Let $(X,\mu,f)$ be a two-sided full shift system. Assume that there is $t \in \mathbb{N}$ such that for every $n \in \mathbb{N}$ and $x \in X$, we can define $T(x)=f^{n+m(x)}(x)$, where $m(x) \leq t; $ $m(x) \in \mathbb{N}$ and that depends on $x$.

It is well-known that $h_{\mu}(f^n)=nh_{\mu}(f)$, where $h_{\mu}$ is the measure-theoretic entropy.

Can we use the above fact to say $h_{\mu}(T) \leq (n+t)h_{\mu}(f)$?

Let $(X,\mu,f)$ be a two-sided full shift system. Assume that there is $t \in \mathbb{N}$ such that for every $n \in \mathbb{N}$ and $x \in X$, we can define $T(x)=f^{n+m(x)}(x)$, where $m(x) \leq t; $ $m(x) \in \mathbb{N}$ and that depends on $x$. I also assume that $T$ is measure preserving.

It is well-known that $h_{\mu}(f^n)=nh_{\mu}(f)$, where $h_{\mu}$ is the measure-theoretic entropy.

Can we use the above fact to say $h_{\mu}(T) \leq (n+t)h_{\mu}(f)$?

Edit: As it was mentioned in comments, $T$ is not necessarily measure preserving. I add the assumption that $T$ is measure preserving.

Let $(X,\mu,f)$ be a full shift system. Assume that there is $t \in \mathbb{N}$ such that for every $n \in \mathbb{N}$ and $x \in X$, we can define $T(x)=f^{n+m(x)}(x)$, where $m(x) \leq t; $ $m(x) \in \mathbb{N}$ and that depends on $x$.

It is well-known that $h_{\mu}(f^n)=nh_{\mu}(f)$, where $h_{\mu}$ is the measure-theoretic entropy.

Can we use the above fact to say $h_{\mu}(T) \leq (n+t)h_{\mu}(f)$?

Let $(X,\mu,f)$ be a two-sided full shift system. Assume that there is $t \in \mathbb{N}$ such that for every $n \in \mathbb{N}$ and $x \in X$, we can define $T(x)=f^{n+m(x)}(x)$, where $m(x) \leq t; $ $m(x) \in \mathbb{N}$ and that depends on $x$.

It is well-known that $h_{\mu}(f^n)=nh_{\mu}(f)$, where $h_{\mu}$ is the measure-theoretic entropy.

Can we use the above fact to say $h_{\mu}(T) \leq (n+t)h_{\mu}(f)$?