Let $X$ be a separable metric space, $\phi\in C(X,X)$ be a topologically transitive dynamical system, and $V$ be a non-empty open subset of $X$, and $\nu$ be a locally-positive and atomless Borel probability measure on $X$. 


Then, for every $\delta \in (0,1)$, does there exist:


* $\{V\_i\}\_{i \in \mathbb{N}}$ are open subsets of $V$ satisfying $\nu\left(
V-\cup\_{n \in \mathbb{N}} V\_n
\right)=0$
* A sequence $\{N\_i\}\_{i \in \mathbb{N}}$,
*such that the following holds:*
$$
\nu\left(X - \cup\_{i \in \mathbb{N}} \phi^{-N\_i}[V\_i] \right)<\delta
.
$$


*In intuitive words: there exists an open cover of a non-empty open set be reversed into an almost-everywhere cover of the entire space by appropriately reversing the dynamical sytem?*


**Auxiliary Definitions:**


* A Borel measure $\nu$ on $X$ is said to be *locally-positive* iff for every non-empty open subset $U\subseteq X$, $\nu(U)>0$. *For example, if $X$ has more than two points then the Dirac is not such a measure.*
* $\phi$ is said to be *topologically transitive* iff for every two non-empty open subsets $U,V\subseteq X$ there exists some $N\in \mathbb{N}$ such that
$$
\phi^N(U)\cap V \neq \emptyset.
$$



Reversal of open cover with topologically transitive dynamical system
**No**, even if we assume $\nu$ to be invariant under $\phi$.


Let $X = \{0,1\}^\mathbb{Z}$ be the set of two-way infinite binary sequences with the prodiscrete topology, and let $\phi$ be the left shift on $X$.
Let $\nu = (\mu\_1 + \mu\_2)/2$ where $\mu\_1$ is the uniform Bernoulli measure on $X$ and $\mu\_2$ is an atomless $\phi$-invariant probability measure on some proper subshift of $X$.
For simplicity, let's choose $\mu\_2$ as the Parry measure on the shift of finite type $Y \subsetneq X$ where $0 0$ is forbidden.
Let $V = \{ x \in X : x\_0 = x\_1 = 0 \}$ be the set of sequences with an occurrence of the forbidden word $0 0$ at the origin.
These definitions satisfy your requirements: $\phi$ is well-known to be transitive, $\nu$ gives positive measure to each nonempty clopen set (which form a basis of the topology) and has no atoms, and $V$ is a nonempty open set.


Consider an open cover $(V\_i)\_{i \in \mathbb{N}}$ of $V$ and a sequence $(N\_i)\_{i \in \mathbb{N}}$ of integers.
For each $i$ the translate $\phi^{-N\_i} V\_i$ is disjoint from $Y$, so $\nu(X - \bigcup\_{i \in \mathbb{N}} \phi^{-N\_i} V\_i) \geq \nu(Y) = 1/2$.


**But yes**, if we strengthen the assumptions further.


In my counterexample the ergodic decomposition of $\nu$ features a positive-weight measure $\mu\_2$ which is not locally positive.
Let's thus assume that $\nu$ has an ergodic decomposition as an integral $\nu = \int\_{E(M\_\phi)} x \, d\mu(x)$ over the $\phi$-ergodic probability measures on $X$ and $\mu$-a.e. $x \in E(M\_\phi)$ is locally positive.
Then $x(V) > 0$ holds for those measures $x$.
Since they are ergodic, this implies $x(\bigcup\_{i \in \mathbb{N}} \phi^{-i} V) = 1$, so that $\nu(\bigcup\_{i \in \mathbb{N}} \phi^{-i} V) = \int x(\bigcup\_{i \in \mathbb{N}} \phi^{-i} V) \, d\mu(x) = 1$.
Then $V\_i = V$ and $N\_i = i$ give the sequence you're looking for, for every $\delta > 0$.
Note that even if I didn't use transitivity in this proof, it's implied by the existence of a locally positive ergodic measure.


Depending on your application, the assumptions of $\phi$-invariance and local positivity of the ergodic decomposition may be too strong.
In the context of dynamical systems invariance seems natural, but by itself it's not enough.


