Stochastic Stability of the Classical Lorenz Flow Under Impulsive Type Forcing

Abstract

We introduce a novel type of random perturbation for the classical Lorenz flow in order to better model phenomena slowly varying in time such as anthropogenic forcing in climatology and prove stochastic stability for the unperturbed flow. The perturbation acts on the system in an impulsive way, hence is not of diffusive type as those already discussed in Keller (Attractors and bifurcations of the stochastic Lorenz system Report 389, Institut für Dynamische Systeme, Universität Bremen, 1996), Kifer (Random Perturbations of Dynamical Systems. Birkhäuser, Basel, 1988), and Metzger (Commun. Math. Phys. 212, 277–296, 2000). Namely, given a cross-section $\mathcal {M}$ for the unperturbed flow, each time the trajectory of the system crosses $\mathcal {M}$ the phase velocity field is changed with a new one sampled at random from a suitable neighborhood of the unperturbed one. The resulting random evolution is therefore described by a piecewise deterministic Markov process. The proof of the stochastic stability for the umperturbed flow is then carried on working either in the framework of the Random Dynamical Systems or in that of semi-Markov processes.

Appendices

Part IV

Appendix

Here we give examples of the cross-section $\mathcal {M}$ and of the maps $T_{\eta }$ and $R_{\eta }$ discussed in the paper, as well as some comments on the results achieved in our previous paper [18]. We also present the proof of Proposition 2.

The Poincaré Section $\mathcal {M}$

Although what stated in Part I and Part II of the paper are not directly affected by a particular choice of $\mathcal {M},$ to set up the problem in a way easy to visualize we found useful to refer to the following examples.

Let us consider (2) with the parameter $\gamma ,\zeta ,\beta$ defining the classical Lorenz flow and let $c_{0}:=\left( 0,0,-\left( \gamma +\zeta \right) \right)$ be the hyperbolic equilibrium point of (2). If $O:\mathbb {R}^{3}\circlearrowleft$ is such that $O^{\text {t}}D\Phi _{0} ^{t}\left( c_{0}\right) O$ is diagonal, we can distinguish between two cases:

• 1. in the first case we choose $\mathcal {M}\equiv \mathcal {M}^{\prime },$ where

  $$\begin{aligned} \mathcal {M}^{\prime }:\mathcal {=}\left\{ y\in \mathbb {R}^{3}:\left| \left( O^{\text {t}}y\right) _{1}\right| ,\left| \left( O^{\text {t} }y\right) _{2}\right| \le \frac{1}{2},\left( O^{\text {t}}y\right) _{3}=y_{3}=1-\left( \gamma +\zeta \right) \right\} \ ; \end{aligned}$$

  (219)

• 2. in the second, we choose $\mathcal {M}$ to be the Poincaré section for the Lorenz’63 flow given in (2) constructed in [18], namely $\mathcal {M}:=\mathcal {M}^{\prime \prime },$ where

  $$\begin{aligned} \mathcal {M}^{\prime \prime }&:=\left\{ y\in \mathbb {R}^{3}:\left| O^{\text {t}}y_{1}\right| ,\left| O^{\text {t}}y_{2}\right| \le \frac{1}{2},y_{3}\in \left[ -\left( \gamma +\zeta \right) ,1-\left( \gamma +\zeta \right) \right] \ ;\right. \nonumber \\&\left. \left\langle \phi _{0}\left( y\right) ,\nabla \left\| y\right\| ^{2}\right\rangle =0\ ,\ \left\langle \phi _{0}\left( y\right) ,\nabla \left\langle \phi _{0}\left( y\right) ,\nabla \left\| y\right\| ^{2}\right\rangle \right\rangle \le 0\right\} \ , \end{aligned}$$

  (220)

  with $\phi _{0}$ given by (2), which is given by the union of two $C^{2}$ compact manifolds $\mathcal {M}_{1},\mathcal {M}_{2}$ intersecting at $c_{0}$ only and such that, if

  $$\begin{aligned} \mathbb {R}^{3}\ni \left( y_{1},y_{2},y_{3}\right) \longmapsto \mathbf {P} \left( y_{1},y_{2},y_{3}\right) :=\left( -y_{1},-y_{2},y_{3}\right) \ , \end{aligned}$$

  (221)

  $\mathbf {P}\mathcal {M}_{1}=\mathcal {M}_{2}.$

1.1 The Poincaré Map for $\mathcal {M}^{\prime \prime }$

Since no confusion will arise, here we will drop the subscript 0 to refer to the unperturbed one-dimensional maps.

In Sect. 2.2.2 in [18] we showed that the Poincaré surface $\mathcal {M}^{\prime \prime }$ defined in (220) is foliated by curves given by the intersection of the spheres $\left\{ y\in \mathbb {R}^{3}:\left\| y\right\| ^{2}=\mathfrak {r}\right\} ,\mathfrak {r}\in \left[ \mathfrak {r} ^{*},y_{3}^{2}\left( c_{0}\right) \right] ,$ for some $\mathfrak {r} ^{*}>0,$ with the surface

$$\begin{aligned} \left\{ y\in \mathbb {R}^{3}:\left\langle \phi _{0}\left( y\right) ,\nabla \left\| y\right\| ^{2}\right\rangle =0,\left\langle \phi _{0}\left( y\right) ,\nabla \left\langle \phi _{0}\left( y\right) ,\nabla \left\| y\right\| ^{2}\right\rangle \right\rangle \le 0\right\} \ , \end{aligned}$$

(222)

where $\phi _{0}$ is defined in (2). By (221), $\mathbf {P}$ defines an equivalence relation between the points of $\mathcal {M}^{\prime \prime }$ and we can identify $\mathcal {M}_{1}$ with the set $\mathcal {M}_{\mathbf {P}}$ of the corresponding equivalence classes. Moreover, we can identify the interval $\left[ \mathfrak {r}^{*},y_{3}^{2}\left( c_{0}\right) \right]$ with the collection of the equivalence classes of the points of $\mathcal {M}_{1},$ and so of $\mathcal {M}_{\mathbf {P}},$ having the same squared Euclidean distance from the origin, i.e. those belonging to the same leaf of the just mentioned foliation which we denote by $\mathfrak {C}.$ In [33] it has been shown by numerical simultations that $\mathfrak {C}$ is invariant exhibiting an automorphism $\hat{T}:\left[ \mathfrak {r}^{*},y_{3} ^{2}\left( c_{0}\right) \right] \circlearrowleft .$ By construction, the Lorenz-type cusp map of the interval given in [18, Fig. 1], which we denote by $\tilde{T},$ is the representation of $\hat{T}$ as a map of the interval $\left[ 0,1\right] .$ Furthermore, if $c_{i}$ is the critical point of $\phi _{0}$ different from $c_{0}$ having minimal Euclidean distance from the component $\mathcal {M}_{i},i=1,2,$ in Section B of [33] it has also been shown that the k-th branch of the induced map of $\tilde{T}$ on $\left[ u_{0},1\right] ,$ with $u_{0}:=\tilde{T}^{-1}\left( 1\right) ,$ refers to trajectories of the system started at $\mathcal {M}_{i}$ that wind k times around $c_{j},i\ne j,$ before returning on $\mathcal {M}_{i},$ while the trajectories of the points of $\mathcal {M}_{i}$ winding just around $c_{i}$ before returning on $\mathcal {M}_{i}$ correspond to the branch $\tilde{T}\upharpoonleft _{\left[ 0,u_{0}\right] }$ of $\tilde{T}$ (see [33, Fig. 11]). Therefore, from these last observations, the map T (i.e. $\bar{T}_{\eta }:\left[ -1,1\right] \circlearrowleft$ in (225) for $\eta =0$) can be reconstructed from $\tilde{T}$ and consequently also its invariant measure. As a matter of fact, describing $\mathcal {M}_{1}$ as in (198), setting $\mathcal {O}\ni \left( u,v\right) \longmapsto \bar{\mathbf {P}}\left( u,v\right) :=\left( \mathbf {p}\left( u\right) ,\mathbf {p}\left( v\right) \right) ,$ with $\mathbb {R}\ni w\longmapsto \mathbf {p}\left( w\right) :=-w\in \mathbb {R},$ and identifying the unperturbed Poincaré map $R_{0}:\mathcal {M}^{\prime \prime }\circlearrowleft$ with the skew-product $\mathcal {O}\bigvee \bar{\mathbf {P}}\mathcal {O} \ni \left( u,v\right) \longmapsto \left( \bar{T}_{0}\left( u\right) ,\Upsilon _{0}\left( u,v\right) \right) \in \mathcal {O}\bigvee \bar{\mathbf {P}}\mathcal {O},$ it follows that $\mathbf {P}\circ R_{0}=R_{0}\circ \mathbf {P},$ hence, since $\mathbf {P}$ is an involution, $\tilde{T} =\mathbf {p}\circ \bar{T}_{0}\circ \mathbf {p\upharpoonleft }_{\left[ 0,1\right] }$ and, setting $\overline{\Upsilon }:=\mathbf {p}\circ \Upsilon _{0} \circ \bar{\mathbf {P}},$ we get the map $\hat{R}_{0}:\mathcal {M}_{\mathbf {P} }\circlearrowleft ,$ which can be identified with the continuous skew-product map $\mathcal {O}\ni \left( u,v\right) \longmapsto \left( \tilde{T}\left( u\right) ,\overline{\Upsilon }\left( u,v\right) \right) \in \mathcal {O}.$ The same considerations apply to perturbations of the phase velocity field that preserves the same symmetry of the system under $\mathbf {P}$ (see [18, Example 8]). In this case rather than (225) we would have had

$$\begin{aligned} \left[ -1,1\right] \ni u\longmapsto T_{\eta }\left( u\right)&:=\mathbf {1}_{\left[ -1,-u_{0,\eta }\right] }\left( u\right) \tilde{T}_{\eta }\left( -u\right) -\mathbf {1}_{\left[ -u_{0,\eta },0\right] }\left( u\right) \tilde{T}_{\eta }\left( -u\right) \nonumber \\&\quad +\mathbf {1}_{\left[ 0,u_{0,\eta }\right] }\left( u\right) \tilde{T}_{\eta }\left( u\right) -\mathbf {1}_{\left[ u_{0,\eta },1\right] }\left( u\right) \tilde{T}_{\eta }\left( u\right) \in \left[ -1,1\right] \end{aligned}$$

(223)

On the other hand, if the perturbed phase velocity field $\phi _{\eta }$ is not invariant under $\mathbf {P},$ the maps of the interval $\tilde{T}_{1}$ and $\tilde{T}_{2},$ representing respectively the automorphisms, associated with the pertubed flow, of the collections of the equivalence classes of the points of $\mathcal {M}_{1}$ and $\mathcal {M}_{2}$ belonging to the leaves of $\mathfrak {C},$ can be thought as perturbations of $\tilde{T}$ fitting into the perturbing scheme given in Sect. 8.4, if $\eta$ is sufficiently small (see [18, Example 9]).

The One-Dimensional Map $T_{\eta }$

In [7] and [22] it has been proven that, in the case we choose $\mathcal {M}:=\mathcal {M}^{\prime },$ identifying I with $\left[ -\frac{1}{2},\frac{1}{2}\right]$ and, with abuse of notation, still denoting by $\bar{T}_{\eta }:\left[ -\frac{1}{2},\frac{1}{2}\right] \backslash \left\{ 0\right\} \longrightarrow \left[ -\frac{1}{2},\frac{1}{2}\right]$ the corresponding transitive, piecewise continuous map of the interval, there exists $\alpha \in \left( 0,1\right) ,G_{\eta }\in C^{\epsilon \alpha }\left( \left[ -\frac{1}{2},\frac{1}{2}\right] \right)$ such that $\bar{T}_{\eta }$ is locally $C^{1+\alpha }$ on $\left[ -\frac{1}{2},\frac{1}{2}\right] \backslash \{0\}$ and

$$\begin{aligned} \left[ -\frac{1}{2},\frac{1}{2}\right] \backslash \left\{ 0\right\} \ni u\longmapsto \bar{T}_{\eta }^{\prime }\left( u\right) :=\left| u\right| ^{-1+\alpha }G_{\eta }\left( u\right) \in \left[ -\frac{1}{2},\frac{1}{2}\right] \ . \end{aligned}$$

(224)

Moreover, $\bar{T}_{\eta }\left( 0^{\mp }\right) =\pm \frac{1}{2}.$ Namely, in this case, $\bar{T}_{\eta }$ is the classical Lorenz-type map (see e.g. Fig. 3.24 in [5] for a sketch).

Fig. 2

Experimental plots of the unperturbed map $\tilde{T}_{0}$ (in black) and of its perturbations (in grey)

In the case $\mathcal {M}:=\mathcal {M}^{\prime \prime },\Gamma _{0}=\left\{ c_{0}\right\} .$ Hence, we identify I with $\left[ -1,1\right]$ and, again with abuse of notation, we denote by $\bar{T}_{\eta }:\left[ -1,1\right] \circlearrowleft$ the map

$$\begin{aligned} \left[ -1,1\right] \ni u\longmapsto \bar{T}_{\eta }\left( u\right)&:=\mathbf {1}_{\left[ -1,-u_{0,\eta }^{2}\right] }\left( u\right) \tilde{T}_{\eta ,2}\left( -u\right) -\mathbf {1}_{\left[ -u_{0,\eta }^{2},0\right] }\left( u\right) \tilde{T}_{\eta ,2}\left( -u\right) \nonumber \\&\quad +\mathbf {1}_{\left[ 0,u_{0,\eta }^{1}\right] }\left( u\right) \tilde{T}_{\eta ,1}\left( u\right) -\mathbf {1}_{\left[ u_{0,\eta }^{1},1\right] }\left( u\right) \tilde{T}_{\eta ,2}\left( u\right) \in \left[ -1,1\right] \ , \end{aligned}$$

(225)

where, for $i=1,2,\tilde{T}_{\eta ,i}:\left[ 0,1\right] \circlearrowleft$ is a transitive, continuous Lorenz-like cusp map of the interval of the type studied in [18], with two branches and a point $u_{0,\eta }^{i} \in \left[ 0,1\right]$ such that $\tilde{T}_{\eta ,i}\left( \left( u_{0,\eta }^{i}\right) ^{-}\right) =\tilde{T}_{\eta ,i}\left( \left( u_{0,\eta }^{i}\right) ^{+}\right) =1.$

In fact, in [33], the paper that inspired our previous work [18], the authors showed that the invariant measure for $\bar{T}_{\eta }$ can be deduced directly from those of the $\tilde{T}_{\eta ,i}$’s, whose local behaviour is therefore the following (compare formulas (52)–(55) in [18] and Fig. 2):

$$\begin{aligned} \tilde{T}_{\eta ,i}\left( u\right) =\left\{ [Arrayl]\begin{array}{ll} a_{\eta ,i}u+b_{\eta ,i}u^{1+c_{\eta ,i}}+o(u^{1+c_{\eta ,i}})\ ;\ a_{\eta ,i}\ ,\ c_{\eta ,i}>1,b_{\eta ,i}>0 &{} u\rightarrow 0^{+}\\ 1-A_{\eta ,i}(u_{0,\eta }-u)^{B_{\eta ,i}}+o((u_{0,\eta }-u)^{B_{\eta ,i} })\ ;\ A_{\eta ,i}>0,B_{\eta ,i}\in \left( 0,1\right) &{} u\rightarrow \left( u_{0,\eta }^{i}\right) ^{-}\\ 1-A_{\eta ,i}^{\prime }(u-u_{0,\eta })^{B_{\eta ,i}^{\prime }}+o((u-u_{0,\eta })^{B_{\eta ,i}^{\prime }})\ ;\ A_{\eta ,i}^{\prime }>0,B_{\eta ,i}^{\prime } \in \left( 0,1\right) &{} u\rightarrow \left( u_{0,\eta }^{i}\right) ^{+}\\ a_{\eta ,i}^{\prime }\left( 1-u\right) +b_{\eta ,i}^{\prime }\left( 1-u\right) ^{1+c_{\eta ,i}^{\prime }}+o(\left( 1-u\right) ^{1+c_{\eta ,i}^{\prime } })\ ;\ a_{\eta ,i}^{\prime }\in \left( 0,1\right) ,b_{\eta ,i}^{\prime }>0,c_{\eta ,i}^{\prime }>1 &{} u\rightarrow 1^{-} \end{array} \right. \ . \nonumber \\ \end{aligned}$$

(226)

We remark that to prove the stochastic stability of the invariant measure for the evolution defined by the unperturbed map $T_{0}$ we needed supplementary assumptions on $T_{0};$ see Sect. 8.4.

In particular, in the case $\mathcal {M}:=\mathcal {M}^{\prime \prime },$ by construction the stochastic stability of $T_{0}$ will follow from that of $\tilde{T}_{0}.$

Existence of Invariant Measures for the Lorenz-Type Cusp Map

In our previous paper [18] the one-dimensional Lorenz-cusp type map T ($\tilde{T}$ in the present paper) had a branch with first derivative less than one on a open set but still bounded from below by a positive number. We were unable to show that the derivative became globally larger that one for a suitable power of the map and therefore we proceeded differently to prove the statistical stability of the unperturbed invariant measure; namely we induced and we showed that on a (lot of) induced set(s), the derivative of the first return map was uniformly larger than one.

Anyway, the existence of an invariant measure for T follows combining Theorem 2 in [34] and the results in Sect. 4.2 of [12] since one can check by direct computation that the map

$$\begin{aligned} I\ni u\longmapsto \overline{T}\left( u\right) :=W\circ T\circ W^{-1}\left( u\right) \in I\ , \end{aligned}$$

(227)

where W is the distribution function associated to the probability measure on $\left( \left[ 0,1\right] ,\mathcal {B}\left( \left[ 0,1\right] \right) \right)$ with density

$$\begin{aligned} \left[ 0,1\right] \ni x\longmapsto W^{\prime }\left( x\right) :=N_{\bar{\gamma },\bar{\beta }}e^{-\bar{\gamma }x}x^{\bar{\beta }}\left( 1-x\right) ^{\bar{\beta }} \end{aligned}$$

(228)

(see formulas (83) and (84) in [18]) for suitably chosen parameters $\bar{\gamma },\bar{\beta }>0$ is such that $\inf \left| \overline{T} ^{\prime }\right| >1.$

In particular, by (226), for any $\eta \in spt\lambda _{\varepsilon },$ setting $B_{\eta }^{*}:=B_{\eta }\vee B_{\eta }^{\prime }$ and choosing $0<\bar{\beta }<\inf _{\eta \in spt\lambda _{\varepsilon }}\frac{1}{B_{\eta }^{*}}-1,\bar{\gamma }>\sup _{\eta \in spt\lambda _{\varepsilon }}\frac{\bar{\beta } +1}{1-x_{0,\eta }}\log \frac{1}{a_{\eta }^{\prime }},$ for any $\eta \in spt\lambda _{\varepsilon },$ we get $\inf _{\eta \in spt\lambda _{\varepsilon }} \inf \left| \overline{T}_{\eta }^{\prime }\right| >1.$ Hölder continuity of $\frac{1}{\overline{T}_{\eta }^{\prime }}$ follows from (229).

Statistical Stability for Lorenz-Like Cusp Maps

We take the chance to rectify an incorrect statement we made in [18] about the regularity properties of the one-dimensional map T.

Therefore, in this section, we will use the same notation we used in [18].

In that paper we state that the map T was $C^{1+\iota },$ for some $\iota \in \left( 0,1\right) ,$ on the union of the two sets $(0,x_{0}),(x_{0},1),$ where the map was 1 to 1. This is incorrect. What is true is that $T^{-1}$ is $C^{1+\iota },$ for some $\iota \in \left( 0,1\right) ,$ on each open interval $(0,x_{0}),(x_{0},1).$ Indeed, by the result in [4], the stable foliation for the classical Lorenz flow is $C^{1+\alpha }$ for some $\alpha \in \left( 0.278,1\right) ,$ which means, by (54) and (55) in [18], that, for any $x\in (0,x_{0}),T^{^{\prime }}\left( x\right) =\left| x_{0}-x\right| ^{1-B^{\prime }}\left[ 1+G_{1}\left( x\right) \right]$ with $G_{1}\in C^{\alpha B^{\prime }}(0,x_{0})$ and, for any $x\in (x_{0},1),T^{\prime }\left( x\right) =\left| x-x_{0}\right| ^{1-B}\left[ 1+G_{2}\left( x\right) \right]$ with $G_{2}\in C^{\alpha B}(x_{0},1).$ In particular this implies that for any couple of points x, y belonging either to $(0,x_{0})$ or to $(x_{0},1)$

$$\begin{aligned} |T^{\prime }(x)-T^{\prime }(y)|\le C_{h}\left| T^{\prime }(x)\right| \left| T^{\prime }(y)\right| \left| x-y\right| ^{\iota }\ , \end{aligned}$$

(229)

where $\iota \in (0,1-B^{*}],$ with $B^{*}:=B\vee B^{\prime },$ and the constant $C_{h}$ is independent of the location of x and y.⁵

We now detail the modifications that these corrections induce on some of the proofs of the results given in [18], all the statements of our results remaining unchanged.

• Distortion The proof of the boundedness of the distortion was sketched in the footnote (1) of [18] by using arguments given in [15]. In particular, in the initial formula (5) in [15] we need now to replace the term $\left| \frac{D^{2}T(\xi )}{DT(\xi )}\right| |T^{q}\left( x\right) -T^{q}\left( y\right) |,$ where $\xi$ is a point between $T^{q}\left( x\right)$ and $T^{q}\left( y\right) ,$ with $\frac{1}{|DT(\xi )|}C_{h}|DT(T^{q}\left( x\right) )||DT(T^{q}\left( y\right) )||T^{q}\left( x\right) -T^{q}\left( y\right) |^{\iota }$ which is smaller than $C_{h}\left( |DT(T^{q}\left( x\right) )|\vee |DT(T^{q}\left( y\right) )|\right) |T^{q}\left( x\right) -T^{q}\left( y\right) |^{\iota }$ by monotonicity of $\left| DT\right| .$ The key estimate (11) in [15] will reduce in our case to the bound of the quantity $\sup _{\xi \in [b_{i+1},b_{i}]}|DT\left( \xi \right) ||b_{i}-b_{i+1}|.$ By using for DT the expressions given in the formulas (54) and (55) of [18], and for the $b_{i}$ the scaling given in formula (75) of the same paper, we immediately get that the above quantity is of order $\frac{1}{(\alpha ^{\prime })^{i}},$ which is enough to pursue the argument about the estimate of the distortion presented in [15].

• Perturbation In order to prove the statistical stability of the invariant measure $\mu _{T}$ for the evolution given by the map T,  the perturbed map $T_{\epsilon }$ must satisfy at least the same regularity properties required for T. Therefore, in [18, Sect. 3.2]:

  • Assumption A should be replaced by the assumption that there exists $\iota _{\epsilon }\in \left( 0,1\right)$ such that $T\upharpoonleft _{(0,x_{\epsilon ,0})},T\upharpoonleft _{(x_{\epsilon ,0},1)}$ are $C^{1+\iota _{\epsilon }}$ rather than assuming the stronger requirement that $T_{\epsilon }$ is $C^{1+\iota _{\epsilon }}$ on $(0,x_{\epsilon ,0})\cup (x_{\epsilon ,0},1);$

  • Assumption C should be replaced by the requirement that the multiplicative Hölder constant $C_{h}^{\epsilon }$ of $D\left( T_{\epsilon }^{-1}\right)$ will converge to $C_{h}$ when $\epsilon \rightarrow 0.$

• We have then to modify the bounds (92), (99) and (114) in [18] which are all of the form $|DT_{\epsilon }(a)-DT_{\epsilon }(a_{\epsilon })|,$ with a $\epsilon$-close to $a_{\epsilon }.$ We have $|DT_{\epsilon }(a)-DT_{\epsilon }(a_{\epsilon })|\le C_{h}^{\epsilon }|DT_{\epsilon }(a)||DT_{\epsilon }(a_{\epsilon })||a-a_{\epsilon }|.$ By the continuity and the monotonicity of $DT_{\epsilon }$ we can replace $a_{\epsilon }$ in $|DT_{\epsilon }(a_{\epsilon })|$ with a or with another given point between a and $x_{0};$ finally we use the limit (88) in Assumption B to conclude.

Proof of Proposition 2

Proof

The invariance of $\mu _{\overline{\mathbf {R}}}$ under $\overline{\mathbf {R}}$ follows by (68), since

$$\begin{aligned} \mu _{\overline{\mathbf {R}}}\left( \psi \circ \overline{\mathbf {R}}\right) :=\lim _{n\rightarrow \infty }\int \mu _{\mathbf {T}}\left( du,d\omega \right) \inf _{x\in q^{-1}\left( u\right) }\psi \circ \overline{\mathbf {R}} ^{n+1}\left( x,\omega \right) =\mu _{\overline{\mathbf {R}}}\left( \psi \right) \ . \end{aligned}$$

(230)

Hence, since

$$\begin{aligned} \lim _{n\rightarrow \infty }\int \mu _{\mathbf {T}}\left( du,d\omega \right) \inf _{x\in q^{-1}\left( u\right) }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega \right)&\le \lim _{n\rightarrow \infty }\int \mu _{\mathbf {T}}\left( du,d\omega \right) \left( \left( \mathbf {1}_{q^{-1}\left( u\right) }\circ p\right) \psi \right) \circ \overline{\mathbf {R}}^{n}\left( x,\omega \right) \nonumber \\&\le \lim _{n\rightarrow \infty }\int \mu _{\mathbf {T}}\left( du,d\omega \right) \sup _{x\in q^{-1}\left( u\right) }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega \right) \ , \end{aligned}$$

(231)

it is enough to prove that

$$\begin{aligned} \lim _{n\rightarrow \infty }\int \mu _{\mathbf {T}}\left( du,d\omega \right) \inf _{x\in q^{-1}\left( u\right) }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega \right) =\lim _{n\rightarrow \infty }\int \mu _{\mathbf {T}}\left( du,d\omega \right) \sup _{x\in q^{-1}\left( u\right) }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega \right) \ . \end{aligned}$$

(232)

By (48), (32) and the definition of $\bar{R}_{\pi \left( \omega \right) },\forall \omega \in \Omega ,$

$$\begin{aligned} \overline{\mathbf {R}}\left( Q^{-1}\left( u,\omega \right) \right) \subset Q^{-1}\left( \mathbf {T}\left( u,\omega \right) \right) \ . \end{aligned}$$

(233)

Therefore,

$$\begin{aligned} \sup _{x\in q^{-1}\left( u\right) }\psi \circ \overline{\mathbf {R}} ^{n+k}\left( x,\omega \right)&=\sup _{\left( x,\omega ^{\prime }\right) \in Q^{-1}\left( u,\omega \right) }\psi \circ \overline{\mathbf {R}} ^{n+k}\left( x,\omega ^{\prime }\right) \nonumber \\&\le \sup _{\left( x,\omega ^{\prime }\right) \in Q^{-1}\left( \mathbf {T}^{k}\left( u,\omega \right) \right) }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega ^{\prime }\right) \nonumber \\&=\sup _{\left( x,\omega ^{\prime }\right) \in \left\{ \left( y,\omega ^{\prime \prime }\right) \in \mathcal {M}\times \Omega \ :\ Q\left( y,\omega ^{\prime \prime }\right) =\mathbf {T}^{k}\left( u,\omega \right) \right\} }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega ^{\prime }\right) \end{aligned}$$

(234)

and

$$\begin{aligned} \inf _{x\in q^{-1}\left( u\right) }\psi \circ \overline{\mathbf {R}} ^{n+k}\left( x,\omega \right)&=\inf _{\left( x,{\omega }^{\prime }\right) \in Q^{-1}\left( u,\omega \right) }\psi \circ \overline{\mathbf {R} }^{n+k}\left( x,\omega ^{\prime }\right) \nonumber \\&\ge \inf _{\left( x,\omega ^{\prime }\right) \in Q^{-1}\left( \mathbf {T}^{k}\left( u,\omega \right) \right) }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega ^{\prime }\right) \nonumber \\&=\inf _{\left( x,\omega ^{\prime }\right) \in \left\{ \left( y,\omega ^{\prime \prime }\right) \in \mathcal {M}\times \Omega \ :\ Q\left( y,\omega ^{\prime \prime }\right) =\mathbf {T}^{k}\left( u,\omega \right) \right\} }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega ^{\prime }\right) \ . \end{aligned}$$

(235)

Hence, by the invariance of $\mu _{\mathbf {T}}$ under $\mathbf {T},$

$$\begin{aligned}&\int \mu _{\mathbf {T}}\left( du,d\omega \right) \sup _{x\in q^{-1}\left( u\right) }\psi \circ \overline{\mathbf {R}}^{n+k}\left( x,\omega \right) \nonumber \\&\quad \le \int \mu _{\mathbf {T}}\left( du,d\omega \right) \sup _{\left( x,\omega ^{\prime }\right) \in \left\{ \left( y,\omega ^{\prime \prime }\right) \in \mathcal {M}\times \Omega \ :\ Q\left( y,\omega ^{\prime \prime }\right) =\mathbf {T}^{k}\left( u,\omega \right) \right\} }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega ^{\prime }\right) \nonumber \\&\quad =\int \left( \mathbf {T}_{\#}^{k}\mu _{\mathbf {T}}\right) \left( du,d\omega \right) \sup _{\left( x,\omega ^{\prime }\right) \in \left\{ \left( y,\omega ^{\prime \prime }\right) \in \mathcal {M}\times \Omega \ :\ Q\left( y,\omega ^{\prime \prime }\right) =\left( u,\omega \right) \right\} }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega ^{\prime }\right) \nonumber \\&\quad =\int \mu _{\mathbf {T}}\left( du,d\omega \right) \sup _{\left( x,\omega ^{\prime }\right) \in Q^{-1}\left( u,\omega \right) }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega ^{\prime }\right) \nonumber \\&\quad =\int \mu _{\mathbf {T}}\left( du,d\omega \right) \sup _{x\in q^{-1}\left( u\right) }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega \right) \end{aligned}$$

(236)

so that the sequence $\left\{ \int \mu _{\mathbf {T}}\left( du,d\omega \right) \sup _{x\in q^{-1}\left( u\right) }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega \right) \right\} _{n\ge 1}$ is decreasing. On the other hand,

$$\begin{aligned}&\int \mu _{\mathbf {T}}\left( du,d\omega \right) \inf _{x\in q^{-1}\left( u\right) }\psi \circ \overline{\mathbf {R}}^{n+k}\left( x,\omega \right) \nonumber \\&\quad \ge \int \mu _{\mathbf {T}}\left( du,d\omega \right) \inf _{\left( x,\omega ^{\prime }\right) \in \left\{ \left( y,\omega ^{\prime \prime }\right) \in \mathcal {M}\times \Omega \ :\ Q\left( y,\omega ^{\prime \prime }\right) =\mathbf {T}^{k}\left( u,\omega \right) \right\} }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega ^{\prime }\right) \nonumber \\&\quad =\int \left( \mathbf {T}_{\#}^{k}\mu _{\mathbf {T}}\right) \left( du,d\omega \right) \inf _{\left( x,\omega ^{\prime }\right) \in \left\{ \left( y,\omega ^{\prime \prime }\right) \in \mathcal {M}\times \Omega \ :\ Q\left( y,\omega ^{\prime \prime }\right) =\left( u,\omega \right) \right\} }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega ^{\prime }\right) \nonumber \\&\quad =\int \mu _{\mathbf {T}}\left( du,d\omega \right) \inf _{\left( x,\omega ^{\prime }\right) \in Q^{-1}\left( u,\omega \right) }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega ^{\prime }\right) \end{aligned}$$

(237)

so that $\left\{ \int \mu _{\mathbf {T}}\left( du,d\omega \right) \inf _{x\in q^{-1}\left( u\right) }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega \right) \right\} _{n\ge 1}$ is increasing. Since $\forall \omega \in \Omega ,\psi \left( \cdot ,\omega \right) \in C_{b}\left( \mathcal {M} \right)$ and $\forall u\in I,q^{-1}\left( u\right) \subset \mathcal {M}$ is compact, by (233), $\forall \varepsilon ^{\prime }>0,\exists \delta _{\varepsilon ^{\prime }}>0,n_{\varepsilon ^{\prime }}>0$ such that $\forall n\ge n_{\varepsilon ^{\prime }},\omega \in \Omega ,u\in I,{\text {*}}{diam} p\left( \overline{\mathbf {R}}^{n}\left( Q^{-1}\left( u,\omega \right) \right) \right) <\delta _{\varepsilon ^{\prime }}$ and $\forall \left( x,\omega ^{\prime }\right) ,\left( y,\omega ^{\prime }\right) \in \overline{\mathbf {R}}^{n}\left( Q^{-1}\left( u,\omega \right) \right) ,$$\left| \psi \left( x,\omega ^{\prime }\right) -\psi \left( y,\omega ^{\prime }\right) \right| <\varepsilon ^{\prime },$ therefore

$$\begin{aligned}&\left| \int \mu _{\mathbf {T}}\left( du,d\omega \right) \sup _{x\in q^{-1}\left( u\right) }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega \right) -\int \mu _{\mathbf {T}}\left( du,d\omega \right) \inf _{x\in q^{-1}\left( u\right) }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega \right) \right| \nonumber \\&\quad \le \int \mu _{\mathbf {T}}\left( du,d\omega \right) \left| \sup _{\left( x,\omega ^{\prime }\right) \in Q^{-1}\left( u,\omega \right) }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega ^{\prime }\right) -\inf _{\left( x,\omega ^{\prime }\right) \in Q^{-1}\left( u,\omega \right) }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega ^{\prime }\right) \right| \le \varepsilon ^{\prime }\ , \end{aligned}$$

(238)

that is (232) holds.

Thus, the map

$$\begin{aligned} L_{\mathbb {P}}^{1}\left( \Omega ,C_{b}\left( \mathcal {M}\right) \right) \ni \psi \longmapsto \hat{\mu }\left( \psi \right) :=\lim _{n\rightarrow \infty }\int \mu _{\mathbf {T}}\left( du,d\omega \right) \left( \left( \mathbf {1} _{q^{-1}\left( u\right) }\circ p\right) \psi \right) \circ \overline{\mathbf {R}}^{n}\left( x,\omega \right) \in \mathbb {R} \end{aligned}$$

(239)

is a non negative linear functional such that $\hat{\mu }\left( 1\right) =1$ and, by (232),

$$\begin{aligned} \hat{\mu }\left( \psi \right) =\lim _{n\rightarrow \infty }\int \mu _{\mathbf {T} }\left( du,d\omega \right) \inf _{x\in q^{-1}\left( u\right) }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega \right) \ . \end{aligned}$$

(240)

Moreover, $\Omega$ is compact under the product topology, then the space of quasi-local continuous functions $C_{\infty }\left( \Omega ,C_{b}\left( \mathcal {M}\right) \right)$⁶ is dense in $L_{\mathbb {P}}^{1}\left( \Omega ,C_{b}\left( \mathcal {M}\right) \right) ,$ therefore, by the Riesz-Markov-Kakutani theorem there exists a unique Radon measure $\mu _{\overline{\mathbf {R}}}$ on $\left( \mathcal {M}\times \Omega ,\mathcal {B}\left( \mathcal {M}\right) \otimes \mathcal {F}\right)$ such that $\mu _{\overline{\mathbf {R}}}=\hat{\mu }\upharpoonleft _{C_{K}\left( \Omega ,C_{b}\left( \mathcal {M}\right) \right) }.$

The injectivity of the correspondence $\mu _{\mathbf {T}}\longmapsto \mu _{\overline{\mathbf {R}}}$ follows from the fact that, $\forall \varphi \in L_{\mathbb {P}}^{1}\left( \Omega ,C_{b}\left( I\right) \right) ,\varphi \circ Q\in L_{\mathbb {P}}^{1}\left( \Omega ,C_{b}\left( \mathcal {M}\right) \right)$ and

$$\begin{aligned}&\int \mu _{\mathbf {T}}\left( du,d\omega \right) \inf _{x\in q^{-1}\left( u\right) }\varphi \circ Q\circ \overline{\mathbf {R}}^{n}\left( x,\omega \right) =\int \mu _{\mathbf {T}}\left( du,d\omega \right) \inf _{x\in q^{-1}\left( u\right) }\varphi \circ \mathbf {T}^{n}\circ Q\left( x,\omega \right) \nonumber \\&\quad =\int \mu _{\mathbf {T}}\left( du,d\omega \right) \inf _{x\in q^{-1}\left( u\right) }\varphi \circ \mathbf {T}^{n}\left( q\left( x\right) ,\omega \right) =\mu _{\mathbf {T}}\left( \varphi \circ \mathbf {T}^{n}\right) =\mu _{\mathbf {T}}\left( \varphi \right) \ . \end{aligned}$$

(241)

Therefore, if there exist $\mu _{\mathbf {T}}^{\prime }$ invariant under $\mathbf {T}$ such that

$$\begin{aligned} \mu _{\overline{\mathbf {R}}}\left( \psi \right) :=\lim _{n\rightarrow \infty }\int \mu _{\mathbf {T}}^{\prime }\left( du,d\omega \right) \inf _{x\in q^{-1}\left( u\right) }\psi \circ \overline{\mathbf {R}}^{n}\left( x,\omega \right) \ , \end{aligned}$$

(242)

then $\mu _{\mathbf {T}}^{\prime }\left( \varphi \right) =\mu _{\mathbf {T} }\left( \varphi \right) ,$ hence $\mu _{\mathbf {T}}^{\prime }=\mu _{\mathbf {T} }.$

The proof of the ergodicity of $\mu _{\overline{\mathbf {R}}}$ under the hypothesis of the ergodicity of $\mu _{\mathbf {T}}$ is identical to that of Corollary 7.25 in Sect. 7.3.4 of [5]. $\square$