Proof Take $y\in \Sigma $ so that $ \phi ^s (D_{\Pi ^{t_0} y} f^{t_0}_{\mathbf {i}\wedge \mathbf {j}})= \max _{x\in \Sigma } \phi ^s (D_{\Pi ^{t_0} x} f^t_{\mathbf {i}\wedge \mathbf {j}}). $ Let k be the unique integer such that $s\in (k, k+1)$ . Clearly $k\in \{0,1,\ldots , d-1\}$ . For convenience, write

$$ \begin{align*}a:=\phi^k (D_{\Pi^{t_0} y} f^{t_0}_{\mathbf{i}\wedge \mathbf{j}}),\quad b:=\phi^{k+1}(D_{\Pi^{t_0} y} f^{t_0}_{\mathbf{i}\wedge \mathbf{j}}), \end{align*} $$

where $\phi ^s(\cdot )$ stands for the singular value function (see (2.5) for the definition). A direct check shows that

(3.3) $$ \begin{align} \phi^s (D_{\Pi^{t_0} y} f^t_{\mathbf{i}\wedge \mathbf{j}})=a^{k+1-s}b^{s-k}. \end{align} $$

Observe that

$$ \begin{align*} \int_{B(t_0,\delta)} & |\Pi^t({\textbf{i}})-\Pi^t(\mathbf{j}) |^{-s} \;d\eta(t)\\ &=s\int_0^{\infty} r^{-s-1} \eta \{t\in B(t_0,\delta):\; |\Pi^t({\textbf{ i}})-\Pi^t({\textbf{j}}) |< r\}\; dr\\ &\leq sCe^{|\mathbf{i}\wedge \mathbf{j}|\psi(\delta)} \int_0^{\infty} r^{-s-1} Z^{t_0}_{\mathbf{i}\wedge \mathbf{j}}(r) \; dr \quad (\text{by (1.5)})\\ &\leq sCe^{|\mathbf{i}\wedge \mathbf{j}|\psi(\delta)} \int_0^{\infty} r^{-s-1} \min\bigg\{ \frac{r^k}{a}, \frac{r^{k+1}}{b} \bigg\} \; dr \quad (\text{by (1.4)})\\ &\leq sCe^{|\mathbf{i}\wedge \mathbf{j}|\psi(\delta)} \int_0^{b/a} \frac{r^{k-s}}{b} \;dr+ \int_{b/a}^{\infty} \frac{r^{k-s-1}}{a}\; dr\\ &=sCe^{|\mathbf{i}\wedge \mathbf{j}|\psi(\delta)} \bigg(\frac{1}{k+1-s}+\frac{1}{s-k}\bigg)a^{s-k-1}b^{k-s}\\ &=sC\bigg(\frac{1}{k+1-s}+\frac{1}{s-k}\bigg)e^{|\mathbf{i}\wedge \mathbf{j}|\psi(\delta)} (\phi^s (D_{\Pi^{t_0} y}f^{t_0}_{\mathbf{i}\wedge \mathbf{ j}}))^{-1}\quad (\text{by (3.3))}. \end{align*} $$

This proves (3.2) by setting $c=sC(({1}/({k+1-s}))+({1}/({s-k})))$ .

Proof of Proposition 3.1 Fix $t_0\in {\Omega }$ and $\delta \in (0,\delta _0)$ . We first prove part (i). Let $\epsilon>0$ and let s be non-integral so that

(3.4) $$ \begin{align} 0<s<\min\{d, d_{\mu}(t)\}-\frac{\psi(\delta)}{\log (1/\theta)}-2\epsilon. \end{align} $$

To show that (3.1) holds for $\eta $ -a.e. $t\in B(t_0,\delta )$ , it suffices to show that

(3.5) $$ \begin{align} \underline{\dim}_H\Pi^t_*\mu\geq s \quad \text{for } \eta\text{-a.e.~}t\in B(t_0,\delta). \end{align} $$

For this purpose, we write

(3.6) $$ \begin{align} \varphi^s(I)=\max_{x\in \Sigma} \phi^s(D_{\Pi^{t_0}_x} f_I^{t_0}),\quad I\in \Sigma_*. \end{align} $$

We first prove that

(3.7) $$ \begin{align} \lim_{n\to \infty}\frac{\mu([\mathbf{i}|n])}{\varphi^s(\mathbf{i}|n) \exp(-n \psi(\delta))\theta^{n\epsilon}}=0\quad \text{for } \mu\text{-a.e.~}\mathbf{ i}\in \Sigma. \end{align} $$

To see this, according to the definition of $d_{\mu }(t_0)$ (cf. (1.6) and Definition 2.4),

(3.8) $$ \begin{align} h_{\mu}(\sigma)+\lim_{n\to \infty}\frac{1}{n}\int \log \phi^{d_{\mu}(t_0)} (D_{\Pi^{t_0}\sigma^n\mathbf{i}} f^{t_0}_{\mathbf{i}|n})\; d\mu(\mathbf{ i})=0. \end{align} $$

It follows from (3.8), the Shannon–McMillan–Breiman theorem and Kingman’s sub-additive ergodic theorem (see [Reference Walters49, pp. 93 and 231]) that

(3.9) $$ \begin{align} \lim_{n\to \infty}\frac{1}{n}\log \frac{\mu([\mathbf{i}|n])}{\phi^{d_{\mu}(t_0)} (D_{\Pi^{t_0}\sigma^n\mathbf{i}} f^{t_0}_{\mathbf{i}|n})}=0 \quad \text{for } \mu\text{-a.e.~}\mathbf{i}\in \Sigma. \end{align} $$

Observe that for each $\mathbf {i}\in \Sigma $ and $n\in {\Bbb N}$ ,

$$ \begin{align*} \phi^{d_{\mu}(t_0)} (D_{\Pi^{t_0}\sigma^n\mathbf{i}} f^{t_0}_{\mathbf{i}|n}) &\leq \phi^{s} (D_{\Pi^{t_0}\sigma^n\mathbf{i}} f^{t_0}_{\mathbf{i}|n}) \|D_{\Pi^{t_0}\sigma^n\mathbf{i}} f^{t_0}_{\mathbf{i}|n}\|^{d_{\mu}(t_0)-s} \quad \text{(by Lemma 2.2(ii))}\\ &\leq \varphi^{s} (\mathbf{i}|n) \theta^{n(d_{\mu}(t_0)-s)}\quad\quad \text{(by (3.6) and (1.3))}\\ &\leq \varphi^s(\mathbf{i}|n) \exp(-n \psi(\delta))\theta^{2n\epsilon} \quad \text{(by (3.4))}. \end{align*} $$

Combining the above inequality with (3.9) yields (3.7).

By (3.7), we may find a countable disjoint collection of Borel subsets $E_j$ of $\Sigma $ with $\mu (\Sigma \backslash \bigcup _{j=1}^{\infty } E_j)=0$ and numbers $c_j>0$ such that

(3.10) $$ \begin{align} \mu_j([\mathbf{i}|n])\leq c_j \varphi^s(\mathbf{i}|n) \exp(-n \psi(\delta))\theta^{n\epsilon} \quad \text{for all }\mathbf{i}\in \Sigma,\; n\in {\Bbb N}, \end{align} $$

where $\mu _j$ stands for the restriction of $\mu $ to $E_j$ defined by $\mu _j(A)=\mu (E_j\cap A)$ . Clearly,

(3.11) $$ \begin{align} \mu=\sum_{j=1}^{\infty} \mu_j. \end{align} $$

Hence, to prove (3.5), it suffices to show that for each j,

(3.12) $$ \begin{align} \underline{\dim}_H\Pi^t_*\mu_j\geq s \quad \text{for } \eta\text{-a.e.~}t\in B(t_0,\delta). \end{align} $$

By the potential theoretic characterization of the Hausdorff dimension (see e.g. [Reference Falconer16, Theorem 4.13]), it is enough to show that for each j and $\eta $ -a.e. $t\in B(t_0,\delta )$ , $\Pi ^t_*\mu _j$ has finite s-energy:

$$ \begin{align*}I_s(\Pi^t_*\mu_j):=\iint \frac{ d \Pi^t_*\mu_j (x)d \Pi^t_*\mu_j (y)}{|x-y|^{s}}<\infty. \end{align*} $$

Integrating over $B(t_0,\delta )$ with respect to $\eta $ and using Fubini’s theorem,

$$ \begin{align*} \int_{B(t_0,\delta)}I_s(\Pi^t_*\mu_j)\; d\eta(t)&=\int_{B(t_0,\delta)} \iint \frac{ d \Pi^t_*\mu_j (x)d \Pi^t_*\mu_j (y)}{|x-y|^{s}} d\eta(t)\\ &=\int_{B(t_0,\delta)} \iint \frac{ d\mu_j (\mathbf{i})\,d \mu_j (\mathbf{j})}{|\Pi^t(\mathbf{i})-\Pi^t(\mathbf{j})|^{s}} d\eta(t)\\ &= \iint \int_{B(t_0,\delta)} \frac{d\eta(t)}{|\Pi^t(\mathbf{i})-\Pi^t(\mathbf{j})|^{s}} \ d\mu_j (\mathbf{i})\,d \mu_j (\mathbf{j})\\ &\leq \iint c e^{|\mathbf{i}\wedge \mathbf{j}|\psi(\delta)}(\varphi^s(\mathbf{i}\wedge \mathbf{j}))^{-1} d\mu_j (\mathbf{i})\,d \mu_j (\mathbf{ j})\quad \text{(by (3.2), (3.6))}\\ &\leq c\int \sum_{n=0}^{\infty} e^{n\psi(\delta)} (\varphi^s(\mathbf{j}|n))^{-1} \mu_j([\mathbf{j}|n])d \mu_j (\mathbf{j})\\ &\leq cc_j\int \sum_{n=0}^{\infty} \theta^{n\epsilon} d \mu_j (\mathbf{j})\quad \text{(by (3.10))}\\ &<\infty. \end{align*} $$

It follows that $I_s(\Pi ^t_*\mu _j)<\infty $ for $\eta $ -a.e. $t\in B(t_0,\delta )$ . This completes the proof of part (i).

Next we prove part (ii). Take a small $\epsilon>0$ so that

(3.13) $$ \begin{align} d_{\mu}(t_0)>d+\frac{\psi(\delta)}{\log (1/\theta)}+2\epsilon. \end{align} $$

Then for every $\mathbf {i}\in \Sigma $ and $n\in {\Bbb N}$ ,

$$ \begin{align*} \phi^{d_{\mu}(t_0)} (D_{\Pi^{t_0}\sigma^n\mathbf{i}} f^{t_0}_{\mathbf{i}|n}) &\leq \phi^{d} (D_{\Pi^{t_0}\sigma^n\mathbf{i}} f^{t_0}_{\mathbf{i}|n}) \|D_{\Pi^{t_0}\sigma^n\mathbf{i}} f^{t_0}_{\mathbf{i}|n}\|^{d_{\mu}(t_0)-d} \quad \text{(by Lemma 2.2(ii))}\\ &\leq \varphi^{d} (\mathbf{i}|n) \theta^{n(d_{\mu}(t_0)-d)}\quad\quad \text{(by (3.6) and (1.3))}\\ &\leq \varphi^d(\mathbf{i}|n) \exp(-n \psi(\delta))\theta^{2n\epsilon} \quad \text{(by (3.13))}. \end{align*} $$

Combining the above inequality with (3.9) yields that

$$ \begin{align*}\lim_{n\to \infty}\frac{\mu([\mathbf{i}|n])}{\varphi^d(\mathbf{i}|n) \exp(-n \psi(\delta))\theta^{n\epsilon}}=0\quad \text{for } \mu\text{-a.e.~}\mathbf{i}\in \Sigma. \end{align*} $$

Hence, there exist finite positive measures $\nu _j$ and numbers $a_j$ ( $j\geq 1)$ such that $\mu =\sum _{j=1}^{\infty } \nu _j$ and

(3.14) $$ \begin{align} \nu_j([\mathbf{i}|n])\leq a_j \varphi^d(\mathbf{i}|n) \exp(-n \psi(\delta))\theta^{n\epsilon} \quad \text{for all }\mathbf{i}\in \Sigma,\; n\in {\Bbb N}. \end{align} $$

Since $\mu =\sum _{j=1}^{\infty } \nu _j$ , to show that $\Pi ^t_*\mu \ll \mathcal L_d$ for $\eta $ -a.e. $t\in B(t_0,\delta )$ , it suffices to show that for each j,

(3.15) $$ \begin{align} \Pi^t_*\nu_j\ll \mathcal L_d \quad \text{for } \eta\text{-a.e.~}t\in B(t_0,\delta). \end{align} $$

To this end, fix j. We will follow a standard approach (introduced by Peres and Solomyak in [Reference Peres and Solomyak40]). In particular, it suffices to show that

$$ \begin{align*}I:=\int_{B(t_0,\delta)} \int \liminf_{r\to 0} \frac{\Pi^t_*\nu_j(B_{{\Bbb R}^d}(x,r)) }{r^d}\; d\Pi^t_*\nu_j(x)\,d\eta(t) <\infty, \end{align*} $$

where $B_{{\Bbb R}^d}(x,r)$ stands for the closed ball in ${\Bbb R}^d$ centered at x of radius r. Observe that by (1.4) and (3.6),

(3.16) $$ \begin{align} Z^{t_0}_{\boldsymbol{\omega}}(r)\leq \inf_{x\in \Sigma} \frac{r^d}{ \phi^d (D_{\Pi^{t_0} x}f^t_{\boldsymbol{\omega}} ) }=\frac{r^d}{\varphi^d(\boldsymbol{\omega})}\quad \text{for }\boldsymbol{\omega}\in \Sigma_*,\; r>0. \end{align} $$

Applying Fatou’s Lemma and Fubini’s Theorem,

$$ \begin{align*} I&\leq \liminf_{r\to 0}\frac{1}{r^d} \int_{B(t_0,\delta)} \int \Pi^t_*\nu_j(B_{{\Bbb R}^d}(x,r)) \; d\Pi^t_*\nu_j(x)\,d\eta(t)\\ &=\liminf_{r\to 0}\frac{1}{r^d} \int_{B(t_0,\delta)} \iint {\textbf{1}}_{\{ (x,y): \; |x-y|\leq r\}} \; d\Pi^t_*\nu_j(x)\,d\Pi^t_*\nu_j(y)\,d\eta(t)\\ &=\liminf_{r\to 0}\frac{1}{r^d} \int_{B(t_0,\delta)} \iint {\textbf{1}}_{\{(\mathbf{i}, \mathbf{j}):\; |\Pi^t(\mathbf{i})-\Pi^t(\mathbf{j})|\leq r\}} \; d\nu_j(\mathbf{i})\,d\nu_j(\mathbf{j})\,d\eta(t) \\ &=\liminf_{r\to 0}\frac{1}{r^d} \iint \int {\textbf{1}}_{\{ t\in B(t_0,\delta):\; |\Pi^t(\mathbf{i})-\Pi^t(\mathbf{j})|\leq r\}} \; d\eta(t)\,d\nu_j(\mathbf{i})\,d\nu_j(\mathbf{j}) \\ &=\liminf_{r\to 0}\frac{1}{r^d} \iint \eta\{t\in B(t_0,\delta):\; |\Pi^t(\mathbf{i})-\Pi^t(\mathbf{j})|\leq r\} \; d\nu_j(\mathbf{i})\,d\nu_j(\mathbf{j}). \end{align*} $$

By (1.5) and (3.16), we obtain that

$$ \begin{align*} I &\leq \liminf_{r\to 0}\frac{1}{r^d} \iint C e^{|{\textbf{i}}\wedge {\textbf{j}}| \psi(\delta)} Z_{{\textbf{i}}\wedge {\textbf{j}}}^{t_0}(r) \; d\nu_j(\mathbf{i})\,d\nu_j(\mathbf{j})\\ &\leq \iint \frac{C e^{|{\textbf{i}}\wedge {\textbf{j}}| \psi(\delta)}}{\varphi^d(\mathbf{i}\wedge \mathbf{j})} \; d\nu_j(\mathbf{i})\,d\nu_j(\mathbf{j})\\ &\leq \int \sum_{n=0}^{\infty} \frac{C e^{n \psi(\delta)}}{\varphi^d( \mathbf{j}|n)}\nu_j ([\mathbf{j} |n])\; d\nu_j(\mathbf{j})\\ &\leq a_jC\int \sum_{n=0}^{\infty} \theta^{n\epsilon}\; d\nu_j(\mathbf{j}) \quad \text{(by ({3.14}))}\\ &<\infty, \end{align*} $$

which completes the proof of part (ii).

Proof of Theorem 1.2(i) Let $\mu $ be a $\sigma $ -invariant ergodic measure on $\Sigma $ . We first show that for $\eta $ -a.e. $t\in {\Omega }$ , $\Pi ^t_*\mu $ is exact dimensional with dimension equal to $\min \{d, d_{\mu }(t)\}$ . Recall that $\overline {\dim }_P\Pi ^t_*\mu \leq \min \{d, d_{\mu }(t)\}$ for each $t\in {\Omega }$ (see Theorem 2.7(ii)). Hence, it is sufficient to show that for $\eta $ -a.e. $t\in {\Omega }$ , $\underline {\dim }_H \Pi ^t_*\mu \geq \min \{d, d_{\mu }(t)\}$ . Suppose on the contrary that this is false. Then there exist $k\in {\Bbb N}$ and $A\subset {\Omega }$ with $\eta (A)>0$ such that

(3.17) $$ \begin{align} \underline{\dim}_H \Pi^t_*\mu< \min\{d, d_{\mu}(t)\}-\frac2k\quad \text{for all } t\in A. \end{align} $$

Take a number $\delta \in (0, \delta _0)$ small enough such that

(3.18) $$ \begin{align} \frac{ \psi(\delta)}{\log (1/\theta)}<\frac1k. \end{align} $$

Since ${\Omega }$ is a separable metric space, it has a countable dense subset

$$ \begin{align*} Y=\{y_n: \; n\in {\Bbb N}\}. \end{align*} $$

Notice that by (1.3) and Lemma 2.6,

$$ \begin{align*}0\leq d_{\mu}(t)\leq d(t)\leq \frac{ \log \ell}{ \log (1/\theta)}\quad \text{for each }t\in {\Omega}. \end{align*} $$

Due to this fact, for each $n\in {\Bbb N}$ , we may pick $y_n^*\in B(y_n,\delta /2)$ so that

(3.19) $$ \begin{align} d_{\mu}(y_n^*)\geq \sup_{t\in B(y_n, \delta/2)}d_{\mu}(t)-\frac{1}{k}. \end{align} $$

Let $Y^*=\{y_{n}^*:\; n\in {\Bbb N}\}$ . Clearly, $Y^*$ is countable. We claim that

(3.20) $$ \begin{align} \sup_{y^*\in B(t, \delta)\cap Y^*} d_{\mu}(y^*)\geq d_{\mu}(t)-\frac1k \quad \text{for all } t\in {\Omega}. \end{align} $$

To see this, let $t\in {\Omega }$ . Since Y is dense in ${\Omega }$ , there exists an integer m such that $\rho (y_m, t)\leq \delta /2$ . Then

(3.21) $$ \begin{align} y_m^*\in B(y_m, \delta/2)\subset B(t,\delta). \end{align} $$

Meanwhile, by (3.19),

$$ \begin{align*}d_{\mu}(y_m^*)\geq \sup_{t'\in B(y_m, \delta/2)}d_{\mu}(t')-\frac{1}{k}\geq d_{\mu}(t)-\frac1k, \end{align*} $$

where in the last inequality, we use the fact that $t\in B(y_m, \delta /2)$ (since $\rho (y_m, t)\leq \delta /2$ ). This proves (3.20), since $y^*_{m}\in B(t,\delta )\cap Y^*$ by (3.21).

Set for $n\in {\Bbb N}$ ,

(3.22) $$ \begin{align} {\Omega}_n=\{t\in B(y_n^*,\delta):\; d_{\mu}(y^*_n)\geq d_{\mu}(t)-1/k\}. \end{align} $$

By (3.20), ${\Omega }=\bigcup _{n=1}^{\infty } {\Omega }_n$ . Define

$$ \begin{align*}E_n=\{t\in B(y_n^*,\delta):\; \underline{\dim}_H \Pi^t_*\mu<\min\{d, d_{\mu}(y_n^*)\}-1/k\},\quad n\in {\Bbb N}. \end{align*} $$

By (3.17), we see that $A\cap {\Omega }_n\subset E_n$ for each $n\in {\Bbb N}$ . However, by Proposition 3.1(i) and (3.18), for each $n\in {\Bbb N}$ and $\eta $ -a.e. $t\in B(y_n^*,\delta )$ ,

$$ \begin{align*}\underline{\dim}_H \Pi^t_*\mu\geq \min\{d, d_{\mu}(y_n^*)\}-\frac{ \psi(\delta)}{\log (1/\theta)}> \min\{d, d_{\mu}(y_n^*)\}-\frac1k. \end{align*} $$

It follows that $\eta (E_n)=0$ for each $n\in {\Bbb N}$ . Hence

$$ \begin{align*}\eta(A)=\eta\bigg(A\cap \bigg( \bigcup_{n=1}^{\infty} {\Omega}_n\bigg)\bigg)\leq \sum_{n=1}^{\infty} \eta(A\cap {\Omega}_n)\leq \sum_{n=1}^{\infty} \eta(E_n)=0, \end{align*} $$

leading to a contradiction. This proves the statement that for $\eta $ -a.e. $t\in {\Omega }$ , $\Pi ^t_*\mu $ is exact dimensional with dimension $\min \{d, d_{\mu }(t)\}$ .

Next we prove that $\Pi ^t_*\mu \ll \mathcal L_d$ for $\eta $ -a.e. $t\in \{t'\in {\Omega }:\; d_{\mu }(t')>d\}$ . Again we use a contradiction. Suppose on the contrary that this result is false. Then there exist $k\in {\Bbb N}$ and $A'\subset {\Omega }$ with $\eta (A')>0$ such that

(3.23) $$ \begin{align} d_{\mu}(t)>d+\frac2k \quad \text{and}\quad \Pi^t_*\mu\not\ll \mathcal L_d\quad \text{for all }t\in A'. \end{align} $$

Set

$$ \begin{align*}F_n=\{t\in B(y_n^*,\delta):\; \Pi^t_*\mu\not \ll \mathcal L_d\}, \quad n\in {\Bbb N}. \end{align*} $$

Clearly, $A'\cap {\Omega }_n\subset F_n$ for each $n\in {\Bbb N}$ . Since ${\Omega }=\bigcup _{n=1}^{\infty } {\Omega }_n$ and $\eta (A')>0$ , there exists $m\in {\Bbb N}$ so that $\eta (A'\cap {\Omega }_m)>0$ . Hence $A'\cap {\Omega }_m\neq \emptyset $ . Pick $t\in A'\cap {\Omega }_m$ . By (3.22), (3.23) and (3.18),

$$ \begin{align*}d_{\mu}(y_m^*)\geq d_{\mu}(t)-\frac{1}{k}>d+\frac{1}{k}>d+\frac{ \psi(\delta)}{\log (1/\theta)}.\end{align*} $$

Hence $\eta (F_m)=0$ by Proposition 3.1(ii). Since $A'\cap {\Omega }_m\subset F_m$ , it follows that $\eta (A'\cap {\Omega }_m)=0$ , leading to a contradiction.

Proof of Theorem 1.2(ii) By Lemma 2.6, for each $t\in {\Omega }$ , we can find a $\sigma $ -invariant ergodic measure $\mu _t$ on $\Sigma $ such that

$$ \begin{align*}d(t)=d_{\mu_t}(t). \end{align*} $$

Moreover, $0\leq d(t)\leq (\log \ell /\log (1/\theta ))$ .

Next we prove that $\dim _H K^t=\dim _BK^t=\min \{d, d(t)\}$ for $\eta $ -a.e. $t\in {\Omega }$ . By Theorem 2.7, $\overline {\dim }_B K^t\leq \min \{d, d(t)\}$ for every $t\in {\Omega }$ . Hence it is sufficient to show that

$$ \begin{align*}\dim_H K^t\geq \min\{d, d(t)\} \quad \text{for } \eta\text{-a.e.~}t\in {\Omega}. \end{align*} $$

Suppose on the contrary that this statement is false. Then there exist $k\in {\Bbb N}$ and $H\subset {\Omega }$ with $\eta (H)>0$ such that

(3.24) $$ \begin{align} \dim_H K^t<\min\{d, d(t)\}-2/k \quad \text{for all } t\in H. \end{align} $$

Take a number $\delta \in (0,\delta _0)$ such that (3.18) holds. Since $d(t)$ is uniformly bounded from above, similar to the construction of $Y^*$ in the proof of part (i), we can construct a countable dense subset $Y'=\{y^{\prime }_n\}_{n=1}^{\infty }$ of ${\Omega }$ such that

(3.25) $$ \begin{align} \sup_{y'\in B(t, \delta)\cap Y'} d(y')\geq d(t)-\frac1k \quad \text{for all } t\in {\Omega}. \end{align} $$

Write

(3.26) $$ \begin{align} {\Omega}_n'=\{t\in B(y^{\prime}_n,\delta):\; d(y^{\prime}_n)\geq d(t)-1/k\}\quad \text{for }n\in {\Bbb N}. \end{align} $$

By (3.25), ${\Omega }=\bigcup _{n=1}^{\infty } {\Omega }_n'$ . Notice that for each $n\in {\Bbb N}$ ,

$$ \begin{align*} {\Omega}_n'\cap H&\subset \{t\in B(y_n', \delta):\; \dim_H K^t<\min\{d, d(y_n')\}-1/k\}\\ &\subset \{t\in B(y_n', \delta):\; \underline{\dim}_H \Pi^t_*(\mu_{y_n'})<\min\{d, d_{\mu_{y_n'}} (y_n')\}-1/k\}\\ &\subset \bigg\{t\in B(y_n', \delta):\; \underline{\dim}_H \Pi^t_*(\mu_{y_n'})<\min\{d, d_{\mu_{y_n'}} (y_n')\}-\frac{ \psi(\delta)}{\log (1/\theta)}\bigg\}, \end{align*} $$

where we have used the facts that $\dim _H K^t\geq \underline {\dim }_H \Pi ^t_*(\mu _{y_n'})$ and $d(y_n')=d_{\mu _{y_n'}} (y_n')$ in the second inclusion, and (3.18) in the last inclusion. Hence $\eta ({\Omega }_n'\cap H)=0$ for each n by applying Proposition 3.1(i). It follows that $\eta (H)\leq \sum _{n=1}^{\infty } \eta ({\Omega }_n'\cap H)=0$ , leading to a contradiction. This completes the proof of the statement that $\dim _H K^t=\min \{d, d(t)\}$ for $\eta $ -a.e. $t\in {\Omega }$ .

Finally, we prove that $\mathcal L_d(K^t)>0$ for $\eta $ -a.e. $t\in \{t'\in {\Omega }:\; d(t')>d\}$ . Suppose on the contrary that this result is false. Then there exist $k\in {\Bbb N}$ and $H'\subset {\Omega }$ with $\eta (H')>0$ such that

(3.27) $$ \begin{align} d(t)>d+\frac2k \quad \text{and}\quad \mathcal L_d(K^t)=0\quad \text{for all }t\in H'. \end{align} $$

Set

$$ \begin{align*}F_n'=\{t\in B(y_n',\delta):\; \mathcal L_d(K^t)=0\}, \quad n\in {\Bbb N}. \end{align*} $$

Clearly, $H'\cap {\Omega }_n'\subset F^{\prime }_n$ for each $n\in {\Bbb N}$ . Since ${\Omega }=\bigcup _{n=1}^{\infty } {\Omega }_n'$ and $\eta (H')>0$ , there exists $m\in {\Bbb N}$ so that $\eta (H'\cap {\Omega }_m')>0$ . Hence $H'\cap {\Omega }_m'\neq \emptyset $ . Taking $t\in H'\cap {\Omega }_m'$ and applying (3.26), (3.27) and (3.18) gives

$$ \begin{align*}d_{\mu_{y_m'}}(y_m')=d(y_m')\geq d(t)-\frac1k>d+\frac{1}{k}>d+\frac{ \psi(\delta)}{\log (1/\theta)}.\end{align*} $$

Now by Proposition 3.1(ii), $\Pi ^t_*(\mu _{y_m'})\ll \mathcal L_d$ for $\eta $ -a.e. $t\in B(y_m', \delta )$ . This implies that $\eta (F_m')=0$ . Since $H'\cap {\Omega }_m'\subset F_m'$ , it follows that $\eta (H'\cap {\Omega }_m')=0$ , leading to a contradiction.

Proof By the assumptions (i) and (ii), we may pick two open connected sets V and W (for instance, we may let V and W be the $\epsilon $ -neighborhood and $2\epsilon $ -neighborhood of S, respectively, for a sufficiently small $\epsilon>0$ ), such that

$$ \begin{align*} S\subset V\subset \overline{V}\subset W\subset \overline {W}\subset U, \text{ and} \end{align*} $$

$$ \begin{align*} f_i(\overline{V})\subset V \text{ and } f_i(\overline{W})\subset W \quad \text{for all } i. \end{align*} $$

Then by continuity, we can pick a small $r_0$ such that for all $\boldsymbol {\mathfrak {t}}=(\mathbf {t}_1,\ldots , \mathbf {t}_{\ell })\in {\Bbb R}^{\ell d}$ with $|\boldsymbol {\mathfrak {t}}|<r_0$ ,

$$ \begin{align*}f_i^{\boldsymbol{\mathfrak{t}}}(\overline{V})\subset V \text{ and } f_i^{\boldsymbol{\mathfrak{t}}}(\overline{W})\subset W \quad \text{for all } i, \end{align*} $$

where $f_i^{\boldsymbol {\mathfrak {t}}}:=f_i+\mathbf {t}_i$ . Fix this $r_0$ and set $\Delta =\{\boldsymbol {\mathfrak {s}}\in {\Bbb R}^{\ell d}:\; |\boldsymbol {\mathfrak {s}}|<r_0\}$ . In what follows, we prove that the family ${\mathcal F}^{\boldsymbol {\mathfrak {t}}}$ , $\boldsymbol {\mathfrak {t}}\in \Delta $ , satisfies the GTC with respect to $\mathcal L_{\ell d}$ on $\Delta $ .

For $i=1,\ldots , \ell $ , define $g_i: \overline {W}\to {\Bbb R}$ by

$$ \begin{align*}g_i(z)=\log \|f_i'(z)\|.\end{align*} $$

Then $g_i$ is continuous on $\overline {W}$ for each i. Define $\gamma :\; (0,\infty )\to (0,\infty )$ by

$$ \begin{align*}\gamma(u)=\max_{1\leq i\leq \ell} \sup\{|g_i(x)-g_i(y)|:\; x,y\in \overline{W},\; |x-y|\leq u\}. \end{align*} $$

That is, $\gamma $ is a common continuity modulus of $g_1,\ldots , g_{\ell }$ . Clearly $\lim _{u\to 0}\gamma (u)=0$ . Notice that for $\boldsymbol {\mathfrak {t}}\in \Delta $ , $y\in W$ and $\boldsymbol {\omega }\in \Sigma _n$ ,

$$ \begin{align*}\log \|(f^{\boldsymbol{\mathfrak{t}}}_{{\boldsymbol{\omega}}} )'(y)\|=\sum_{k=1}^n g_{\omega_k}(f^{\boldsymbol{\mathfrak{t}}}_{\sigma^k{\boldsymbol{\omega}}}(y)). \end{align*} $$

Using similar arguments (with minor changes) as in Step 1 and Step 2 of the proof of Proposition 4.3, we can show that the following two properties hold.

1. (a) Write for $n\in {\Bbb N}$ ,

   (5.1) $$ \begin{align} C_n:=\sup\bigg\{ \frac{\|(f^{\boldsymbol{\mathfrak{t}}}_{{\boldsymbol{\omega}}} )'(y)\|}{\|(f^{\boldsymbol{\mathfrak{t}}}_{{\boldsymbol{\omega}}} )'(z)\|}:\; \boldsymbol{\mathfrak{t}}\in \Delta,\; y,z\in W, \;{\boldsymbol{\omega}}\in \Sigma_n\bigg\}. \end{align} $$
   Then $\lim _{n\to \infty }({1}/{n})\log C_n=0$ .

2. (b) For $y\in W$ , $\boldsymbol {\mathfrak {s}},\boldsymbol {\mathfrak {t}}\in \Delta $ , $n\in {\Bbb N}$ and ${\boldsymbol {\omega }}\in \Sigma _n$ ,

   (5.2) $$ \begin{align} \frac{\|(f^{\boldsymbol{\mathfrak{t}}}_{{\boldsymbol{\omega}}} )'(y)\|}{\|(f^{\boldsymbol{\mathfrak{s}}}_{{\boldsymbol{\omega}}} )'(y)\|}\leq \exp\bigg(n\gamma\bigg(\frac{|\boldsymbol{\mathfrak{t}}-\boldsymbol{\mathfrak{s}}|}{1-\theta}\bigg)\bigg), \end{align} $$
   where $\theta :=\max _{1\leq i\leq \ell }\rho _i<1$ .

Let $\mathcal H$ denote the collection of $C^1$ injective conformal mappings $h: W\to W$ such that $h(\overline {V})\subset V$ . The following fact is known (for a proof, see e.g. part 3 of the proof of [Reference Patzschke39, Lemma 2.2]): there exists a constant $D\in (0,1)$ depending on V and W, such that

(5.3) $$ \begin{align} D\cdot \Big(\inf_{z\in W}\| h'(z)\|\Big) \cdot |x-y|\leq |h(x)-h(y)| \quad \text{for all }h\in \mathcal H,\; x,y\in V. \end{align} $$

Now fix $\boldsymbol {\mathfrak {s}}\in \Delta $ and $\delta \in (0,r_0)$ . Let $\mathbf {i},\mathbf {j}\in \Sigma $ with $\mathbf {i}\neq \mathbf {j}$ . Set

$$ \begin{align*}\boldsymbol{\omega}=\mathbf{i}\wedge \mathbf{j}\quad \text{and}\quad n=|\boldsymbol{\omega}|.\end{align*} $$

Write $\mathbf {a}=\sigma ^n \mathbf {i}$ and $\mathbf {b}=\sigma ^n \mathbf {j}$ . Clearly $a_1\neq b_1$ . Fix $y\in S$ . We claim that for $r>0$ ,

(5.4) $$ \begin{align} &\{\boldsymbol{\mathfrak{t}} \in B_{{\Bbb R}^{\ell d}}(\boldsymbol{\mathfrak{s}},\delta)\cap \Delta:\! |\Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{ i})-\Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{j})|< r\}\nonumber\\ &\text{}\; \subset \{\boldsymbol{\mathfrak{t}}\in B_{{\Bbb R}^{\ell d}}(\boldsymbol{\mathfrak{s}},\delta)\cap \Delta:\! \Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{ a})-\Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{b})\in (D_yf^{\boldsymbol{\mathfrak{s}}}_{\boldsymbol{\omega}})^{-1} B_{{\Bbb R}^d}(0,c(n,\delta)r) \}, \end{align} $$

where

$$ \begin{align*}c(n,\delta):=D^{-1}C_n \exp\bigg(n\gamma\bigg(\frac{\delta}{1-\theta}\bigg)\bigg)>1, \end{align*} $$

in which D is the constant from (5.3).

To show (5.4), let $\boldsymbol {\mathfrak {t}} \in B_{{\Bbb R}^{\ell d}}(\boldsymbol {\mathfrak {s}},\delta )\cap \Delta $ so that $|\Pi ^{\boldsymbol {\mathfrak {t}}}(\mathbf {i})-\Pi ^{\boldsymbol {\mathfrak {t}}}(\mathbf {j})|<r$ . Notice that

$$ \begin{align*} \begin{split} |\Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{i})-\Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{ j})|&=|f^{\boldsymbol{\mathfrak{t}}}_{\boldsymbol{\omega}}(\Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{ a}))-f^{\boldsymbol{\mathfrak{t}}}_{\boldsymbol{\omega}}(\Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{b}))|\\ &\geq D\cdot \Big(\inf_{z\in W}\| (f^{\boldsymbol{\mathfrak{t}}}_{\boldsymbol{\omega}})'(z)\|\Big) \cdot |\Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{ a})-\Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{b})|\quad \text{(by (5.3))}\\ &\geq D(C_n)^{-1} \exp\bigg(-n\gamma\bigg(\frac{\delta}{1-\theta}\bigg)\bigg)\cdot\| (f^{\boldsymbol{\mathfrak{s}}}_{\boldsymbol{\omega}})'(y)\|\cdot |\Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{a})-\Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{b})|, \end{split} \end{align*} $$

where, in the last inequality, we have used (5.1) and (5.2). It follows that

$$ \begin{align*}|\Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{a})-\Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{b})|\leq \| (f^{\boldsymbol{\mathfrak{s}}}_{\boldsymbol{\omega}})'(y)\|^{-1}\cdot D^{-1}C_n \exp\bigg(n\gamma\bigg(\frac{\delta}{1-\theta}\bigg)\bigg)\cdot r. \end{align*} $$

Since $(f^{\boldsymbol {\mathfrak {s}}}_{\boldsymbol {\omega }})'(y)=D_yf^{\boldsymbol {\mathfrak {s}}}_{\boldsymbol {\omega }}$ is a scalar multiple of an orthogonal matrix, the above inequality implies that

$$ \begin{align*}\Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{a})-\Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{b})\in (D_yf^{\boldsymbol{\mathfrak{s}}}_{\boldsymbol{\omega}})^{-1} B_{{\Bbb R}^d}(0,c(n,\delta)r). \end{align*} $$

from which (5.4) follows.

By (5.4) and Lemma 4.7 (which is also valid in this context),

$$ \begin{align*} \begin{split} \mathcal L_{\ell d}&\{\boldsymbol{\mathfrak{t}} \in B_{{\Bbb R}^{\ell d}}(\boldsymbol{\mathfrak{s}},\delta)\cap \Delta:\; |\Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{i})-\Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{j})|< r\}\\ &\leq \mathcal L_{\ell d} \{\boldsymbol{\mathfrak{t}}\in B_{{\Bbb R}^{\ell d}}(\boldsymbol{\mathfrak{s}},\delta)\cap \Delta:\; \Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{a})-\Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{b})\in (D_yf^{\boldsymbol{\mathfrak{s}}}_{\boldsymbol{\omega}})^{-1} B_{{\Bbb R}^d}(0,c(n,\delta)r) \}\\ &\leq \widetilde{C}\cdot\min\bigg\{\frac{c(n,\delta)^k r^k}{\phi^k(D_yf^{\boldsymbol{\mathfrak{s}}}_{\boldsymbol{\omega}})}:\; k=0,1\ldots,d\bigg\}\\ &\leq \widetilde{C}c(n,\delta)^d\cdot\min\bigg\{\frac{ r^k}{\phi^k(D_yf^{\boldsymbol{\mathfrak{s}}}_{\boldsymbol{\omega}})}:\; k=0,1\ldots,d\bigg\}\\ &= \widetilde{C}D^{-d} (C_n)^d \exp\bigg(nd\gamma\bigg(\frac{\delta}{1-\theta}\bigg)\bigg) \min\bigg\{\frac{r^k}{\phi^k(D_yf^{\boldsymbol{\mathfrak{s}}}_{\boldsymbol{\omega}})}:\; k=0,1\ldots,d\bigg\}. \end{split} \end{align*} $$

Since $y\in S$ is arbitrary and $\Pi ^{\boldsymbol {\mathfrak {s}}}(\Sigma )\subset S$ , recalling

$$ \begin{align*} Z_{{\boldsymbol{\omega}}}^{\boldsymbol{\mathfrak{s}}}(r)= \inf_{x\in \Sigma} \min\bigg \{ \frac{r^k}{ \phi^k (D_{\Pi^{\boldsymbol{\mathfrak{s}}} x}f^{\boldsymbol{\mathfrak{s}}}_{{\boldsymbol{\omega}}} ) }:\; k=0, 1,\ldots, d\bigg\}, \end{align*} $$

it follows that

$$ \begin{align*} \mathcal L_{\ell d}& \{\boldsymbol{\mathfrak{t}} \in B_{{\Bbb R}^{\ell d}}(\boldsymbol{\mathfrak{s}},\delta)\cap \Delta:\; |\Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{i})-\Pi^{\boldsymbol{\mathfrak{t}}}(\mathbf{j})|< r\} \\ & \leq \widetilde{C}D^{-d} (C_n)^d \exp\bigg(nd\gamma\bigg(\frac{\delta}{1-\theta}\bigg)\bigg)Z_{\boldsymbol{\omega}}^{\boldsymbol{\mathfrak{s}}}(r)\\ &\leq c_{\delta} e^{n\psi(\delta)}Z_{\boldsymbol{\omega}}^{\boldsymbol{\mathfrak{s}}}(r), \end{align*} $$

where

$$ \begin{align*}c_{\delta}:= \sup_{n\in {\Bbb N}}\widetilde{C}D^{-d} (C_n)^d e^{-n\delta}<\infty,\quad \psi(\delta):=\delta+d\gamma\bigg(\frac{\delta}{1-\theta}\bigg). \end{align*} $$

Since $\lim _{u\to 0}\gamma (u)=0$ , we see that $\lim _{\delta \to 0} \psi (\delta )=0$ . Thus, $({\mathcal F}^{\boldsymbol {\mathfrak {t}}})_{\boldsymbol {\mathfrak {t}}\in \Delta }$ satisfies the GTC.

Proof The proof of (i) is direct and simple. We leave it to the reader as an exercise. Part (ii) is just a consequence of (i), using the fact that the set of singular values (including the multiplicity) of M are precisely the union of those of $A_i$ , $i=1,\ldots , n$ .

Proof of Proposition 6.1 Write

$$ \begin{align*}\mathcal F_k^{t_k}=\{f_{i,k}^{t_k}\}_{i=1}^{\ell},\quad t_k\in {\Omega}_k,\; k=1,\ldots,n.\end{align*} $$

For $\boldsymbol {\omega }=\omega _1\ldots \omega _m\in \Sigma _*$ , write $f_{{\boldsymbol {\omega }},k}^{t_k}=f_{\omega _1,k}^{t_k}\circ \cdots \circ f_{\omega _m,k}^{t_k}$ . Let $\Pi ^{t_k}_k$ denote the coding map associated with the IFS $\mathcal F_k^{t_k}$ , and $\Pi ^{(t_1,\ldots , t_n)}$ the coding map associated with the IFS $\mathcal F^{(t_1,\ldots , t_n)}$ . According to the GTC assumption on the families $(\mathcal F_k^{t_k})_{t_k\in {\Omega }_k}$ , $k=1,\ldots , n$ , there exist $\delta _0>0$ and a function $\psi :\; (0, \delta _0)\to [0,\infty )$ with $\lim _{\delta \to 0}\psi (\delta )=0$ such that for every $\delta \kern1.5pt{\in}\kern1.5pt (0, \delta _0)$ and $(s_1,\ldots , s_n)\kern1.5pt{\in}\kern1.5pt {\Omega }_1\times \cdots \times {\Omega }_n$ , there is $C\kern1.5pt{=}\kern1.5pt C(\delta ,s_1,\ldots , s_n)\kern1.5pt{>}\kern1.5pt 0$ satisfying the following: for each $k\in \{1,\ldots , n\}$ , distinct $\mathbf {i},\mathbf { j}\in \Sigma $ and $r>0$ ,

(6.2) $$ \begin{align} \eta_k&\{t_k\in B_{{\Omega}_k}(s_k, \delta):\! |\Pi^{t_k}_k({\textbf{ i}})-\Pi^{t_k}_k({\textbf{j}}) |< r\}\nonumber\\ &\leq C e^{|{\textbf{i}}\wedge {\textbf{j}}| \psi(\delta)} \inf_{x\in \Sigma} \min\bigg\{ \frac{r^p}{\phi^p(D_{\Pi^{t_k}_kx}f_{\mathbf{i}\wedge \mathbf{j}, k}^{t_k})}:\! p=0,1,\ldots, q_k \bigg\}, \end{align} $$

where $B_{{\Omega }_k}(s_k, \delta )$ stands for the closed ball in ${\Omega }_k$ of radius $\delta $ centered at $s_k$ . Writing $\mathbf {t}=(t_1,\ldots , t_n)$ , $\mathbf {s}=(s_1,\ldots , s_n)$ and using (6.2),

$$ \begin{align*} \eta_1&\times \cdots\times \eta_n\{ \mathbf{t}\in B_{{\Omega}}(\mathbf{s}, \delta): \; |\Pi^{\mathbf{t}}({\textbf{i}})-\Pi^{\mathbf{t}}({\textbf{j}}) |< r\}\\ &\leq \prod_{k=1}^n\eta_k\{t_k\in B_{{\Omega}_k}(s_k, \delta): \; |\Pi^{t_k}_k({\textbf{ i}})-\Pi^{t_k}_k({\textbf{j}}) |< r\}\\ &\leq \prod_{k=1}^n \bigg(C e^{|{\textbf{i}}\wedge {\textbf{j}}| \psi(\delta)} \inf_{x\in \Sigma} \min\bigg\{ \frac{r^p}{\phi^p(D_{\Pi^{s_k}_kx}f_{\mathbf{i}\wedge \mathbf{j}, k}^{s_k})}:\; p=0,1,\ldots, q_k \bigg\}\bigg)\\ & \leq C^n e^{n|{\textbf{i}}\wedge {\textbf{j}}| \psi(\delta)}\inf_{x\in \Sigma}\prod_{k=1}^n \min\bigg\{ \frac{r^p}{\phi^p(D_{\Pi^{s_k}_kx}f_{\mathbf{i}\wedge \mathbf{j}, k}^{s_k})}:\; p=0,1,\ldots, q_k \bigg\}\\ & =C^n e^{n|{\textbf{i}}\wedge {\textbf{j}}| \psi(\delta)}\inf_{x\in \Sigma}\min \bigg\{ \frac{r^p}{\phi^p(D_{\Pi^{\mathbf{t}}x}f_{\mathbf{i}\wedge \mathbf{j}}^{\mathbf{s}})}:\; p=0,1,\ldots, q_1+\cdots+q_n \bigg\}\\ & =C^n e^{n|{\textbf{i}}\wedge {\textbf{j}}| \psi(\delta)}Z_{\mathbf{i}\wedge\mathbf{j}}^{\mathbf{s}}(r), \end{align*} $$

where we have used (6.1) in the second last equality. Hence, the family $\mathcal F^{\mathbf {t}}$ , $\mathbf {t}\in {\Omega }$ , satisfies the GTC with respect to the measure $\eta _1\times \cdots \times \eta _n$ , where the involved constant and the function in the definition of the GTC are $C^n$ and $n\psi (\cdot )$ , respectively.

Proof of Theorem 1.6 This follows directly by combining Theorems 4.2, 5.1 and Proposition 6.1.

Question 7.1. Is it true that for every $C^1$ IFS $\mathcal F=\{f_i\}_{i=1}^{\ell }$ on ${\Bbb R}^d$ satisfying (1.8), there is a neighborhood $\Delta $ of ${\textbf {0}}$ in ${\Bbb R}^{\ell d}$ such that for $\mathcal L_{\ell d}$ -a.e. $\boldsymbol {\mathfrak {t}}=({\textbf {t}}_1,\ldots , {\textbf {t}}_{\ell })\in \Delta $ ,

$$ \begin{align*}\dim_HK^{\boldsymbol{\mathfrak{t}}}=\dim_BK^{\boldsymbol{\mathfrak{t}}}=\min\{\dim_S \mathcal F^{\boldsymbol{\mathfrak{t}}}, \; d\}, \end{align*} $$

where $K^{\boldsymbol {\mathfrak {t}}}$ is the attractor of the IFS ${\mathcal F}^{\boldsymbol {\mathfrak {t}}}=\{f_i+{\textbf {t}}_{i}\}_{i=1}^{\ell }$ ?

Question 7.2. Do we have

$$ \begin{align*}\dim_HK=\dim_BK=\min\{\dim_S \mathcal F, \; d\} \end{align*} $$

for the attractor K of a ‘generic’ $C^1$ IFS $\mathcal F$ on ${\Bbb R}^d$ (in an appropriate sense)?