# American Institute of Mathematical Sciences

## Journals

DCDS
Discrete & Continuous Dynamical Systems - A 2018, 38(4): 2065-2078 doi: 10.3934/dcds.2018084
Given a dynamical system
 $(Ω,Σ,μ, τ)$
with
 $μ$
a non-atomic probability measure and
 $τ$
an invertible measure preserving ergodic transformation, we prove that the maximal operator, considered by I. Assani, Z. Buczolich and R. D. Mauldin in 2005,
 ${N^*}f\left( x \right) = \mathop {\sup }\limits_{\alpha > 0} \alpha \# \left\{ {k \ge 1:\frac{{\left| {f\left( {{\tau ^k}x} \right)} \right|}}{k} > \alpha } \right\}$
satisfies that
 ${N^*}:\left[ {L \log_3 L (μ)} \right] \longrightarrow L^{1, ∞}(μ)$
is bounded where the space
 $\left[ {L \log_3 L (μ)} \right]$
is defined by the condition
 $\Vert f\Vert_{\left[ {L \log_3 L (μ)} \right]} = ∈t_0^1 \frac{\sup\limits_{t≤q y}tf_μ^*(t)}{y} \log_3 \frac 1y dy < ∞,$
with
 $\log_3 x = 1+\log_+\log_+\log_+ x$
and
 $f^*_μ$
the decreasing rearrangement of
 $f$
with respect to
 $μ$
. This space is near
 $L \log_3 L (μ)$
, which is the optimal Orlicz space on which such boundedness can hold. As a consequence, the space
 $\left[ {L \log_3 L (μ)} \right]$
satisfies the Return Times Property for the Tail; that is, for every
 $f∈\left[ {L \log_3 L (μ)} \right]$
, there exists a set
 $X_0$
so that
 $μ(X_0) = 1$
and, for all
 $x_0∈ X_0$
, all dynamical systems
 $(Y,\mathcal{C},ν, S)$
and all
 $g∈ L^1(ν)$
, the sequence
 $R_ng(y) = \frac1nf(τ^nx_0)g(S^ny) \overset{n\to∞}\longrightarrow 0,\;\;\;\;\;\; ν\text{-a.e. } y∈ Y.$
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