In Joyce Carol Oates version of "The Lady with the Pet Dog," yes, there is a dog; the story shares quite a number of similarities with Chekov's version, but Oates makes it her own as well.
Different from Chekov's version, the chronology (order of the story) jumps around. The story begins at a concert, which is actually the middle of the story. The reader has to pay close attention to figure out the importance and relationships of the characters. Both Anna and her lover (who is unnamed) are there, though no one notices the connection between them. Seeing that Anna doesn't feel well, her husband takes her home.
In the story's second section, it is set six months earlier, when Anna resumes her affair. She tries to think seriously about her predicament, but her thoughts are jumbled. She is terribly unhappy and does not want to be with her husband, but she knows there is no future with her lover. When she does return home, she not only contemplates, but she attempts, suicide.
The third section provides clarity for the reader: a deeper understanding of the plot and its characters. It relays the story's events, in chronological order, from start to finish. Anna is at her family's home in Nantucket—however she is there without her husband. She meets a man and his blind son, and their dog. The man sketches pictures of her with his dog. This is where the dog comes in, but it is NOT her dog. This, then, is the starting point for the adulterous affair between Anna and her lover.