Jap k^h

aeDk

This gives:

Vïiï{E X U) * X Iao) = +8.

Using techniques as in [27, Theorem 2], we obtain {E) = +8. □ Proposition 6. Let A > 0. For {q,t, s) e R3, we have:

^^{E) ^ V^{E).

Proof. By Proposition 2, the measures ^ and v satisfy the doubling condition. Therefore, the required inequality follows from [23, Theorem 1]. □

Proposition 7. Assume that A > 0. We have:

{q,t) = Z {q,t), and B^ {q,t) = f {q,t).

Proof. We will prove the first assertion. The proof of the second assertion is very similar and is therefore omitted. Let s < r{q,t). We can find k0 e N, such that for any k * k0 we get s < rk{q,t). This gives:

2 » {J* Y v {J* Y \Ja Is * 2 » {J° Y u {J° ) \J"\Tk ^ " L

aeDk aeDk

Then

liming ß (Ja Y U (Ja Y \Ja\S > 0.

aeDk

By using Proposition 4, we have W^1f{E) > 0, for all s < r{q,t), which implies that bß>v{q,t) ^ r{q,t).

Now fix s > t{q, t). We can find a sequence {ki)i, such that rki{q, t) < s, for any sufficiently large i. Let F Q E, and a p Dk., such that F x Ja ^ 0;

s

then we can choose xi e FxJ*(xi). This shows that (B\Ja(,x.)\(xi)) is a centered (cmax)ki-covering of F. By Proposition 2, and B^\Jiylv. \ (xi) x E c J*, for all a e Dki, we can choose positive constants Vo(A) > 0 and Vi(A) > 0, such that

V(B\J* (Xi)\(x)) ^ \B\Ja (Xi)\(x)) ^ V( (A)

»(Ja (xi)) ^ »{BA\Jr (xH)\(xi)) ^ 0 ,

and

v{B\J.(xi)^) ^ v{B\j„(xi)^) ^ V((A)

y{J*(xi)) ^ "(BA\Ja(xi)\(xi)) ^ '

Then there is a constant C( , , > 0 with

2 Jo {xi)\Pxi))qV (.B\Ja{xi)\(xi jf (2\Ja M)*

^ C(q,t,s)^(J*(xt))qu(j*(xl))t\J*(x,)\s ^ ^C(q ,t,s) S P (J* Yv(J° )\J*\S ^

*eDki

^ C (q ,t,s) S » (J* )q" (J* )\J* \ Tki(q'l) = C (q ,t,s).

*eDki

We now deduce that V^,o( F) ^ C(q,t,s), and (E) < +8, for all

s > r(q,t). Finally, we conclude that b^>v(q,t) ^ r(q,t). □

>

Proposition 8. Assume that A > 0, , ^Bt^) exists, and that

liming v (J* )q » (J* j \J* \T{q ,f) > 0, for all (q, t ) e R2.

<reDk

If (a, /3) = — (^, ^), then H^T{q,t) (K(a,/3)j > 0. Proof. By using Proposition 4, we have

^tfPq*)(E) ^ A liminf y v (J*)qu(J*)t\J*\T-Pq ft) > 0.

aeDk

Now, for a = (a,3) e R2, consider the following sets:

r log jv(Br(x))) ^ . log (v(Br(x))) !

ta = \x; limsup-^-- > a, or limsup-^-- > /3>,

<- r^o log r r^o log r J

F* = ix; liminf —givi—r( ^ < a, or liminf ——r( ^ < /), I r^o log r r^o log r J

j-i2 [ v log (MBr O^) r . plog(V(—r (x))) ) F a = ix; limsup--- > a, or limmi--- < P(,

(. r^o log r r^o log r )

and

j-,3 ( r ■ rlOg{M(Br (x)) r iOg^U(Br (x))

F3 = ix; liminf-^-- < a, or limsup-^-- > p >.

I r^o log r r^0 log r )

The a-subadditivity of the mutual Hausdorff measure allows this:

nj,t,T(q,t)(Fa) = 0, for every a > , and p > , (5)

Uff(q' f)( F3) = 0, for every a < -M^, and p <J-I^lA, (6) ^ 3 dq dt

V^T(q,t)(f2) = 0, for every a > , and p < , (7)

and

Klz{q't)(F*) = 0, for every a < —MM, and / > J-^A. (8) ^ a dq dt

The proof of (6)-(8) follows by using the same ideas as in the proof of (5). So, we will prove (5). Let a > — dTpq ^ and / > — ; then, there is h > 0 with

r(q — h,t) < r(q,t) + ah, and r(q,t — h) < r(q,t) + /h. It follows from Proposition 4 that

liminf y v (J*)q~h" (J*f\J*\~{q't)+ah

aeD,

liminf y v (J*YHJ*)h V*\T{q't)+i)h " 0.

*eDn

Let £ > 0; there are a sequence (n(i))i and i0 e N, such that if i ^ z0, we get

y v (J* ) q~h" (J* )t\J* \rM 't)+ah < ^ (9)

*eDn(i)

and

2 » {Ja )9 * {Ja ) t~H \Ja I^^ < £■ (10)

aeDn(i)

We need to show that Uf^" ,t](F) = 0, for any F c Fa. Let x e F Q Fa, then there exists rx > 0 with ^(Brx (x)) < or v(Brx (x)) < rf. We have

^{Brx {x))qu{Brx {x))\2rx)^^ ^

'»{Brx {x)) i-hv {Brx {x)f{2rx) ^ ,t)+ah, or

{^{Brx {x))qu{Brx {x))t-h{2rx)T(9 *)+iih.

(11)

If {Br* {x*))i and {Br** {x**))i are, respectively, a centered ^-covering of F and E\F, then {Br. {xi))i = (^Br* {x*) y Br** {x**)^ is a centered

^-covering of E. It follows from Proposition 3 and (3) that for all {q,t,s) e R3 :

2 MBr* {x*))qu{Br* {x*))t{2r*)s ^ 2 {xi))q* {Br, {Xi))t{2ri)s ^

^ V [Ja{i)\n(i){Xi )) 9 V [Ja{ï)\n{ï){Xi])t \Ja(i)\n(i){Xi)\S,

where k1,k2 and k3 are suitable constants. Let K = k1k2k3; using (9) and (10), we have:

2 MBr? {x*)) q-hU {Br* {x*))\2 T*y-(q * ï+ah ^ eK 2 Xq-h,t,r_ {q ,t )+ah{Bn {Xi)),

or

2 MBr? {x*)qU{Br* {x*))t-h{2r*y-(q**>+lh ^ eK 2 Xq,t-h,r_(q,t)+l3h{Bn {Xi)). It follows from (11) that

Kiïq,t){F) ^ 2» K* №))№)) {2r*f(q,t) ^

^ £^2 maX (Xq-h,t,r(q,t)+ah {Bn {Xi)) , Xq,t-h,r(q,t)+?h {Bn {Xi))) .

We deduce that

,q,t, T_(q ,t)

1 \F) < max{% q-h,t,T(q,t) + ah{Bn {Xi)) ,Xq,t-h,r(q,t) + flh {Bn {Xi))).

Since (Bri(xi))i is a centered ^-covering of the set E, then by using Besicovitch's covering theorem, there exists £ = ^(n) finite sub-families

(Bri. (x^Dj,..., (Bra (x^j))j fulfilling the following: for each ie{1, 2,..., £}, 3 ?

the family (Brij (x^))j is a ^-packing of E, and E c [^J [^J Brij (xij). Which

i"i j

implies that

* \ F) ^

^ e K SS max (Xg-h,t,T(g ,t )+ah (-Bnj (xij)) ,Xq, t-h,r (g, t)+^h(yBrij (xij= i"i j

= £K Smax( xg-h,t, r(g, t)+ah

(U Bri, , Xg,t-h,r(g,t)+fih(KU B rij (xij)jl ^ i=i j j

^ £K£ —> 0 as e ^ 0,

which implies that \F) = 0, then, V^f(q\Fa) = 0. □

Proof of Theorem 2. By using the convexity and differentiability of the function r, we have

i) The function t* is concave.

ii)

{(a,p) e E \ Z'(a,p) > . ( - ,(R»>-

iii)

r*(a,p ) =

aq + pt + r(q, t), for (q, t) e E2, (a,p)e ((MM, MM)) (E2),

-8, for (a,p) R (-(^, ^)) (E2).

(12)

It follows now from [23], [24] that

Uq,?t(q^ (K(a,p)) ^ Vqa+tr+T(q't)-s (K(a,p)),

for any 0 < 8 < qa + tft + r(q, t). By Proposition 8, we get

nga+tl3+Tpg,Q-S (K(a,p)) > 0,

and this gives

dimH(K(a, ft)) ^ qa + tft + r(q,t) - 8, for any 0 < 8 < qa + tft + r(q,t).

Letting 8 ^ 0 yields that

dimH (K (a, ft)) ^ qa + tft + r (q,t). (13)

Now, by using [23], [24], and Proposition 7, we obtain

dimH(K(a, ft)) < qa + tft + r(q,t). (14)

It follows from (12)-(14) that

dimH(K(a, ft)) = qa + tft + r(q,t) = r*(a,ft).

The desired result follows immediately from Proposition 7.

The proof of the second statement is very identical to the proof of the first. The third follows from the first and the second statements.

Acknowledgment. We would like to thank the anonymous referees for their valuable comments and suggestions that led to the improvement of the manuscript. The first author was supported by Analysis, Probability & Fractals Laboratory (No: LR18ES17).

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Received September 4, 2022. In revised form, December 6, 2022. Accepted December 10, 2022. Published online February 1, 2023.

Bilel Selmi

Analysis, Probability and Fractals Laboratory LR18ES17 Faculty of sciences of Monastir Department of mathematics 5000-Monastir-Tunisia

E-mail: bilel.selmi@fsm.rnu.tn, bilel.selmi@isetgb.rnu.tn

Nina Yu. Svetova Petrozavodsk State University 33 Lenina pr., Petrozavodsk 185910, Russia E-mail: nsvetova@petrsu.ru