Citation: Bernardo Lopez Abel, María I Martínez-Soto, Maria Luz Couce. Integrative cardiology-state of the art of mind body therapies for the treatment of cardiovascular disease and risk factors[J]. AIMS Medical Science, 2018, 5(1): 80-89. doi: 10.3934/medsci.2018.1.80
[1] | Xuesong Si, Chuanze Niu . On skew cyclic codes over $ M_{2}(\mathbb{F}_{2}) $. AIMS Mathematics, 2023, 8(10): 24434-24445. doi: 10.3934/math.20231246 |
[2] | Ismail Aydogdu . On double cyclic codes over $ \mathbb{Z}_2+u\mathbb{Z}_2 $. AIMS Mathematics, 2024, 9(5): 11076-11091. doi: 10.3934/math.2024543 |
[3] | Fatma Zehra Uzekmek, Elif Segah Oztas, Mehmet Ozen . $ (\theta_i, \lambda) $-constacyclic codes and DNA codes over $ \mathbb{Z}_{4}+u\mathbb{Z}_{4}+u^{2}\mathbb{Z}_{4} $. AIMS Mathematics, 2024, 9(10): 27908-27929. doi: 10.3934/math.20241355 |
[4] | Wei Qi . The polycyclic codes over the finite field $ \mathbb{F}_q $. AIMS Mathematics, 2024, 9(11): 29707-29717. doi: 10.3934/math.20241439 |
[5] | Hatoon Shoaib . Double circulant complementary dual codes over $ \mathbb{F}_4 $. AIMS Mathematics, 2023, 8(9): 21636-21643. doi: 10.3934/math.20231103 |
[6] | Shakir Ali, Amal S. Alali, Kok Bin Wong, Elif Segah Oztas, Pushpendra Sharma . Cyclic codes over non-chain ring $ \mathcal{R}(\alpha_1, \alpha_2, \ldots, \alpha_s) $ and their applications to quantum and DNA codes. AIMS Mathematics, 2024, 9(3): 7396-7413. doi: 10.3934/math.2024358 |
[7] | Turki Alsuraiheed, Elif Segah Oztas, Shakir Ali, Merve Bulut Yilgor . Reversible codes and applications to DNA codes over $ F_{4^{2t}}[u]/(u^2-1) $. AIMS Mathematics, 2023, 8(11): 27762-27774. doi: 10.3934/math.20231421 |
[8] | Hongfeng Wu, Li Zhu . Repeated-root constacyclic codes of length $ p_1p_2^t p^s $ and their dual codes. AIMS Mathematics, 2023, 8(6): 12793-12818. doi: 10.3934/math.2023644 |
[9] | Jianying Rong, Fengwei Li, Ting Li . Two classes of two-weight linear codes over finite fields. AIMS Mathematics, 2023, 8(7): 15317-15331. doi: 10.3934/math.2023783 |
[10] | Yang Pan, Yan Liu . New classes of few-weight ternary codes from simplicial complexes. AIMS Mathematics, 2022, 7(3): 4315-4325. doi: 10.3934/math.2022239 |
A finite collection of strictly contractive maps on the real line is called an iterated function system (IFS). Let $ \Phi = \{ \varphi_a \}_{a \in \Lambda} $ be an IFS and $ p = (p_a)_{a \in \Lambda} $ be a probability vector. It is well-known that there exists a unique Borel probability measure $ \nu $, called the invariant measure, such that
$ \nu = \sum\limits_{a \in \Lambda} p_a \cdot \varphi_a \nu, $ |
where $ \varphi_a \nu $ is the push-forward of $ \nu $ under the map $ \varphi_a : \mathbb{R} \to \mathbb{R} $.
When the construction does not involve complicated overlaps (say, under the strong separation condition), the invariant measures are relatively easy to understand. For example, if the strong separation condition holds, then the invariant measure $ \nu $ is supported on a Cantor set and is singular, and the dimension of $ \nu $ is given by
$ dimν=hχ, $
|
where $ h = h(p) $ is the entropy and $ \chi = \chi(\Phi, p) $ is the Lyapunov exponent.
In this paper we consider IFS with inverses (i.e., IFS that contain inverse maps). IFS with inverses were first introduced by the author in [5], motivated by the Furstenberg measure. See also [6]. We show that the invariant measures for IFS with inverses agree with the invariant measures for associated graph-directed IFS under the suitable choice of weight. The main results of [5] and [6] follow directly from our result.
The paper is organized as follows: In section 2, we recall IFS with inverses and state the main result. Section 3 is devoted to preliminary lemmas. In section 4 we prove the main result.
Let $ G $ be the free group of rank $ r \geq 2 $, and let $ W $ be a free generating set of $ G $. Let $ \Lambda $ be a set that satisfies
$ W⊂Λ⊂W∪W−1, $
|
where $ W^{-1} = \{ a^{-1} \}_{a \in W} $. Let $ \mathcal{E}^{*} = \bigcup_{n \geq 1} \Lambda^n $ and $ \mathcal{E} = \Lambda^{\mathbb{N}} $. For $ \omega = \omega_0 \omega_1 \cdots $ we denote $ \omega|_n = \omega_0 \cdots \omega_{n-1} $. For $ \omega, \xi \in \mathcal{E} \cup \mathcal{E}^{*} $ we denote by $ \omega \wedge \xi $ their common initial segment. For $ \omega \in \mathcal{E}^{*} $ and $ \xi \in \mathcal{E} \cup \mathcal{E}^{*} $, we say that $ \omega $ precedes $ \xi $ if $ \omega \wedge \xi = \omega $.
Let $ p = (p_a)_{a \in \Lambda} $ be a non-degenerate probability vector, and let $ \mu $ be the associated Bernoulli measure on $ \mathcal{E} $. We say that a (finite or infinite) sequence $ \omega \in \mathcal{E}^{*} \cup \mathcal{E} $ is reduced if $ \omega_{i} \omega_{i+1} \neq a a^{-1} $ for all $ i \geq 0 $ and $ a \in \Lambda $. Let $ \Gamma^{*} $ (resp. $ \Gamma $) be the set of all finite (resp. infinite) reduced sequences. For $ \omega \in \Gamma^{*} $ we denote the associated cylinder set in $ \Gamma $ by $ [\omega] $. Define the map
$ red:E∗→Γ∗ $
|
in the obvious way, i.e., $ \mathrm{red}(\omega) $ is the sequence derived from $ \omega $ by deleting all occurrences of consecutive pairs $ aa^{-1} \ (a \in \Lambda) $. Let $ \overline{\mathcal{E}} \subset \mathcal{E} $ be the set of all $ \omega $ such that the limit
$ limn→∞red(ω|n) $
|
(1) |
exists. For example, for any $ a \in \Lambda $ we have $ a a a \cdots \in \overline{\mathcal{E}} $ and $ a a^{-1} a a^{-1} \cdots \notin \overline{\mathcal{E}} $. By abuse of notation, for $ \omega \in \overline{\mathcal{E}} $ we denote the limit (1) by $ \mathrm{red}(\omega) $. The following is well-known (see, e.g., chapter 14 in [3]):
Lemma 2.1. There exists $ 0 < \ell \leq 1 $ ($\textrm{drift}$ or $\textrm{speed}$) such that
$ \lim\limits_{n \to \infty} \frac{1}{n} \left| \mathrm{red}( \omega|_{n} ) \right| = \ell $ |
for $ \mu $-a.e. $ \omega \in \mathcal{E} $. In particular, $ \overline{\mathcal{E}} $ has full measure.
Denote
$ \Lambda^{\star} = \left\{ (a, b) \in \Lambda^2 : a \neq b^{-1} \right\}. $ |
For $ a \in \Lambda $, write $ \mathbb{R}_a = \mathbb{R} \times \{a\} $. We freely identify $ \mathbb{R}_a $ with $ \mathbb{R} $ below. Let $ X_a \subset \mathbb{R}_a \ (a \in \Lambda) $ be open intervals and write $ X = \bigcup_{a \in \Lambda} X_a $. Assume that there exist $ 0 < \gamma < 1 $ and $ 0 < \theta \leq 1 $ such that for all $ (a, b) \in \Lambda^{\star} $, the map $ \varphi_{ab} : X_b \to X_a $ is $ C^{1 + \theta} $ and satisfies
(ⅰ) $ \mkern 1.5mu\overline{\mkern-1.5mu{ \varphi_{ab} (X_{b}) }\mkern-1.5mu}\mkern 1.5mu \subset X_{a} $;
(ⅱ) $ 0 < | \varphi'_{ab}(x) | < \gamma $ for all $ x \in X_b $;
(ⅲ) $ \varphi_{ab}^{-1} : \varphi_{ab}(X_b) \to X_b $ is $ C^{1 + \theta} $.
We say that $ \Phi = \{ \varphi_{ab} \}_{(a, b) \in \Lambda^{\star}} $ is an IFS with inverses. For $ \omega = \omega_0 \cdots \omega_n \in \Gamma^{*} $, we denote
$ \varphi_{\omega} = \varphi_{ \omega_ 0\omega_1 } \circ \cdots \circ \varphi_{ \omega_{n-1} \omega_{n} }. $ |
Let $ \Pi: \Gamma \to X $ be the natural projection map, i.e.,
$ \Pi(\omega) = \bigcap\limits_{n \geq 1} \varphi_{\omega |_{n + 1}}( \mkern 1.5mu\overline{\mkern-1.5mu{ X_{ \omega_n} }\mkern-1.5mu}\mkern 1.5mu ). $ |
Define $ \Pi^{\mathcal{E}} : \overline{\mathcal{E}} \to X $ by $ \Pi^{\mathcal{E}} = \Pi \circ \mathrm{red} $. Define the measure $ \nu = \nu(\Phi, p) $ by $ \nu = \Pi^{\mathcal{E}} \mu $ (i.e., the push-forward of the measure $ \mu $ under the map $ \Pi^{\mathcal{E}} : \overline{\mathcal{E}}\to X $). We call $ \nu $ an invariant measure. It is easy to see that if $ \Lambda = W $, then the measure $ \nu $ is an invariant measure of an IFS. Let $ \chi = \chi(\Phi, p) $ be the Lyapunov exponent, and $ h_{RW} = h_{RW}(p) $ be the random walk entropy. See section 3 in [5] for the precise definition. Fix $ x_{a} \in X_a $ for each $ a \in \Lambda $. For $ \omega \in \overline{\mathcal{E}} $ and $ n \in \mathbb{N} $ we denote $ x_{\omega, n} = x_{j} $, where $ j = j(\omega, n) \in \Lambda $ is the last letter of $ \mathrm{red}(\omega|_n) $.
Proposition 2.1 (Proposition 3.1 in [5]). We have
$ \chi = -\lim\limits_{n \to \infty} \frac{1}{n} \log \big| \varphi'_{ \mathrm{red} ( \omega|_n ) } ( x_{ \omega, n } ) \big| $ |
for $ \mu $-a.e. $ \omega $.
Notice that an IFS with inverses $ \Phi = \{ \varphi_{ab} \}_{(a, b) \in \Lambda^{\star}} $ does not have any explicit inverse map. The next example illustrates why we call $ \Phi $ an IFS with inverses. For more detail, see Example 2.1 and Appendix in [5].
Example 2.1. Let $ r = 2 $, $ W = \{ 0, 1 \} $ and $ \Lambda = \{ 0, 1, 1^{-1} \} $. For $ 0 < k, l < 1 $, define
$ f0(x)=kx, f1(x)=(1+l)x+1−l(1−l)x+1+l. $
|
Let $ f_{1^{-1}} = f^{-1}_1 $. It is easy to see that we have $ f_0(0) = 0 $, $ f_1(-1) = -1 $, $ f_1(1) = 1 $ and $ f'_0(0) = k $, $ f'_1(1) = l $. It is well-known that there exists a unique Borel probability measure $ \nu $ that satisfies
$ ν=∑a∈Λpafaν. $
|
The above measure is called a Fustenberg measure. See., e.g., [2]. Let
$ Y0=(−k,k), Y1=(f1(−k),1) and Y−1=(−1,f−1(k)). $
|
Then we have
$ fa(Y∖Ya−1)⊂Ya, $
|
for all $ a \in \Lambda $, where $ Y = \bigcup_{a \in \Lambda} Y_a $ and $ Y_{0^{-1}} = \emptyset $. Notice that the sets $ \{ Y_a \}_{a \in \Lambda} $ are not mutually disjoint if and only if $ k > f_1(-k) $, which is equivalent to
$ \sqrt{l} > \frac{ 1 - k }{ 1 + k }. $ |
It is easy to see that there exist open intervals $ X_0, X_1, X_{1^{-1}} \subset \mathbb{R} $ such that
$ Ya⊂Xa and ¯fa(X∖Xa−1)⊂Xa $
|
for all $ a \in \Lambda $, where $ X = \bigcup_{a \in \Lambda} X_a $ and $ X_{0^{-1}} = \emptyset $. Then $ \{ f_a |_{X_b} \}_{ (a, b) \in \Lambda^{\star} } $ is an IFS with inverses, and the associated invariant measure agrees with $ \nu $. For the proof, see the Appendix in [5].
Denote
$ \dim \nu = \inf \left\{ \dim_{H} Y : \nu(\mathbb{R} \setminus Y) = 0 \right\}. $ |
Proposition 2.2 (Proposition 3.3 in [5]). Assume that for all $ a \in \Lambda $, the sets $ \{ \mkern 1.5mu\overline{\mkern-1.5mu{\varphi_{ab}(X_b)}\mkern-1.5mu}\mkern 1.5mu \}_{b \in \Lambda_a^{\star}} $ are mutually disjoint, where
$ \Lambda_a^{\star} = \{ b \in \Lambda : (a, b) \in \Lambda^{\star} \}. $ |
Then we have
$ \dim \nu_a = \frac{h_{RW}}{\chi} $ |
for all $ a \in \Lambda $.
Given an IFS with inverses $ \Phi = \{ \varphi_{ab} \}_{(a, b) \in \Lambda^{\star}} $, one can naturally associate a graph-directed IFS by restricting transitions from $ a $ to $ a^{-1} $ for all $ a \in W $. For the precise definitions of graph-directed IFS, see section 1.7 in [1].
Let $ \widetilde{P} = (\widetilde{p}_{ab}) $ be a $ |\Lambda| \times |\Lambda| $ stochastic matrix that satisfies $ \widetilde{p}_{ab} > 0 \ (a \neq b^{-1}) $ and $ \widetilde{p}_{ab} = 0 \ (a = b^{-1}) $. Let $ \widetilde{p} = (\widetilde{p}_1, \cdots, \widetilde{p}_{N}) $ be the unique row vector satisfying $ \widetilde{p} \widetilde{P} = \widetilde{p} $. Let $ \widetilde{\mu} $ be the probability measure on $ \Gamma $ associated with $ \widetilde{P} $ and $ \widetilde{p} $. Define the measure $ \widetilde{\nu} = \widetilde{\nu}(\Phi, \widetilde{P}) $ by $ \widetilde{\nu} = \Pi \widetilde{\mu} $. For $ a \in \Lambda $, denote $ \widetilde{\nu}_a = \widetilde{\nu}|_{X_a} $. It is easy to see that
$ \widetilde{\nu}_a = \sum\limits_{(a, b) \in \Lambda^{\star}} \widetilde{p}_{ab} \cdot \varphi_{ab} \widetilde{\nu}_b. $ |
Let $ \widetilde{h} = \widetilde{h}(\widetilde{P}) $ be the entropy and $ \widetilde{\chi} = \widetilde{\chi}(\Phi, \widetilde{P}) $ be the Lyapunov exponent, i.e.,
$ \widetilde{h} = -\sum\limits_{(a, b) \in \Lambda^{\star}} \widetilde{p}_{a} \widetilde{p}_{ab} \log \widetilde{p}_{ab}, $ |
and
$ \widetilde{\chi} = -\lim\limits_{n \to \infty} \frac{1}{n} \log \big| \varphi'_{ \omega|_n } ( x_{\omega_n} ) \big| $ |
for $ \widetilde{\mu} $-a.e. $ \omega $. Under the separation condition, we obtain the following. The argument is classical, so we omit the proof. See, e.g., the proof of (2.6) in [4].
Proposition 2.3. For every $ a \in \Lambda $, assume that the sets $ \{ \mkern 1.5mu\overline{\mkern-1.5mu{ \varphi_{ab}(X_b) }\mkern-1.5mu}\mkern 1.5mu \}_{b \in \Lambda_a^{\star}} $ are mutually disjoint. Then we have
$ \dim \widetilde{\nu}_{a} = \frac{ \widetilde{h} }{ \widetilde{\chi} } $ |
for all $ a \in \Lambda $.
Our main result is the following:
Theorem 2.1. Let $ \Phi = \{ \varphi_{ab} \}_{(a, b) \in \Lambda^{\star}} $ be an IFS with inverses. Then there exists a stochastic matrix $ \widetilde{P} $ such that
$ \nu = \widetilde{\nu}, \ h_{RW} = \ell \widetilde{h} \ and \ \chi = \ell \widetilde{\chi}. $ |
Since the graph directed IFS has essentially the same structure as IFS, by the above theorem most of the results of IFS can be immediately extended to IFS with inverses. For example, the main results of [5] and [6] follow directly from the above result.
Define $ \mu_{\mathrm{red}} $ by $ \mathrm{red} \, \mu $, i.e., the push-forward of the measure $ \mu $ under the map $ \mathrm{red} : \mathcal{E} \to \Gamma $. From below, for $ n > 0 $, which is not necessarily an integer, we interpret $ \omega|_n $ to be $ \omega|_{\lfloor n \rfloor } $. The following lemma is immediate.
Lemma 3.1. We have
$ \chi = -\lim\limits_{n \to \infty} \frac{1}{n} \log \big| \varphi'_{ \omega|_{\ell n} } ( x_{ \omega, \ell n } ) \big| $ |
for $ \mu_{\mathrm{red}} $-a.e. $ \omega \in \Gamma $.
Proof. Let $ \omega \in \Gamma $, and let $ \eta \in \mathcal{E} $ be such that $ \omega = \mathrm{red} (\eta) $. We can assume that $ \eta $ satisfies
$ \lim\limits_{n \to \infty} \frac{1}{n} \left| \mathrm{red}( \eta|_{n} ) \right| = \ell $ |
and
$ \chi = -\lim\limits_{n \to \infty} \frac{1}{n} \log \big| \varphi'_{ \mathrm{red} ( \eta|_n ) } ( x_{ \eta, n } ) \big|. $ |
Let $ \epsilon > 0 $, and let $ n \in \mathbb{N} $ be sufficiently large. Then, since $ \omega|_{(\ell - \epsilon)n} $ precedes $ \mathrm{red}(\eta|_n) $ and $ | \mathrm{red}(\eta|_n) | < (\ell + \epsilon)n $, we have
$ -\log \big| \varphi'_{ \omega|_{(\ell - \epsilon)n} } ( x_{ \omega_{(\ell - \epsilon)n} } ) \big| < n \chi < -\log \big| \varphi'_{ \omega|_{(\ell - \epsilon)n} } ( x_{ \omega_{(\ell - \epsilon)n} } ) \big| - 2 \epsilon n \cdot \log \lambda_{\min}, $ |
where
$ \lambda_{\min} = \min \{ | \varphi'_{ab}(x) | : (a, b) \in \Lambda^{\star}, \, x \in X_b \}. $ |
The result follows from this.
For $ \omega \in \Gamma^{\ast} $, we denote
$ \mathcal{E}_{\omega} = \left\{ \upsilon \in \overline{\mathcal{E}} : \omega \text{ precedes } \mathrm{red}(\upsilon) \right\} $ |
and
$ \hat{\mathcal{E}}_{\omega} = \left\{ \upsilon \in \overline{\mathcal{E}} : \text{there exists } n \in \mathbb{N} \text{ s.t.} \ \mathrm{red}(\upsilon|_{n}) = \omega \right\}. $ |
Notice that $ \mathcal{E}_{\omega} \subset \hat{\mathcal{E}}_{\omega} $. For $ a \in \Lambda $, write
$ q_{a} = {1−μ(Ea−1)(a−1∈Λ)1(a−1∉Λ) $
|
and $ p_{a} = \mu(\hat{\mathcal{E}}_a) $. We next prove the following crucial lemma.
Lemma 3.2. Let $ \omega \in \Gamma^{\ast} $ and $ a \in \Lambda $ be such that $ \omega a \in \Gamma^{\ast} $. Then we have
$ p_{a} = \frac{ \mu( \hat{\mathcal{E}}_{\omega a} ) }{ \mu( \hat{\mathcal{E}}_{\omega} ) }. $ |
Proof. Fix such $ \omega \in \Gamma^{\ast} $ and $ a \in \Lambda $. Notice that
$ \hat{\mathcal{E}}_{\omega} = \bigsqcup\limits_{i = |\omega|}^{\infty} \hat{\mathcal{E}}_{\omega}^{(i)}, $ |
where
$ \hat{\mathcal{E}}_{\omega}^{(i)} = \left\{ \upsilon \in \overline{\mathcal{E}} : \mathrm{red}( \upsilon|_i ) = \omega, \ \mathrm{red}( \upsilon|_k ) \neq \omega \ (|\omega| \leq k < i) \right\}. $ |
Then, since
$ \hat{\mathcal{E}}_{\omega a} \cap \hat{\mathcal{E}}^{(i)}_{\omega} = \big\{ \upsilon \in \overline{\mathcal{E}} : \sigma^{i}(\upsilon) \in \hat{\mathcal{E}}_{a} \big\} \cap \hat{\mathcal{E}}_{\omega}^{(i)}, $ |
we have
$ \mu( \hat{\mathcal{E}}_{\omega a} \cap \hat{\mathcal{E}}^{(i)}_{\omega} ) = p_a \cdot \mu( \hat{\mathcal{E}}_{\omega}^{(i)} ). $ |
Therefore,
$ μ(ˆEωa)=μ(∞⨆i=|ω|ˆEωa∩ˆE(i)ω)=∞∑i=|ω|pa⋅μ(ˆE(i)ω)=pa⋅μ(ˆEω). $
|
Similarly, we have the following:
Lemma 3.3. For $ a \in \Lambda $ and $ \omega = \omega_0 \cdots \omega_n \in \Gamma^{\ast} $ with $ \omega_n = a $, we have
$ \mu( \mathcal{E}_{\omega} ) = q_a \cdot \mu( \hat{ \mathcal{E} }_{\omega} ). $ |
Proof. Fix such $ a \in \Lambda $ and $ \omega \in \Gamma^{\ast} $. Notice that
$ \hat{\mathcal{E}}_{\omega} = \bigsqcup\limits_{i = |\omega|}^{\infty} \hat{\mathcal{E}}_{\omega}^{(i)}, $ |
where
$ \hat{\mathcal{E}}_{\omega}^{(i)} = \left\{ \upsilon \in \overline{\mathcal{E}} : \mathrm{red}( \upsilon|_i ) = \omega, \ \mathrm{red}( \upsilon|_k ) \neq \omega \ (|\omega| \leq k < i) \right\}. $ |
Then, since
$ \mathcal{E}_{\omega} \cap \hat{\mathcal{E}}^{(i)}_{\omega} = \big\{ \upsilon \in \overline{\mathcal{E}} : \sigma^{i}(\upsilon) \notin \bigcup\limits_{b \in \Lambda \setminus \{a^{-1}\}} \mathcal{E}_{b} \big\} \cap \hat{\mathcal{E}}_{\omega}^{(i)}, $ |
we have
$ \mu( \mathcal{E}_{\omega} \cap \hat{\mathcal{E}}^{(i)}_{\omega} ) = q_a \cdot \mu( \hat{\mathcal{E}}_{\omega}^{(i)} ). $ |
Therefore,
$ μ(Eω)=μ(∞⨆i=|ω|Eω∩ˆE(i)ω)=∞∑i=|ω|qa⋅μ(ˆE(i)ω)=qa⋅μ(ˆEω). $
|
In this section we prove Theorem 2.1. Denote
$ \widetilde{p}_a = p_a q_a \text{ and } \widetilde{p}_{ab} = \frac{p_b q_b}{q_a}. $ |
Let $ \widetilde{P} = (\widetilde{p}_{ab}) $. By Lemma 3.2 and Lemma 3.3, we obtain the following.
Proposition 4.1. For all $ \omega = \omega_0 \cdots \omega_{n} \in \Gamma^{\ast} $, we have
$ \mu_{\mathrm{red}}( [ \omega ] ) = \widetilde{p}_{\omega_0} \widetilde{p}_{\omega_0 \omega_1} \cdots \widetilde{p}_{\omega_{n-1} \omega_n}. $ |
The above proposition implies that $ \mu_{\mathrm{red}} = \widetilde{\mu} $. Therefore, we obtain $ \nu = \widetilde{\nu} $. By Lemma 3.1, we have $ \chi = \ell \widetilde{\chi} $. It remains to show the following lemma. Notice that $ h_{RW} $, $ \ell $ and $ \widetilde{h} $ all depend only on $ \Lambda $ and $ p $.
Lemma 4.1. We have
$ h_{RW} = \ell \widetilde{h}. $ |
Proof. Let $ \Phi' = \{ \phi_{ab} \}_{(a, b) \in \Lambda^{\star}} $ be an IFS with inverses that satisfies the separation condition. Then, by Proposition 2.2 and Proposition 2.3 we obtain $ h_{RW} = \ell \widetilde{h} $.
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.
The author would like to thank the anonymous referees for the careful reading and all the helpful suggestions and remarks. Y. T. was supported by JSPS KAKENHI grant 2020L0116.
The author declares no conflict of interest.
[1] | WHO (1948) Preamble to the Constitution of WHO as adopted by the International Health Conference, In: Official Records of the World Health Organization, New York: 19 June–22 July 1946 signed on 22 July 1946 by the representatives of 61 States and entered into force on 7 April 1948, 100. |
[2] |
Münzel T, Daiber A, Steven S, et al. (2017) Effects of noise on vascular function, oxidative stress, and inflammation: Mechanistic insight from studies in mice. Eur Heart J 38: 2838–2849. doi: 10.1093/eurheartj/ehx081
![]() |
[3] | Kemppainen LM, Kemppainen TT, Reippainen JA, et al. (2017) Use of complementary and alternative medicine in Europe: Health-related and sociodemographic determinants. Scand J Public Health 2017: 1403494817733869. |
[4] |
Rice BI (2001) Mind-body interventions. Diabetes Spectrum 14: 213–217. doi: 10.2337/diaspect.14.4.213
![]() |
[5] | Bazarko D, Cate RA, Azocar F, et al. (2013) The impact of an innovative Mindfulness-Based Stress Reduction Program on the Health and Well-Being of Nurses Employed in a Corporate Setting. J Workplace Behav Health 28: 107–133. |
[6] |
Greeson JM (2009) Mindfulness research update: 2008. Complement Health Pract Rev 14: 10–18. doi: 10.1177/1533210108329862
![]() |
[7] | Barnes PM, Bloom B, Nahin RL (2008) Complementary and alternative medicine use among adults and children: United States, 2007. Natl Health Stat Report 12: 1–23. |
[8] | National Center for Complementary and Integrative Health. Available from: https://nccih.nih.gov/health/yoga. |
[9] |
AAP Section of integrative medicine (2016) Mind-body therapies in children and youth. Pediatr 138: e20161896. doi: 10.1542/peds.2016-1896
![]() |
[10] |
Montgomery GH, Hallquist MN, Schnur JB, et al. (2010) Mediators of a brief hypnosis intervention to control side effects in breast surgery patients: Response expectancies and emotional distress. J Consult Clin Psychol 78: 80–88. doi: 10.1037/a0017392
![]() |
[11] | The relaxation response. Benson H, Kliper MZ. Avon Books Pub1975;p:1-221 |
[12] | Mayden KD (2012) Mind-body therapies: Evidence and implications in advanced oncology practice. J Adv Pract Oncol 3: 357–373. |
[13] | Piepoli MF, Hoes AW, Agewall S, et al. (2017) 2016 European guidelines on cardiovascular disease prevention in clinical practice. The Sixth Joint Task Force of the European Society of Cardiology and Other Societies on Cardiovascular Disease Prevention in Clinical Practice. Eur Heart J 37: 2315–2381. |
[14] |
Goff DC Jr, Lloyd-Jones DM, Bennett G, et al. (2014) 2013 ACC/AHA guideline on the assessment of cardiovascular risk: A Report of the American College of Cardiology/American Heart Association Task Force on Practice Guidelines. Circulation 129: S49–S73. doi: 10.1161/01.cir.0000437741.48606.98
![]() |
[15] |
Lee M, Pittler M, Guo R, et al. (2007) Qigong for hypertension: A systematic review of randomized clinical trials. J Hypertens 25: 1525–1532. doi: 10.1097/HJH.0b013e328092ee18
![]() |
[16] |
Guo X, Zhou B, Nishimura T, et al. (2008) Clinical effect of qigong practice on essential hypertension: A meta-analysis of randomized controlled trials. J Altern Complement Med 14: 27–37. doi: 10.1089/acm.2007.7213
![]() |
[17] |
Elliot WJ, Izzo JL, White WB, et al. (2004) Graded blood pressure reduction in hypertensive outpatients associated with use of a device to assist with slow breathing. J Clin Hypertens 6: 553–559. doi: 10.1111/j.1524-6175.2004.03553.x
![]() |
[18] |
Schein MH, Gavish B, Herz M, et al. (2001) Treating hypertension with a device that slows and regularises breathing: A randomised, double-blind controlled study. J Hum Hypertens 15: 271. doi: 10.1038/sj.jhh.1001148
![]() |
[19] |
Anderson JW, Liu C, Kryscio RJ (2008) Blood pressure response to transcendental meditation: A meta-analysis. Am J Hypertens 21: 310–316. doi: 10.1038/ajh.2007.65
![]() |
[20] |
Greenhalgh J, Dickson R, Dundar Y (2010) Biofeedback for hypertension: A systematic review. J Hypertens 28: 644–652. doi: 10.1097/HJH.0b013e3283370e20
![]() |
[21] |
Yucha CB, Clark L, Smith M, et al. (2001) The effect of biofeedback in hypertension. Appl Nurs Res 14: 29–35. doi: 10.1053/apnr.2001.21078
![]() |
[22] |
Nakao M, Yano E, Nomura S, et al. (2003) Blood pressure-lowering effects of biofeedback treatment in hypertension: A meta-analysis of randomized controlled trials. Hypertens Res 26: 37–46. doi: 10.1291/hypres.26.37
![]() |
[23] |
Paul-Labrador M, Polk D, Dwyer JH, et al. (2006) Effects of a randomized controlled trial of transcendental meditation on components of the metabolic syndrome in subjects with coronary heart disease. Arch Intern Med 166: 1218–1224. doi: 10.1001/archinte.166.11.1218
![]() |
[24] |
Khatri D, Mathur KC, Gahlot S, et al. (2007) Effects of yoga and meditation on clinical and biochemical parameters of metabolic syndrome. Diabetes Res Clin Pract 78: 9–10. doi: 10.1016/j.diabres.2007.05.002
![]() |
[25] |
Cohen BE, Chang AA, Grady D, et al. (2008) Restorative yoga in adults with metabolic syndrome: A randomized, controlled pilot trial. Metab Syndr Relat Disord 6: 223–229. doi: 10.1089/met.2008.0016
![]() |
[26] | Liu X, Miller YD, Burton NW, et al. (2011) Qi-gong mind-body therapy and diabetes control. A randomized controlled trial. Am J Prev Med 41: 152–158. |
[27] |
Weigensberg MJ, Lane CJ, Ávila Q, et al. (2014) Imagine HEALTH: Results from a randomized pilot lifestyle intervention for obese Latino adolescents using interactive guided imagery SM. BMC Complementary Altern Med 14: 28. doi: 10.1186/1472-6882-14-28
![]() |
[28] |
Younge JO, Gotink RA, Baena CP, et al. (2015) Mind–body practices for patients with cardiac disease: A systematic review and meta-analysis. Eur J Prev Cardiol 22: 1385–1389. doi: 10.1177/2047487314549927
![]() |
[29] |
Krucoff MW, Crater SW, Gallup D, et al. (2005) Music, imagery, touch and prayer as adjuncts to interventional cardiac care: The Monitoring and Actualization of Noetic Trainings (MANTRA) II randomized study. Lancet 366: 211–217. doi: 10.1016/S0140-6736(05)66910-3
![]() |
[30] |
Wang X, Pi Y, Chen P, et al. (2016) Traditional Chinese exercise for cardiovascular diseases: Systematic review and meta-analysis of randomized controlled trials. J Am Heart Assoc 5: e002562. doi: 10.1161/JAHA.115.002562
![]() |
[31] | Cramer H, Lauche R, Paul A, et al. (2015) Mind–body medicine in the secondary prevention of coronary heart disease: A systematic review and meta-analysis. Dtsch Arztebl Int 112: 759–767. |
[32] |
Schneider RH, Grim CE, Rainforth MV, et al. (2012) Stress reduction in the secondary prevention of cardiovascular disease: Randomized, controlled trial of transcendental meditation and health education in Blacks. Circ Cardiovasc Qual Outcomes 5: 750–758. doi: 10.1161/CIRCOUTCOMES.112.967406
![]() |
[33] |
Yeh GY, Wood MJ, Lorell BH, et al. (2004) Effects of tai chi mind-body movement therapy on functional status and exercise capacity in patients with chronic heart failure: a randomized controlled trial. Am J Med 117: 541–548. doi: 10.1016/j.amjmed.2004.04.016
![]() |